Digital Pre- and Post-Equalizers for In

Transcription

UNIVERSITÄT STUTTGART
Digital Pre- and Post-Equalizers for
In-Car Data Transmission over
Plasti Optial Fibers
Von der Fakultät Informatik, Elektrotechnik und Informationstechnik
der Universität Stuttgart zur Erlangung der Würde eines
Doktor-Ingenieurs (Dr.-Ing.) genehmigte Abhandlung
Vorgelegt von
Yixuan Voigt
(geb. Yixuan Wang)
aus Zhuhai, China
Hauptberichter:
Prof. Dr.-Ing. J. Speidel
Mitberichter:
Prof. Dr.-Ing. M. Berroth
Tag der mündlichen Prüfung:
15. Juli 2014
Institut für Nachrichtenübertragung der Universität Stuttgart
2014
Aknowledgements
This dissertation presents the most important outcome of my research activities conducted under the supervision of Professor Joachim Speidel at the institute of telecommunications, University of Stuttgart, Germany.
Foremost, I would like to express my sincerest appreciation to Professor Speidel, for
offering thorough and extremely good feedback on this dissertation and for all his excellent mentorship in the last four years. His influence on me was paramount - not only
academically but also personally. He encouraged me to grow as a precision engineering
expertise and always supported me to attend international academic conferences which
is extremely rewarding in broadening my perspective. He is also a great role model for
me in the way that he works hard and behaves gracefully. I owe my deepest gratitude to
him.
Special thanks go to my advisory committee members Professor Berroth, Professor
Roth-Stielow, Professor ten Brink and Professor Kallfass for their caring and concern
about this dissertation.
Special thanks also naturally go to all my colleagues at the institute of telecommunications. Thank you for being not only wonderful colleagues but also my friends. Thank
you for being my great support when I am far away from home. Thank you for teaching
me German cultures and Swabian dialect, and of course for training me as a "kicker"
player. Thank you for the beautiful doctoral hat and the unique poem. Thank you for
all the beautiful memories (travel to Berlin, Chinese hotpot, Kara-Okay...). Because of
you, I felt for the first time that Stuttgart is my second hometown.
I also want to thank my family and my girls-group. Their prays really helped me going
through hard times. To my mom Ning and father Jianfeng, thank you for always telling
me that I could achieve anything when I give effort, for actually believing it, for trusting
me to do things on my own, for helping me when I couldn’t, and for sending constant
love and support across the world for the past 12 years!
Last but least, I would like to give my grateful thank to my husband, Simon. He is really
a sweet husband, a big helper and my best friend. The writing of this dissertation would
not have been completed without him. I am grateful to Simon not just because he has
given up so much time to make my PhD a priority in our lives, but because he considers
me as important as himself. Those beautiful flowers and little surprises from him have
cheered me up through my ups and downs. I feel so lucky to marry him.
i
ii
Stuttgart, 2014
Yixuan Voigt
Contents
Nomenclature
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List of Symbols
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Abstract
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Kurzfassung
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1 Introduction
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Problem Statement . . . . . . . . . . . . . . . . . . .
1.3 Previous Works and Suggested Solutions . . . . . . . .
1.4 An Insight into the Considered Equalization Strategies
1.5 Outline of the Dissertation . . . . . . . . . . . . . . .
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2 System Description
2.1 The Channel Model . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Modulation behavior of the standard plastic optical fiber . .
2.1.2 Optical channel modeling . . . . . . . . . . . . . . . . . .
2.1.3 Receiver noise model and the receiver-side SNR estimation
2.1.3.1 Receiver noise model . . . . . . . . . . . . . . .
2.1.3.2 Analytical SNR estimation . . . . . . . . . . . .
2.1.3.3 Numerical SNR estimation . . . . . . . . . . . .
2.1.4 Experimental setup and results . . . . . . . . . . . . . . . .
2.1.4.1 Experimental setup . . . . . . . . . . . . . . . .
2.1.4.2 The measured magnitude responses . . . . . . . .
2.2 Overview of the Transmission System . . . . . . . . . . . . . . . .
2.2.1 The discrete system model . . . . . . . . . . . . . . . . . .
2.2.2 Channel impairments . . . . . . . . . . . . . . . . . . . . .
2.2.3 The matched filter bound for band-limited channel . . . . .
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3 Equalization of the Optical Channel
3.1 General Equalization Techniques . . . . . . . . . . . . . . . . . . . . .
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CONTENTS
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4 Tomlinson-Harashima Precoded Systems
4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Principles of the Tomlinson-Harashima Precoding for Channels without
Pre-Cursors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Tomlison-Harashima Precoding for Channels with Pre- and Post-Cursors
4.4 Design of the Feedforward Equalizer . . . . . . . . . . . . . . . . . . .
4.4.1 During the start-up . . . . . . . . . . . . . . . . . . . . . . . .
4.4.2 After the start-up . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Discussion of THP Losses . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Evaluation and Simulation Results . . . . . . . . . . . . . . . . . . . .
4.6.1 Comparison of THP-FFE, FFE and DFE at 2 Gbit/s and 3 Gbit/s
4.6.2 Effects of a decreased channel bandwidth . . . . . . . . . . . .
4.6.3 Performance of the adaptive filter . . . . . . . . . . . . . . . .
4.6.4 Evaluation of the THP losses . . . . . . . . . . . . . . . . . . .
4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.1.1
3.1.2
3.2
3.3
3.4
Optimal equalization architectures . . . . . . . . . . . . . .
FIR equalizers . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2.1 Design of FIR ZF-FFE . . . . . . . . . . . . . .
3.1.2.2 Design of FIR MMSE-FFE . . . . . . . . . . . .
3.1.3 Digital adaptive equalizers . . . . . . . . . . . . . . . . . .
3.1.3.1 Least mean square (LMS) algorithm . . . . . . .
3.1.3.2 Normalized least mean square (NLMS) algorithm
3.1.3.3 Recursive least square (RLS) algorithm . . . . . .
3.1.3.4 Comparison of the adaptive algorithms . . . . . .
Design of Transmission Systems with Pre- and/or Post-Equalizers .
3.2.1 Design of adaptive equalizers . . . . . . . . . . . . . . . .
3.2.1.1 Structure of the adaptive equalizers . . . . . . . .
3.2.1.2 Configurations of the adaptive equalizers . . . . .
3.2.2 Design of a pre-equalizer . . . . . . . . . . . . . . . . . . .
Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Performances with post-equalization . . . . . . . . . . . . .
3.3.2 Performances with joint pre- and post-equalization . . . . .
3.3.3 Performances with regard to the MPAM modulation order .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 Bidirectional Decision Feedback Equalization
5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 The System Model . . . . . . . . . . . . . . . . . . . . . .
5.3 Finite-length BiDFE Algorithms . . . . . . . . . . . . . . .
5.3.1 Symbol-wise arbitrated bidirectional arbitrated DFE
5.3.2 Block-wise arbitrated trellis-based conflict resolution
5.3.3 The novel trellis-based BiDFE algorithm . . . . . .
5.4 Comparison of Computational Complexities . . . . . . . . .
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CONTENTS
5.5
5.6
Simulation Results of the BiDFEs . . . . . . . . . . . . . . . . . . . .
5.5.1 Symbol-spaced BiDFEs . . . . . . . . . . . . . . . . . . . . .
5.5.2 Fractionally-spaced BiDFEs . . . . . . . . . . . . . . . . . . .
5.5.3 Evaluation of the computational complexities in the reconstruction and arbitration stage . . . . . . . . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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6 Conclusions
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6.1 Contribution Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.2 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
A Minimum-phase Spectral Factorization
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Bibliography
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Nomenlature
AWGN
BAD
BER
additive white Gaussian noise
bi-directional arbitrated decision feedback equalizer
bit error rate
BiDFE
CBA
bi-directional decision feedback equalizer
contradictory block arbitration
DC
DFE
DMT
direct current
decision feedback equalizer
discrete multi-tone modulation
DSP
DVB
digital signal processing
digital video broadcasting
EDS
EMI
FBF
effective data sequence
electro-magnetic interference
feedback filter
FEC
FET
forward error correction
field-effect transistor
FFE
FFF
FFT/IFFT
feedforward equalizer
feedforward filter
fast Fourier transform/inverse fast Fourier transform
FIR
FS
finite impulse response
fractionally-spaced
FS-DFE
FSE
FS-FFE
fractionally-spaced decision feedback equalizer
fractionally spaced equalizer
fractionally-spaced feedforward equalizer
i.i.d.
IIR
independent and identically distributed
infinite impulse response
ISI
inter-symbol-interference
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viii
NOMENCLATURE
LC-BiDFE
LD
LE
linear combining bidirectional DFE
laser diode
linear equalizer
LED
LLR
light emitting diode
log-likelihood ratio
LMS
MF
MFB
least mean square
matched filter
matched filter bound
ML
MLSE
maximum likelihood
maximum likelihood sequence estimation
MMSE
MOST
MPAM
minimum mean squared error
media oriented system transport
M-ary pulse amplitude modulation
MSE
NLMS
mean squared error
normalized least mean square
OFDM
PD
PIN PD
orthogonal frequency division multiplexing
photo-diode
positive-intrinsic-negative photo-diode
POF
PSD
RLS
plastic optical fiber
power spectral density
recursive least square
PAM
PMMA
pulse amplitude modulation
poly-methylmetacrylate
QAM
RE
RS
quadrature amplitude modulation
recursive equalizer
Reed-Solomon
SER
SINR
symbol error rate
signal-to-interference-plus-noise ratio
SI-POF
SNR
SS
step-index plastic optical fiber
signal-to-noise ratio
symbol-spaced
SS-DFE
SSE
symbol-spaced decision feedback equalizer
symbol spaced equalizer
SS-FFE
THP
TBCR
symbol-spaced feedforward equalizer
Tomlinson-Harashima precoding
trellis-based conflict resolution
TB-BiDFE
VSLMS
trellis-based bi-directional DFE
variable step size LMS
ix
NOMENCLATURE
WLAN
ZF
wireless local area network
zero-forcing
List of Symbols
a[k]
at [k]
M-ary pulse amplitude modulated symbol sequence . . . . . . . . . . . . . . . . . . 27
the training sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
A
Ah
b[n]
linear fiber loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
a real-valued constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
bit sequence after the Reed-Solomon encoder . . . . . . . . . . . . . . . . . . . . . . . . 27
b′ [m]
B
information bit sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
B3dB
B(z)
Ca
3 dB modulation bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
the product of H(z) and W (z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
input capacitance of an amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Cd
CT
capacitance of a photo-diode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
total capacitance of a photo-diode and an amplifier . . . . . . . . . . . . . . . . . . . 17
C
d
e[k]
a selection matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
the minimum distance between two constellation points . . . . . . . . . . . . . . 28
the error between the equalized sample and a reference sample . . . . . . . . 43
Eh
FB (z)
the energy of the channel impulse response . . . . . . . . . . . . . . . . . . . . . . . . . . 30
FF (z)
Fn
G(z)
the discrete-time feedforward filter in the DFE . . . . . . . . . . . . . . . . . . . . . . . 38
amplifier noise figure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
a causal, monic and minimum-phase discrete-time filter . . . . . . . . . . . . . . . 39
G0
h[k]
the amplifier gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
the discrete-time channel impulse response . . . . . . . . . . . . . . . . . . . . . . . . . . 26
ha (t)
he [k]
H(z)
the channel impulse response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
the estimated discrete-time channel impulse response . . . . . . . . . . . . . . . . . 73
Ha (f )
receiver bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
the discrete-time feedback filter in the DFE . . . . . . . . . . . . . . . . . . . . . . . . . . 38
z-transform of h[k] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
the channel frequency response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
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LIST OF SYMBOLS
He (z) the estimated channel transfer function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Hpof (f ) the POF modulation transfer function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
HR (f ) the receive filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
the discrete-time channel impulse response vector . . . . . . . . . . . . . . . . . . . . 28
h
the estimated discrete-time channel impulse response vector . . . . . . . . . . 96
he
H
ia
in
ip
it
a matrix containing the channel information . . . . . . . . . . . . . . . . . . . . . . . . . 42
the amplifier noise current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
the total noise current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
the signal current generated by a photo-diode . . . . . . . . . . . . . . . . . . . . . . . . 16
the thermal noise current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
is
I[k]
Idark
the shot noise current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
the inter-symbol-interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
the dark current generated by a photo-diode . . . . . . . . . . . . . . . . . . . . . . . . . 18
Ip
i
the average signal current generated by a photo-diode . . . . . . . . . . . . . . . . 18
the desired discrete-time channel impulse response . . . . . . . . . . . . . . . . . . . 71
I
J
Jo
an identity matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
the mean-squared-error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
the minimum MSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
kB
K
Lc
the Boltzmann’s constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
the oversampling factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
length of the discrete-time channel impulse response . . . . . . . . . . . . . . . . . 26
Lcontr
Le
the mean length of the conflict events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
length of the estimated discrete-time channel impulse response . . . . . . . . 96
Lpof
M
N
the fiber length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
N0
Ncontr
the modulation order of MPAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
the length of the data sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
the single-sided PSD of the AWGN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Nm
Ny
n[k]
the mean number of the conflict events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
an arbitrary integer number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
the length of the received sample sequence . . . . . . . . . . . . . . . . . . . . . . . . . . 94
AWGN sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
n′ [k]
nf
the colored noise sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
the length of a FIR feedforward filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
nb
p
Po
the length of a FIR feedback filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
the pre-cursors ISI vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
the root mean square optical power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
the symbol error rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Ps
PS,T HP the THP symbol error rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
LIST OF SYMBOLS
xiii
PS,M AP M the MPAM symbol error rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Pb
the bit error rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
the elementary charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
q
Q
r[k]
the Q-function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
the reference sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Ra
Rb
RT
the amplifier input resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
R0
Ry
the responsivity of a PIN photo-diode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
the detector bias resistor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
the parallel resistance of Rb and Ra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
the correlation matrix of y[k] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
s[k]
tc
t0
it equals either a[k] or x[k] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
T
T0
the sampling duration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
the absolute temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Ts
v[k]
w
the symbol duration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
the effective data sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
w[k]
wo
W
the coefficient vector of a time-variant FIR filter . . . . . . . . . . . . . . . . . . . . . 45
the optimum coefficient vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
half of the window length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
W (z)
WD (z)
a linear post feedforward equalizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
x[k]
x[kk21 ]
x̂F [k]
the central time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
the sampling delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
the coefficient vector of W (z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
a discrete-time filter in FF (z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
the transmit signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
indicate a sequence (x[k1 ], x[k1 + 1], · · · , x[k2 ]) . . . . . . . . . . . . . . . . . . . . . 99
the decided symbol of the forward DFE in a BiDFE . . . . . . . . . . . . . . . . . . 95
x̂R [k]
y
the decided symbol of the reverse DFE in a BiDFE . . . . . . . . . . . . . . . . . . . 95
the received sample vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
y[kk21 ]
ỹ
y ′[k]
indicate a sequence (y[k1], y[k1 + 1], · · · , y[k2]) . . . . . . . . . . . . . . . . . . . . . 99
the time-reversed received sample vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
the received sample after the received filter . . . . . . . . . . . . . . . . . . . . . . . . . . 34
ŷ F
ŷ R
the estimated received sample by the forward DFE in a BiDFE . . . . . . . . 96
the estimated received sample by the reverse DFE in a BiDFE . . . . . . . . . 96
ŷ F
ξ2
ξa
the estimated received sample by the forward DFE in a BiDFE . . . . . . . . 96
the Euclidean distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
the mean power of the MPAM data sequence . . . . . . . . . . . . . . . . . . . . . . . . 28
ξx
γSN R
the mean power of the x[k] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
the signal-to-noise ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
xiv
LIST OF SYMBOLS
γSIN R the signal-to-interference-plus-noise ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
γmodulo the THP modulo loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
γprecoding the THP precoding loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
the forgetting factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
λ
the largest eigenvalue of the correlation matrix Ry . . . . . . . . . . . . . . . . . . . 46
λm ax
the step size of LMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
the step size of NLMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
µ
µ̃
µa
the mean value of a[k] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
µs
µx
the mean value of s[k] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
the mean value of x[k] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Φnn
Φn′ n′
σ
PSD of the AWGN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
PSD of the colored noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
standard deviation of the POF impulse response . . . . . . . . . . . . . . . . . . . . . . 13
σn
σn′
standard deviation of the AWGN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
standard deviation of the colored noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
variance of a[k] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
σa2
σˆ2
n
σs2
σx2
τ
the estimated variance of the AWGN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
variance of s[k] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
variance of x[k] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
the transmission delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Abstrat
Lately, a hot topic in the automobile industry is the development of the in-vehicle
infotainment communication network based on the media oriented system transport
(MOST) standard, where a cost-effective optical physical layer composed of light emitting diodes (LED), plastic optical fibers (POF) and positive-intrinsic-negative photodiodes (PIN PD) is used by the in-car network. The latest MOST150 standard has
specified a transmission speed of 150 Mbit/s, while the next MOST generation is targeted at multi-Gbit/s. Obviously, the very limited bandwidth of the current physical
layer will weigh on the future MOST generations. However, it is important to evaluate
the potential of the current physical layer, for the reason that the car-manufacturers may
continue using the low-cost and easily operable POFs and LEDs.
The objective of this dissertation is to increase the data-rate for the next MOST generation from 150 Mbit/s to 2 ∼ 3 Gbit/s, based upon the current MOST150 optical
physical layer. The main emphasis lies in investigating electronic signal processing
techniques to detect the multi-level pulse-amplitude modulated (MPAM) signal transmitted through the noisy dispersive POF-based optical channel. To be specific, four
different transmission schemes are studied respectively: the post-equalization scheme
using either linear or decision-feedback equalizer, the joint pre- and post-equalization
scheme, the non-linear Tomlinson-Harashima precoding (THP) scheme, and the bidirectional decision feedback equalization (BiDFE) scheme. In the BiDFE scheme, a
novel trellis-based BiDFE (TB-BiDFE) equalizer is proposed. Their performances are
investigated by means of theoretical analysis and computer simulations. As will be
shown, with the help of electronic equalizers and error-correcting code, the final bitrate is able to reach 3 Gbit/s over a 10 m standard step-index POF, despite the use of a
low-cost LED transmitter.
xv
Kurzfassung
In den letzten Jahren verstärkte sich der Trend im Automobilbereich, optische Netzwerke für Information, Kommunikation und Unterhaltung (Infotainment) im Fahrzeug
einzusetzen. Eine erste weltweite Standardisierung erfolgte mit dem Media Oriented
System Transport und einer Bitrate von 25 Mbit/s (MOST25). Kürzlich wurde das
System MOST150 spezifiziert, das eine höhere Bitrate von 150 Mbit/s erlaubt. Beide
Systeme setzen auf der Bit-Transportschicht (Physical Layer) kostengünstige optische
Komponenten, wie lichtemittierende Dioden (LED), optische Plastikfasern (POF) und
preisgünstige Photodioden ein. Durch die stürmischen Entwicklungen des Internets, des
digitalen Fernsehens mit hoher Auflösung, der Navigation und dem vielfältigen Einsatz
von Videokameras in Fahrzeugen wird der künftige Bitratenbedarf sehr stark ansteigen.
In der vorliegenden Dissertation werden daher Übertragungsverfahren untersucht, die
eine um etwa Faktor 20 höhere Bitrate erlauben sollen, d.h. ca. 3 Gbit/s zur Verfügung stellen. Da die Systeme weiterhin kosteneffizient bleiben müssen, wird als harte
Randbedingung formuliert, dass derselbe Physical Layer wie bei MOST150 Anwendung finden kann. Besonders die begrenzte Bandbreite der optischen Verbindung stellt
für hohe Bitraten einen kritischen Erfolgsfaktor dar. Da spezielle optische Komponenten in den nächsten Jahren immer noch deutlich teurer sein werden als die Elektronik,
wird in dieser Arbeit der Schwerpunkt auf elektronische Verfahren der Codierung, Modulation und Signalverarbeitung im Sender und Empfänger gelegt. Aus Gründen des
Aufwands kommen statt komplexer Modulationsverfahren nur M-stufige Puls-Amplituden-Modulation (MPAM) in die engere Wahl. Darüber hinaus bilden sich vier verschiedene Verfahren heraus, die näher untersucht werden: Erstens, die digitale Vorentzerrung im Sender, bei der entweder ein linearer oder ein entscheidungsrückgekoppelter Entzerrer verwendet wird; zweitens, eine Kombination aus einem Vorentzerrer beim
Sender und einem Entzerrer beim Empfänger; drittens, das nichtlineare Vorentzerrungs-
xvii
xviii
KURZFASSUNG
und Codierverfahren nach Tomlinson-Harashima und viertens ein bidirektionaler entscheidungsrückgekoppelter Entzerrer (BiDFE) im Empfänger. Dabei wird auch ein
neuartiges Trellis-basiertes BiDFE-Verfahren vorgeschlagen. Die Leistungsfähigkeit
aller Varianten wird, wenn möglich analytisch, in jedem Fall auch durch Rechnersimulation ermittelt. Kanalmodelle werden durch Messungen im Labor bestätigt.
In der vorliegenden Arbeit wird gezeigt, dass durch effiziente Modulations-, Codierund Entzerrungsverfahren aufwandsgünstige elektronische Schaltungen möglich werden, um auf einem bestehenden Physical Layer einer Standard-Stufenindex-Plastikfaser
von etwa 10 m Länge Bitraten bis ca. 3 Gbit/s zu erzielen.
CHAPTER 1
Introdution
1.1
Motivation
Since 30 years when the first in-vehicle networks were built to exchange control information between car devices, car-makers are dedicated to improve the transmission
speed of these networks. One of the well-known networks is the controller area network (CAN), which operates at a transmission rate of up to 1 Mbit/s in order to transmit
control signals [1]. Over the last few years, the rapidly developed automobile infotainment applications greatly increased the number of automobile devices involved, for
example, many cars are equipped with the high-grade sound system, navigation system,
telephone systems, DVD changers and the voice operation [2]. These devices are in
great demand of interacting with each other and exchanging not only control but also
audio and video signals. However, CAN is no longer adequate to support the required
fast communication among them because of its narrow bandwidth. Then, the media
oriented systems transport (MOST) infotainment backbone was designed to meet this
specific need. MOST backbone is based on a cost-effective optical physical layer that
consists of red light emitting diodes (LED), standard poly-methylmetacrylate step-index
plastic optical fibers (PMMA SI-POF) and PIN photo-diodes (PD). The optical physical
layer connects various infotainment automobile devices in a single-fiber unidirectional
or in a two-fiber bidirectional optical ring.
One of the most obvious benefits of adopting an optical physical layer for the invehicle communications network is its non-sensitivity to the electro-magnetic interference (EMI) generated by many noisy electronic devices in the automobile environment.
In contrast to the commonly used glass fiber, plastic optical fiber (POF) possesses many
unique advantages and is especially suited for short-range transmission: apart from
1
2
CHAPTER 1 Introdution
having lower-cost and lighter-weight, the large core diameter of POF enables an easier
installation and more robust coupling of light sources into the fiber. Moreover, POF
is more mechanically flexible as well as simpler to fabricate [3]. These characteristics
make POF a quite appropriate transmission media for the MOST networks. In addition
to POF, MOST utilizes the LED transmitter, which is also better suited for automobile applications than the laser diode, thanks to their features like less expensive, less
sensitive to temperature variations, simpler to modulate and more reliable [4].
The very first MOST generation was the MOST25 infotainment backbone, which was
equipped in BMW 7-series in the year 2001 and is able to transfer data 25 times faster
than the CAN. Later on, many other car manufactures such as Mercedes, Audi, and
Volvo followed [2]. Since then, POF-based in-vehicle communication networks have
been well accepted and have become popular. The first upgrade of MOST25 was the
MOST50, where the transmission speed was accelerated to 50 Mbit/s. It was then
upgraded to the state-of-the-art MOST150 operating at 150 Mbit/s. As a matter of fact,
despite MOST150 is 3 times faster than MOST50, it is still incompetent for supporting
those bandwidth-demanding infotainment applications, such as parallel transmission
of HD-videos, multiple side and front view cameras, camera-based driver assistance
systems or high-speed WLAN. With the purpose of serving these applications, a 2 to
3 Gbit/s transmission capability is of great interest for the next MOST generation, which
means, a 10 to 20 times improvement in the current transmission capability is needed.
Equally important as increasing the data-rate, the migration from MOST150 to its future
generation should be smooth for cost reasons [2]. Upgrading the current optical physical layer to the Gbit/s range by glass fibers and lasers is more expensive, compared to
the use of sophisticated electrical signal processing techniques at the transceivers on the
basis of the already existing MOST150 optical networks with POFs and LEDs. On account of this consideration, this dissertation centers on designing and investigating various electrical signal processing techniques, so as to meet the need for a cost-effective
high-speed physical layer in the automobile communications network. With these techniques, it is expected that the speed limit on the MOST150 can be successfully raised
to 3 Gbit/s, despite the use of bandwidth-limited LED and POF.
1.2
Problem Statement
Although MOST150 optical physical channel has the potential for high-speed data
transmission over short distances, we are faced with a few problems.
3
From a communication point of view, the first problem to consider is the channel bandwidth that is a measure of the information-carrying capacity of the given channel. In the
MOST150 optical physical layer, the LED transmitter might decrease the signal bandwidth and distort the signal through non-linearity. Similarly, the transmission media
POF is bandwidth-limited, and so is the PD receiver. Consider the cascade of LED,
POF and PD as an overall channel, its bandwidth is below 140 MHz for a 10∼15 m
long POF. If such a channel is used for 2∼3 Gbit/s data transmission, it is not surprising that the current detected optical pulse will be heavily corrupted by previous and
post received optical pulses arising from the inter-symbol-interference (ISI) distortion,
and become indistinguishable at a conventional receiver. Hence, the first task we are
confronted with is to overcome the ISI term.
Another problem that influences the transmission quality is the channel noise composed
of thermal noise and shot noise. Because noise may disturb the received signal amplitude and increase the bit error rate (BER), it should be handled simultaneously with
ISI.
In brief, the intended transmission at multi-Gbit/s over the MOST150 optical physical layer will be corrupted by strong ISI and noise, whose negative impacts must be
mitigated to ensure a reliable transmission.
1.3
Previous Works and Suggested Solutions
There are various previous researches dedicated on improving the data-rate for shortrange optical communications systems by the use of electronic signal processing techniques. Previous works on this topic mainly offer three solutions: one solution is to
use a linear prefilter/peaking at the transmitter for enlarging the transmission bandwidth, another solution is to use optimized bandwidth-utilization techniques such as
the base-band orthogonal frequency division multiplexing (OFDM) technique, and the
third solution is to use post-equalization techniques together with the multi-level pulse
amplitude modulation signaling. For example, a 1.25 Gbit/s transmission utilizing prefiltered four-level pulse amplitude modulation (4PAM) signal and fractionally-spaced
(FS) post-equalization is presented in [5]. Nevertheless, the prefilter scheme becomes
impractical for transmissions at multi-Gbit/s, because the amplitude range of the predistorted input signal increases along with the data-rate and consequently, a large direct
current (DC) offset is required in the LED transmitter, which leads to a large power
consumption and a decrease in the receiver’s dynamic range. In [6], 1 Gbit/s transmission was demonstrated using the discrete multi-tone modulation (DMT) or the so-called
4
base-band OFDM. However, it is well known that DMT technique requires high linear
transceivers over a wide input optical power range, which is very difficult for the circuit
design especially at a high rate of speed. Besides, the quadrature amplitude modulation
(QAM) modulator/demodulator and fast Fourier transform/inverse fast Fourier transform (FFT/IFFT) components needed by DMT noticeably raise the system cost and
complexity. Furthermore, most systems employing the third solution operated at either
lower speed or, if around 3 Gbit/s, costly laser diodes must be used as fast transmitters. Up to now, there is no report to our knowledge of utilizing a simple optical set-up
based on LED and PIN for high data-transfer rates in the multi-Gbit/s range, since little effort was done previously to design an advanced equalizer whose performance is
good enough to combat the strong ISI generated. Therefore, this dissertation intends to
achieve this goal.
Despite of that, before looking for solutions in the electronic domain, solutions in the
optical domain are looked into. To cope with ISI, there are approaches that use lenses
or similar principles to restrict the modes excited in a fiber. Because the number of
modes involved in a transmission is reduced, the modal dispersion becomes less problematic and the bandwidth of the fiber can be increased. This approach, however, might
have a large power loss because all higher-order modes will be rejected. Moreover,
even if the bandwidth of POF is successfully extended, the bandwidth of LED will still
restrict the speed of the overall optical link. Another drawback of the optical-domain
implementation is that the installation and maintenance of the optical devices might be
expensive and inconvenient. By contrast, the use of conventional optical transceivers
together with electronic signal processing techniques can be quite advantageous. In one
aspect, it is more flexible: if later on there is a change in the fiber length, the same
optical transceivers can be kept without redesigning the system layout but by adjusting
the corresponding electronic signal processing part. In another aspect, it is more stable: after the transmission link is established, most slowly time-varying factors in the
channel such as the operating temperature or aging degradation of the optical devices
can be captured and compensated by a digital adaptive equalizer. For the reasons mentioned above, the optical-domain approaches is finally dropped, and this dissertation
concentrates on developing solutions in the electrical domain.
For simplicity and cost-reasons, the LED transmitter is modulated by intensity modulation, where the information is described by the intensity of a carrier light. Correspondingly, the PD receiver performs a non-coherent direct detection. In addition, with the
intention to reduce the equalization complexity, the bandwidth efficiency is enhanced
by means of modulation techniques. Among which, the classical multi-level pulse amplitude modulation scheme (MPAM) is chosen, for it is probably the most simple and
5
adequate method that can be combined with the LED intensity modulation to increase
the spectrum efficiency. For example, by use of 8PAM, the bandwidth efficiency is
improved three-fold, so the remaining equalization part can be less complex. A short
introduction about the equalizer design will be discussed in the upcoming section.
1.4
An Insight into the Considered Equalization
Strategies
Among all kinds of equalizers, maximum-likelihood sequence estimation (MLSE) [7]
is the most powerful joint equalization and detection technique in the field of combating
ISI. It minimizes probability of errors by applying the trellis-based Viterbi algorithm [8].
However, the complexity of the Viterbi algorithm grows exponentially with the channel
order and the order of the modulation scheme. For the considered transmission, the
MLSE equalizer is nearly impractical to be implemented because of its high complexity.
By contrast, the linear feedforward equalizer (FFE) is well known for its simplicity
but with sub-optimal performance. Thanks to the linear transversal filter structure, its
complexity in proportion to the filter length grows only linearly with the channel order.
For a channel introducing weak to moderate ISI, its performance is often sufficient.
However, the linear equalizer enhances the noise in the process of suppressing ISI.
So eventually, as the channel distortion becomes severe, the performance of a linear
equalizer can be limited by the noise enhancement in an obvious manner. Another
widely used sub-optimal equalizer is the decision-feedback equalizer (DFE) [9, 10].
It improves the performance of a linear equalizer by employing a non-linear structure,
where a feedback filter (FBF) and a decision device are used in addition to a feedforward
filter (FFF). Assume correct decisions, the previous ISI can then be subtracted from the
current symbol by feeding back the previously decided symbols through the FBF. The
FFF suppresses the contribution of the precursor ISI, although it enhances the noise
at the same time, the noise magnification is not that severe as in the case of a linear
equalizer. Besides, the noise can be eliminated at the decision device at the cost of
decision errors.
Both FFE and DFE can apply the zero-forcing (ZF) or minimum-mean squared error
(MMSE) [11, 12] criterion to calculate their coefficients based on the channel information. ZF aims at completely removing the ISI regardless of possible noise enhancement,
while MMSE criterion provides a better trade-off between the noise enhancement and
the ISI elimination. A pre-condition of performing these criteria is to know the channel information. Yet in reality, the channel characteristics are usually unknown, and in
6
many cases, are time-variant. In such situations, the equalizer coefficients shall be able
to converge automatically to a ZF or a MMSE solution, irrespective of what their initial values are. Meanwhile, the equalizers shall be adapted such that any time variation
in the channel will be compensated. Therefore, the application of adaptive equalizers
is also an emphasis of this dissertation. Among the most popular adaptive algorithms,
the least mean square (LMS) algorithm and its variations are considered due to their
simplicity. Compared to LMS, the recursive least squares (RLS) algorithm has a faster
convergence rate and more fidelity, yet at the expense of an increased computational
load. In typical applications, the adaptive equalizer starts with a training mode to gather
information about the channel, and later on switches to the decision-directed mode to
follow the moderate variations during the transmission.
The deduction of the DFE design typically presupposes correct past decisions. However, once a decision error is made, this error will propagate in the FBF, which increases
the error probabilities of the subsequent decisions and may eventually cause error bursts.
This well-known error propagation effect is more serious when the tap weights and/or
the number of the feedback taps become larger [13]. In order to overcome the problem, one of the commonly used precoding techniques at the transmitter [14, 15, 16, 17],
the Tomlinson [18]-Harashima [19] precoding (THP), can be applied. It requires channel information at the transmitter, so an up-link is a requisite for sending the estimated
channel information backwards from the receiver. In this dissertation, THP is combined
with a feedforward equalizer at the receiver to construct, depending on the FFE type, the
zero-forcing THP (ZF-THP) [20, 19], the MMSE-THP [21] or the adaptive THP-FFE.
DFE has another drawback except for the error propagation, that is, the sub-optimal
performance in comparison to the matched filter bound (MFB) [22]. The MFB is defined as the signal-to-noise ratio (SNR) at the receiver output under the assumption that
a matched filter receiver is used and only one symbol is transmitted. Through sending a
single symbol, the transmission is not affected by ISI despite of a band-limited channel,
so the use of the matched filter can provide the maximum output SNR and the minimum
symbol error rate (SER). Under this circumstance, the MFB provides a lower bound on
SER [23, 24] and a pseudo-bound on BER. It is thus a good figure of merit to evaluate
the BER performance of an equalizer.
To cope with the limitation of combating severe ISI and the error propagation problem
in a DFE, we are motivated to search for an alternative suboptimal nonlinear equalizer
with low computational complexity. The new type of equalizer is the bi-directional DFE
(BiDFE) [25, 26, 27], which improves the performance of a single DFE without paying
7
Equalization
Computational complexity
Low
Linear
Performance
Poor
Impairments
Decision Device
Optimization
Moderate
Error Propagation
Filter Length
Optimization
Figure 1.1:
High
MLSE/MAP
Low
DFE
Good
Gap from MFB
Bidirectional
DFE
Non-causal
DFE
The equalization problem
the price for an up-link as in the case of THP. This approach brings in the receive diversity by use of two post DFEs in parallel at the receiver: it uses a reverse-mode DFE
for equalizing the time-reversed received signal and simultaneously, a forward-mode
DFE for equalizing the received signal as in the conventional way. BiDFE is able to
mitigate the error propagation effect because in the two DFEs, decisions are made in
opposite directions and thus the decision errors will propagate in opposite directions,
consequently the most erroneous locations in the decisions of both DFEs are different.
Moreover, even without decision errors, the performance of a BiDFE is superior to a
single DFE because the noises at the outputs of both DFEs (before the decision device)
exhibit a low correlation with each other. That means, a diversity combining of the
decisions from both DFEs can be performed for improving the reliability of the results.
As proposed by [28], a weighted linear combination of the soft decisions from both
DFEs resulted in a smaller value of noise enhancement in comparison to either of the
two constituent DFEs. In addition, it has been shown that when both DFEs are allowed
to have infinite length, the linear combined BiDFE (LC-BiDFE) will be capable of attaining the matched filter bound under the ideal feedback assumption. However, in the
presence of error propagation and if the filter length is constrained, the LC-BiDFE then
will perform worse than another variety of BiDFE - the bidirectional arbitrated DFE
(BAD) [29]. BAD combines the decisions of both DFEs using a reconstruction based
arbitration technique with a higher computational cost. To reduce the complexity of
BAD, the trellis-based conflict resolution (TBCR) [30] algorithm is investigated thereafter. Motivated by TBCR, a novel BiDFE technique - the trellis-based bi-directional
DFE (TB-BiDFE) - is proposed in this dissertation, which is able to provide superior
performance than both BAD and TBCR and at the same time being less complex than
BAD.
8
Figure 1.1 summarizes the equalization problems. This dissertation is concerned primarily with the techniques marked with solid boxes. Techniques marked with dashed
boxes are not taken into consideration, because they are inconvenient to implement in
the scope of automotive environment.
1.5
Outline of the Dissertation
Chapter 2 begins with modeling the modulation transfer function of the plastic optical fiber. The overall optical channel model including the LED transmitter and the PD
receiver is subsequently presented with respect to its electrical transfer function and
3 dB modulation bandwidth. Based on the channel model, a proper assessment of the
signal-to-noise ratio, which is later on necessarily required for evaluating the BER performances, is carried out by use of a more comprehensive electronic equivalent receiver
model. The end-to-end transmission system is then described, and the transmission impairments that must be handled by equalization are indicated. After that, the matched
filter bound is deduced to provide a lower bound on the BER.
To manage the transmission impairments, Chapter 3 first outlines the conventional
equalization strategies and a class of adaptive algorithms on the basis of the mean
squared error (MSE) criterion. The adaptive algorithms are essential for compensating
the variations in channel characteristics, or for performing equalization without knowing the precise channel information. To be specific, the least mean square (LMS), the
normalized LMS (NLMS), the variable step size LMS (VSLMS) and the recursive least
squares (RLS) adaptive algorithms are considered. Their characteristics including the
speed of convergence and computational complexity are analyzed as well.
In the second part of Chapter 3, performances of two transmission strategies, which are
the pre- and post-equalization strategy and the post-equalization strategy, are examined
through computer simulations. The equalizers are developed based on either symbolspaced or fractionally-spaced structure and their coefficients are computed via LMS and
RLS adaptive algorithms. The resulting BER curves and the computational complexities
are compared with each other, with the aim to find out a fair trade-off between the
performance and the complexity. Optimization in some key parameters of the adaptive
equalizers is also demonstrated.
The performance of the two transmission strategies in Chapter 3 is limited either by the
increased transmit power or by the error propagation problem. To surpass the limited
9
performance, an alternative transmission strategy employing the nonlinear THP is studied in Chapter 4. In this chapter, THP at the transmitter is used together with a linear
feedforward equalizer at the receiver, whose taps are either optimized by the MMSE
criterion or automatically adjusted by the normalized LMS algorithm. For analysis
purpose, the THP losses in comparison to an ideal DFE without error propagation are
particularly elucidated.
In Chapter 5, a bi-directional DFE used in the electronic part of the receiver is proposed
as the fourth type of transmission strategies. The objective of introducing BiDFE is to
reach the target BER at a lower SNR comparing to the strategies suggested by the previous two chapters, under the constraint of low to moderate complexity. First, the BiDFE
structure and two well-known BiDFE algorithms (BAD and TBCR) are presented. At
the next step, one of the key contributions of this dissertation, the novel TB-BiDFE
algorithm is presented in detail and explained by examples. In the last section, the
simulation results as well as a comparison of the BiDFE complexities are given.
In the end, this dissertation is concluded with Chapter 6.
CHAPTER 2
System Desription
To begin with, it is useful to develop a numerical channel model for the POF based
MOST150 optical physical layer. Based on the channel model, an overview of the digital optical transmission system and its performance specifications, such as bit error rate
(BER), power consumption and complexity, are subsequently provided. This chapter
also contains the study of the noises on the channel, and provides the essential information of the average receiving SNR. Furthermore, a brief description of an experimental
set-up for measuring the modulation transfer function of the optical channel is included.
The closing section of the chapter points out the major transmission impairments and
the necessity of implementing equalization. For that, a performance upper bound for
equalization, called the matched filter bound, is deduced. As it is well beyond the scope
of this chapter to treat the optical communication in depth, we concentrate just on what
is needed for understanding this dissertation.
2.1
The Channel Model
Despite the use of an optical transmission link, the information sinks are automobile electronic devices working with electronic signals. Consequently, the electronicoptical-electronic conversion must be performed. For this reason, we aim to model the
end-to-end optical link to its electrical equivalent channel. Then, we estimate the average receiving SNR at the channel output, which is an important parameter that will
be used by computer simulations and results validation in later chapters. Finally, the
feasibility of the numerical channel model is proven by experimental measurements.
11
12
CHAPTER 2 System Desription
2.1.1 Modulation behavior of the standard plasti optial
ber
The plastic optical fiber is a type of multi-mode waveguide, since there are usually different optical paths propagating along the fiber after an optical pulse is launched by
the transmitter. For a step-index type of POF, light on each path arrives at the receiver
with a different time delay, leading to an overlap-add of multiple copies of the transmitted optical pulse which results in a broadened received optical pulse. This well-known
phenomenon is called the modal dispersion, which depends solely on the POF characteristics and almost determines the modulation bandwidth (or electrical bandwidth)
of the POF. It should be noted that the bandwidth of POF in this dissertation always
refers to the 3 dB bandwidth of its modulation transfer function (or the equivalent lowpass transfer function) rather than the optical frequency transfer function. The modulation bandwidth of POF can be hardly increased once its length, material, structure and
launching condition were specified.
Generally speaking, there exist various types of dispersions in a fiber which limit the
electrical bandwidth. They can be categorized into two types: the propagation path
dependent modal dispersion and the wavelength dependent chromatic dispersion. Table 2.1 summarizes all kinds of them in terms of the fiber type [31]. Because the considered POF is multi-mode and short in length, material and chromatic dispersion can
be ignored for their minor effects on a short fiber. Consequently, the modal dispersion
imposes the most significant constraint on the modulation bandwidth for the considered
POF type.
Table 2.1:
Dispersions in Fibers
modal dispersion
polarization
mode disp.
(singlemode)
hromati dispersion
prole disp.
material disp.
(multimode)
(multi- and
singlemode)
waveguide
disp.
(singlemode)
prole disp.
(multimode)
By taking into account the measured values of modal dispersion, MOST150 oPHY
automotive physical layer sub-specification [32] has specified that the base-band modulation transfer function Hpof (f ) of a standard step-index POF behaves like a Gaussian
low-pass filter:
2
Hpof (f ) = Ae−2(πσf ) e−j2πf τ Lpof ,
0≤f ≤B
(2.1)
13
where A is the linear fiber loss, σ = 0.132/B3dB is the standard deviation, B3dB =
Lpof −0.8747
)
MHz is the 3 dB modulation bandwidth, τ = 4.97 · 10−9 s/m is the
1009 · ( m
transmission delay per meter, Lpof is the fiber length in meters and B is the receiver
bandwidth.
It should be pointed out that except for [32], some literature have reported different
modulation bandwidths for POFs with the same length. Actually, there are many factors affecting the calculation of the theoretical modulation bandwidth, such as the fiber’s
numerical aperture, amount of mode coupling, launch condition, temperature and so
forth. All these factors influence the amount of modal dispersion and thus the modulation bandwidth. For example, an under-filled launch condition tends to produce a
higher bandwidth than an overfilled condition, because only a portion of the modes are
excited in the fiber which results in a smaller total modal dispersion. It has been often
reported that this limited launching effectively increases the POF’s bandwidth. Because
[32] cites the result from measurements carried out according to the MOST150 standard launching condition, we also use (2.1) as the theoretical POF modulation transfer
function throughout the dissertation.
2.1.2 Optial hannel modeling
Figure 2.1 depicts the end-to-end transmission system including the digital signal processing (DSP) component by a set of block diagrams. The optical transmission link,
which is marked by the dashed box, comprises a red light emitting diode (LED) analog transmitter, a common polymethyl methacrylate (PMMA) step-index optical fiber
(POF) with 1 mm core diameter, and a PIN photo diode (PD) receiver applying the
direct detection. The LED transmitter modulates the intensity of light pulses proportional to the electrical signal at the DAC output. The light pulses propagating along
the POF get distorted by fiber dispersions and attenuation, and are finally captured and
converted by the PD receiver to a proportional electric current, which is then converted
to voltage by a subsequent front-end amplifier. If necessary, the voltage signal will be
further amplified by a main amplifier. Note that the optical receiver bandwidth must
be less than one half of the ADC sampling rate in order to meet the Nyquist sampling
criterion. Finally, we define the electrical equivalent representation of the optical link
from the DAC output up to the ADC input as the electrical equivalent channel, which is
abbreviated as channel in the following.
The channel can be seen as linear if both LED and PD work in their linear response
regime. Based on this assumption, the channel frequency response Ha (f ), which is a
14
Electrical domain
Information
Symbols
Transmitter
x[k]
x(t)
Modulator
LED
DAC
Electrical domain
Offline
DSP
Optical link
Receiver
y[k]
y(t)
ADC
Figure 2.1:
POF
Channel
Demodulator
PD
Optial ommuniation system model
cascade of the LED transmitter, POF and the PD receiver, can be decomposed into a
product of three transfer functions: one for the LED transmitter, one for the POF, and
one for the PD receiver. Among the three, the inherent bandwidth of LED can be larger
than 150 MHz depending on the type and the biasing condition, PD provides normally
the greatest bandwidth, yet a POF of length 10 m has a modulation bandwidth of about
135 MHz. Because the transfer functions of LED and PD are normally flat up to the 3
dB bandwidth, and the modulation bandwidth of POF is smaller than the ones of LED
and PD, the channel transfer function Ha (f ) can be approximated by the POF transfer
function in (2.1) up to the 3 dB bandwidth:
Ha (f ) = Hpof (f ),
0 ≤ f ≤ B3dB ,
(2.2)
However, we apply the Gaussian low-pass approximation for the transfer function up to
the receiver bandwidth in the dissertation, as this pessimistic approximation provides us
more conservative simulation results. Finally, the electrical equivalent channel transfer
function is given by:
2
Ha (f ) = Ae−2(πσf ) e−j2πf τ Lpof ,
0 ≤ f ≤ B.
(2.3)
Correspondingly, the electrical equivalent channel impulse response is:
ha (t) = √
)2
A − (t−τ Lpof
2σ 2
e
.
2πσ
(2.4)
ha (t) is a Gaussian impulse with area A (equals to the linear fiber loss) and variance σ 2 ,
and is symmetric to the central time tc = τ · Lpof . Note that σ is a function of Lpof . As a
15
Normalized channel impulse response (10m)
0.04
ha (t)
0.03
0.02
0.01
0
0
0.02
0.04
0.06
Time [us]
0.08
0.1
Normalized modulation transfer function (10m)
0
Ha (f )/Ha (0)
3dB bandwidth 135MHz
−2
−4
−6
−8
0
50
100
Frequency [MHz]
150
200
The normalized hannel impulse response ha (t) and the hannel
transfer funtion Ha (f ) (assuming zero phase shift) based on a 10 m POF
Figure 2.2:
POF used by MOST automotive applications is 10 to 15 meters long [32], we consider
here a 10 m POF whose 3 dB modulation bandwidth is about 135 MHz calculated by
using (2.3). Figure 2.2 shows the normalized theoretical channel impulse response and
its transfer function (assuming zero phase shift) for a 10 m POF .
2.1.3 Reeiver noise model and the reeiver-side SNR estimation
Throughout this dissertation, the commonly used BER in digital communications is utilized as a figure of merit to evaluate the performance of our digital optical transmission
system. BER however, closely relies on the SNR at the receiver which is determined
by the noise quantity if a steady signal power is received. Mostly, noise in an optical
transmission link comes from the receiver and is a mixture of thermal and shot noise.
In this section, noises as well as SNR at the receiver side are investigated.
16
2.1.3.1 Reeiver noise model
During the process of converting an optical signal into an electrical current, PD generates not only the signal current but also additional noise currents. To predict the amount
of noise currents for sake of SNR estimation, a receiver noise model is first established,
whose block diagram is shown in Figure 2.3.
photo-diode
noise filter
ip
G0
y(t)
front-end amplifier
Figure 2.3:
Optial reeiver blok diagram
Without loss of generality, the optical receiver compromises a photo-detector, a frontend amplifier and a noise filter. The functionality of the preamplifier is providing a
low-noise interface for receiving the weak detected photo-current. Ideally, it should
be a current-in/voltage-out amplifier with high bandwidth. Unfortunately, most solidstate field-effect transistor (FET) amplifiers are of the voltage-in/voltage-out or voltagein/current-out variety [33]. To overcome this problem, a bias resistor Rb can be connected in series to the photo-diode, for converting the photo-current into a voltage signal
which can then be amplified by a voltage amplifier. This set-up is known as the highimpedance or low-impedance front-end, depending on the value of Rb . Here, we focus
on analyzing the SNR performance for a low-impedance front-end receiver as plotted in
Figure 2.4, for its wide bandwidth is suitable for high-speed transmission and its worse
noise performance delivers a conservative SNR estimation. The noise filter after the
amplifier is used to reject the noise power outside the receiver bandwidth.
Then, we draw the equivalent circuit of the low-impedance front-end receiver including
various associated noise sources in Figure 2.5. Here, the photo-diode is represented by
a current source ip , a shunt capacitance Cd and a shot noise source is . Rb is the bias
resistor. And the amplifier has an input resistance Ra , an input capacitance Ca as well
as a gain of G0 . The noise current generated by the amplifier at its input is indicated
by ia . The total thermal noise generated by Ra and Rb is represented by it . Hence, the
17
VB
ip
G0
current-to
voltage
converter
(resistor)
Figure 2.4:
Rb
noise filter
Vout
Amplifier
Low-impedane front-end reeiver
i p ′ + i n′
1
Rb
po (t)
Ra
G0
2
Ca
Noise
Filter
Cd
ip (t) is (t)
it (t)
Photo diode
Figure 2.5:
y(t)
ia (t)
Amplifier
Noise free
amplifier
Small signal model of the reeiver
circuit has a total capacitance of
CT = Cd + Ca ,
(2.5)
Rb Ra
,
Rb + Ra
(2.6)
and a total load resistance of
RT =
whose associated thermal noise source is it . Because this front-end has a low-pass
characteristic, its 3 dB bandwidth is determined by the time constant CT · RT , namely:
B≤
1
.
2πRT CT
(2.7)
18
Obviously, RT should be kept small to ensure a large bandwidth B. This is also why
the front-end is a low-impedance type: it allows the thermal noise to dominate and
sacrifices the noise performance to trade for high receiver bandwidth.
The various noise sources associated with the front-end receiver can be described by
their equivalent noise currents, which are summarized as follows:
• Shot noise current is (t) from dark current and quantum noise:
the dark current Idark is referred to as the current generated by a photo-diode
even in the absence of an incident light. And the quantum noise comes from the
fact that the light flows as discrete photons rather than a continuous fluid [34].
The sum of these two noises gives the total shot noise is in a photo-diode. The
mean-square value of the shot noise can be expressed as:
i2s = 2q(Ip + Idark )B,
(2.8)
where q = 1.602 × 10−19 C is the elementary charge, B in Hz is the bandwidth
of the receiver,
q
Ip =
i2p
(2.9)
is the root mean square value of ip (t), and ¯ denotes the mean operation. The
power spectral density (PSD) of is (t) is approximately constant (white).
• Thermal noise current it (t) from the detector bias resistor Rb and the amplifier
input resistance Ra . Its mean square value is:
i2t =
4kB T0 B
,
RT
(2.10)
where kB = 1.38054 × 10−23 J/K is the Boltzmann’s constant, T0 is the absolute
temperature in Kelvin, and RT is the parallel resistance of Rb and Ra given by
(2.6) in Ohm. The PSD of it (t) is constant (white).
• Thermal noise current ia (t) from the front-end amplifier. In general, this noise is
related to the type of the amplifier and its PSD is approximately constant (white)
within the receiver bandwidth. For the ease of calculation, here we adopt the
amplifier noise figure Fn , which is a parameter that can be used to calculate the
thermal noise without considering the internal structure of the amplifier. The
noise figure Fn is referred to as the ratio of the input SNR to the output SNR
exhibited by a device, for a specific input noise temperature T0 .
19
P1
N1
G0
(N3 )
Figure 2.6:
P2
N2
An amplier model
To better explain how to calculate the noise generated by an amplifier by using
the noise figure Fn , an amplifier with gain G0 = P2 /P1 is plotted on Figure 2.6,
where Pν and Nν (ν = 1, 2) represent the signal and the noise mean power at
node 1 or 2, respectively. Now we define the input SNR of the amplifier as P1 /N1 ,
and the output SNR of the amplifier as P2 /N2 . We also define the noise power
generated by the amplifier at its input as N3 . So we could write
N2 = G0 · (N1 + N3 ).
Because Fn is defined as:
P1 /N1
P2 /N2
1 N2
=
·
G0 N1
G0 · (N1 + N3 )
=
G0 · N1
N3
=1+
,
N1
Fn =
Fn equals:
Fn = 1 +
N3
.
N1
(2.11)
In the amplifier circuit in Figure 2.5, we consider the noise power at the amplifier
input N1 = i2t and N3 = i2a . Then using (2.11) and (2.10), we get the mean square
value of ia :
i2a = (Fn − 1) · i2t
=
4kB T0 B(Fn − 1)
.
RT
(2.12)
20
2.1.3.2 Analytial SNR estimation
The signal-to-noise ratio should be estimated at node 2 in Figure 2.5. However, we
could estimate the SNR value at node 1 instead. This approach is feasible because the
block with G0 in Figure 2.5 is noise free, and the SNR value at node 2 is the same as at
node 1.
At node 1, we define ip′ as the signal current generated by ip , and in′ as the total noise
current generated by is , it and ia . We also define the total noise current as:
in = is + it + ia .
(2.13)
Because the noise currents are statistically independent, we obtain:
i2n = i2s + i2t + i2a .
(2.14)
Using (2.8), (2.10) and (2.12), we get
4kB T0 B 4kB T0 B(Fn − 1))
+
RT
RT
4kB T0 Fn B
.
= 2q(Ip + Idark )B +
RT
i2n = 2q(Ip + Idark )B +
(2.15)
Assume that the PSD of the signal ip (t) is Ip2 /2B for |f | ≤ B, and zero elsewhere.
Since the transfer functions between each current source and the current at node 1,
respectively, are the same, and the PSD of the signal and noise currents are constant
(white) for |f | ≤ B, and the signal and the noise currents are statistically independent
with each other, the SNR at node 1 in the frequency range |f | ≤ B results in:
γSN R =
i2p′
i2n′
i2p
=
i2n
,
(2.16)
Inserting (2.9) in (2.16) yields:
γSN R =
Ip2
i2n
.
(2.17)
The signal current ip (t) generated by a photo-diode varies linearly in proportion to the
received optical power po (t):
(2.18)
ip (t) = R0 · po (t),
21
with R0 the responsivity of a PIN photo-diode. With the root mean square optical power
q
p2o (t),
(2.19)
Ip = R0 · Po ,
(2.20)
Po =
we get from (2.18):
Inserting (2.20) and (2.15) into (2.17) yields the electrical SNR for the low-impedance
front-end receiver with an amplifier at node 1:
γSN R =
(R0 Po )2
2q(R0 Po + Idark )B +
4kB T0 Fn B
RT
(with an amplifier)
(2.21)
2.1.3.3 Numerial SNR estimation
To evaluate SNR in (2.21), the receiver bandwidth B should be decided at the first step.
As discussed in Section 2.1.3.1, a suitable receiver bandwidth should be larger than the
bandwidth of LED and POF in cascade, implying that B should be at least 135 MHz,
if a 10 m POF is considered. As B is decided by RT according to (2.7), the maximum
load resistor RT that produces a sufficient receiving bandwidth Bmin = 135 MHz is
then:
max{RT } =
Table 2.2:
peak wavelength
output power
rise and fall time
bandwidth
1
2πBmin CT
(2.22)
.
Transmitter and reeiver basi speiation
LED
(IF : forward urrent)
ondition
IF = 10 mA
IF = 20 mA
IF = 20 mA
typial
650
1.8
4.8
unit
nm
dBm
ns
110
MHz
PD
(VR : reverse voltage, VF : forward voltage)
ondition
typial unit
dark urrent
VR = 10 V, T = 105 ◦ C
300
nm
total apaitane
VF = 1.6 V
2.5
pF
responsivity
λ = 650 nm
0.35 A/W
22
where CT = Cd + Ca is the total capacitance of the photo-diode and the amplifier, and
Cd = 2.5 pF is the capacitance of the photo-diode given by Table 2.2. Table 2.2 also
lists other relevant parameters of LED and PD for the SNR estimation.
Without loss of generality, we assume an amplifier capacity Ca = 2.5 pF and an amplifier noise figure Fn = 5 dB ≈ 3, which are some typical values of a start-of-the-art FET
amplifier [35]. For Bmin = 135 MHz and a total capacitance CT = 5 pF, the maximum
load resistance RT according to (2.22) is 235 Ω. Normally, it is important to select the
total load resistance RT as big as possible in order to keep the associated thermal noise
at a low level. However, for the sake of conservative SNR estimation, a relatively small
value of RT = 150 Ω is chosen, such that the thermal noise power can be relatively
high. Again, according to (2.7), RT = 150 Ω allows a maximum receiver bandwidth of
212 MHz that is higher than the signal bandwidth 135 MHz. Nevertheless, the receiver
bandwidth B should to set to 135 MHz. The reason is that the excessive noise power
outside the signal bandwidth can be rejected by a subsequent low-pass noise filter after
the amplifier. Thus from this point of view, only the signal bandwidth is relevant for the
SNR estimation. Consequently, we calculate γSN R in (2.21) with B = 135 MHz and
RT = 150 Ω.
After determining the receiver bandwidth, the next step is to find out the average received signal power. According to (2.20), its minimum and maximum value can be
calculated via R0 = 0.35 A/W, and a minimum received optical power
Pmin = −22 dBm = 6.3 × 10−6 W
or a maximum received optical power
Pmax = −2 dBm = 6.3 × 10−4 W
defined by [32], respectively. As a result, the minimum average signal current has a
value of:
min{Ip } = R0 · Pmin
= 0.35 × 6.3 × 10−6 A = 2.2 × 10−6 A,
(2.23)
while the maximum average signal current is:
max{Ip } = R0 · Pmax
= 0.35 × 6.3 × 10−4 A = 2.2 × 10−4 A.
(2.24)
23
The following step is to calculate the noise power. Since the mean-squared noise current
contributed by load resistance and amplifier is irrelevant to the received signal power,
they can be calculated explicitly. Applying (2.10), (2.12), RT = 150 Ω, B = 135 MHz,
Fn = 3 and T0 = 293 K, we get:
4kB T0 BFn
RT
5.52 × 10−23 · 293 · 135 × 106 · 3 2
A
=
150
= 4.37 × 10−14 A2 .
i2t + i2a =
(2.25)
In the last step, by inserting (2.23), (2.24) and (2.25) into (2.21), the minimum and
maximum SNR value can be finally evaluated numerically. With Idark = 300 nA, the
minimum and maximum SNR at the receiver yields:
min{γSN R } =
(min{Ip })2
2q(min{Ip } + Idark )B +
4kB T0 Fn B
RT
−6 2
≈
2 · 1.6 ×
≈ 20.4 dB,
and
max{γSN R } =
≈
(2.2 × 10 )
· 2.5 · 135 + 4.37 × 10−14
(max{Ip })2
2q(max{Ip } + Idark )B +
2 · 1.6 ×
≈ 59.5 dB,
(2.26)
10−19
4kB T0 Fn B
RT
−4 2
(2.2 × 10 )
· 2.2 · 135 + 4.37 × 10−14
(2.27)
10−17
respectively. From the calculations above, it is easy to notice that the thermal (AWGN)
noise is the most conspicuous noise source and largely dominates over the shot noise in
the receiver. Because of this effect, an AWGN noise model can be used to describe the
noise generated by the optical receiver.
2.1.4 Experimental setup and results
This section presents the channel magnitude response measured by experimental setup,
with the purpose to verify the previous statement that POF is the limiting factor for the
overall optical channel.
24
2.1.4.1 Experimental setup
The MOST150 standard headers and 1 mm step-index POFs with different lengths are
used in the measurements. These standard headers are embedded with coupling of the
headers to the POF, as well as optical devices (red LED and PIN photo-diode) whose parameters of communications interest are given by Table 2.2. The considered POF length
is 1, 3 and 15 meters respectively, where 15 m is the maximum length allowed between
two communication nodes according to the MOST150 physical layer specification.
The LED is driven by a very simple analog circuit where the information signal directly
modulates the light intensity. The transmitter is not equipped with any special functionality such as analog equalizers/peaking circuit or linearizing circuit. The receive
circuit has a low-impedance front-end structure. The transceivers as well as their electronic driving and receiving circuits are built on to a single-sided copper clad board with
the copper side used as a ground plane. The ground pins of the different components
are directly soldered to the ground plane, while the other pins are air-wired above the
ground plane. The input and output of this board are connected to a network analyzer
to study the base-band transfer function of the optical link. During the measurement,
the network analyzer produces a small swept RF signal imposed on the transmitter and
sweeps the output frequency of the receiver repeatedly over the range of our interest,
i.e., from 10 kHz to 300 MHz. The RF modulation power is varied from -20 dBm (at
small bias) to 0 dBm, and at the same time, the biasing current is adjusted to ensure that
the modulation is in the small signal regime and there is no observable change in the
measured bandwidth.
2.1.4.2 The measured magnitude responses
The outcome of the network analyzer is illustrated in Figure 2.7, where the three colored
curves represent the measured electrical magnitude responses for the cascade of LED,
PD and POF in three lengths, respectively. In addition, the magnitude response of a
theoretical Gaussian low-pass filter with 3 dB bandwidth of 85 MHz is plotted by the
black curve, corresponding to the magnitude response of the theoretical modulation
transfer function of a 15 m POF given by (2.3). Note that the term bandwidth always
refers to the electrical 3 dB bandwidth, which is defined as the frequency where the
electrical level of a sine-modulated signal has dropped by 3 dB.
It can be observed that the measured magnitude response for the 15 m POF is well consistent with the theoretical Gaussian low-pass shape up 160 MHz. Above this frequency,
25
−18
3dB = 105MHz
−21
3dB = 110MHz
Magnitude Response [dB]
−24
−27
3dB = 85MHz
−30
−33
3dB = 85MHz
−36
−39
−42
1m POF
3m POF
15m POF
Gaussian function (3dB 85MHz)
Gaussian function (3dB 110MHz)
−45
−48
0
Figure 2.7:
20
40
60
80
100 120 140 160 180 200 220 240 260 280 300
Frequency [MHz]
The measured magnitude responses between LED input and PD
output using dierent lengths of POF
the Gaussian low-pass filter is no longer a good approximation for the measured channel magnitude response but decays much faster than the measured result. The result
implies, the Gaussian approximation of the POF magnitude response in (2.3) is rather
accurate up to about twice the 3 dB bandwidth. It also implies that the overall channel
bandwidth is limited by POF rather than by optical transceivers.
Moreover, the magnitude responses for 1 or 3 m POF are not really Gaussian shaped.
The reason is that now the LED transmitter somehow dominates the magnitude response
of the cascade. Also it has to be pointed out that significant measurement errors occurred
for frequencies beyond about 180 MHz, mainly due to external interference which is
present even without an incident light.
2.2
Overview of the Transmission System
In the previous section, the POF based optical channel in the continuous time domain is
described. By taking into account that the continuous- and discrete-time domain can be
interchanged without information loss, and the objective is to transfer digital signals, we
want to develop a comprehensive digital transmission system in this section, from introducing the modulation technique and error-correction coding, through exploring the
26
major channel impairments including noise and inter-symbol-interference (ISI), to deduce a performance upper bound for the minimum output BER if ISI can be completely
removed.
2.2.1 The disrete system model
By including DAC and ADC in Figure 2.1 for the channel modeling, the channel impulse response ha (t) given by (2.4) can then be converted into its discrete-time version
h[k], as illustrated by Figure 2.8. For a 10 m POF, the channel impulse response ha (t)
is plotted in Figure 2.2. We select t0 and te such that the very small values of ha (t) for
t < t0 and t > te can be ignored. We then sample ha (t) with T to get the discrete-time
channel impulse response h[k] for the cascade of DAC, the optical link and the ADC:
h[k] =
(
T · ha (t0 + kT ),
0
0 ≤ k ≤ Lc − 1,
k < 0; k ≥ Lc
(2.28)
0
⌉ + 1 is the length of the
where t0 is the sampling delay, k ∈ Z and Lc = ⌈ te −t
T
discrete channel. The sampling duration T considered in the dissertation equals either
the symbol duration Ts or Ts /2 . To compensate for the sampling effect on the numerical
expression, T should be multiplied with ha (t) in (2.28) for converting a continuous-time
system to its discrete-time counterpart. Note that the effects of non-ideal DAC and ADC
conversions are ignored here, because the sampling rates as well as the bandwidths of
DAC and ADC are much higher than the channel bandwidth and the quantization step
size is selected sufficiently small. Neither is the impulse shaping considered, because
the channel itself can be considered as a Gaussian-pulse shaper.
n(t), σn2
x[k]
x(t)
DAC
ha (t), Ha (f )
+
y(t)
ADC
y[k]
n[k]
x[k]
h[k]
Figure 2.8:
+
Disrete hannel model
y[k]
27
source
b′ [m]
b[n]
RS coding
mapper
a[k]
precoding
x[k]
h[k]
+
sink
bˆ′ [m]
RS decoding
b̂[n]
Figure 2.9:
demapper
â[k]
channel
n[k], σn2
y[k]
receiver
Disrete system model
The block diagram of the complete digital transmission model is plotted in Figure 2.9
based on h[k]. At the transmitter, information bits b′ [m] are first encoded by a ReedSolomon (RS) encoder to generate the bit sequence b[n]. Here, RS code (255, 223)
is selected because of its wide use in the field of digital video broadcasting (DVB)
or space communications [36]. RS code (255, 223) is capable of correcting up to
(255 − 223)/2 = 16 erroneous Reed-Solomon symbols in a single codeword of length
255. Since one RS symbol carries 8 bits, RS code can simultaneously correct up to 16
short bursts of bit errors and decrease the BER from 10−3 before decoding to less than
10−9 after decoding [37]. The coding gain brought by the RS code is essential for the
considered digital transmission, because the target BER of 10−9 is hardly achievable by
solely using an equalizer. Using RS coding describes the fact that as long as the BER
at the equalizer output stays below 10−3 , the BER of 10−9 will be reached after the RS
decoder. Consequently, the target BER for an equalizer to achieve is reduced to 10−3 .
Besides, to ensure correct recovery of information bits in the decoding process, a guard
interval longer than the delay introduced by the subsequent channel and receiver, must
be added to each RS codeword at the transmitter.
After RS encoding, b[n] is mapped into symbol sequence a[k] by the M-ary pulse amplitude modulation (MPAM). MPAM is chosen because of its simplicity for optical
communications. If neither precoding nor pre-equalization is used at the transmitter,
the transmit signal x[k] equals a[k]. Otherwise, a[k] is pre-coded or pre-filtered to generate x[k]. At the channel output, a Gaussian noise sample n[k] representing the noise
from the electronic circuitry in the optical receiver is added to the received signal, where
the mean power of the noise is denoted as σn2 . Finally, the received signal can be written
as:
y[k] = x[k] ∗ h[k] + n[k].
(2.29)
28
Figure 2.10:
Unipolar MPAM signal onstellation diagram with d = 2
In case of bipolar MPAM, a[k] takes value from a discrete set {−(M − 1) d2 , −(M −
3) d2 , ..., (M − 1) d2 }, M ≥ 2, where d is the minimum Euclidean distance between
two symbol constellation points. In case of unipolar MPAM, a[k] takes value from
{0, d, ..., (M − 1)d}, M ≥ 2. An example for an unipolar MPAM signal constellation
diagram is given in Figure 2.10 for d = 2. Assume that all constellation points are
chosen with equal probability and a[k] is independent and identically distributed (i.i.d.),
then the mean power for an unipolar MPAM data sequence is derived as:
def
ξa = E[|a[k]|2 ]
=
(2.30)
d2 (M − 1)(2M − 1)
.
6
2.2.2 Channel impairments
Assume the discrete channel impulse response h = {h[k], k = 0, . . . , Lc − 1} has
a length of Lc = L1 + L2 + 1 and a main tap h[L1 ], where L1 and L2 represents,
respectively, the length of the pre- and post-cursors. (2.29) can be decomposed into:
y[k] =
L
c −1
X
i=0
(2.31a)
h[i]x[k − i] + n[k]
= h[L1 ]x[k − L1 ] + I[k] + n[k] .
|
{z
}
|{z}
|{z}
desired signal component
overall ISI
(2.31b)
noise
The first term of (2.31b) corresponds to the desired signal component, the second term
is the overall inter-symbol-interference (ISI), and the third term accounts for the AWGN
with zero mean. The overall ISI I[k]:
I[k] =
L
c −1
X
i=0
i6=L1
=
L
1 −1
X
i=0
|
h[i]x[k − i]
h[i]x[k − i] +
{z
}
ISI from post symbols
L
c −1
X
i=L1 +1
|
h[i]x[k − i],
{z
}
ISI from previous symbols
(2.32)
29
is the sum of the ISI from previous and future transmitted symbols.
The ratio between the desired signal power and the noise power in (2.31b) yields:
γSN R
ξx h2 [L1 ]
,
=
σn2
where ξx is the mean power of x[k]. But the actual signal distortion at the receiver is the
sum of noise and ISI, and the signal-to-interference-plus-noise ratio (SINR) should be
considered:
γSIN R =
ξx h2 [L1 ]
,
σn2 + E[I 2 [k]]
Obviously, γSIN R is always smaller than γSN R . They are equal only at very low SNR
values, where the ISI can be neglected compared to the strong noise. Therefore, at
very low SNRs, the channel can be seen as an approximate “AWGN” channel, and the
matched filter is considered as the optimum receiver.
Conversely, at very high SNR region, we have
γSIN R ≃
ξx h2 [L1 ]
,
E[I 2 [k]]
which is independent of the SNR. In this case, the SINR has a ceiling effect no matter
how much the SNR increases. That means, using a matched filter can no longer bring
forth any obvious benefits, since what it does is just to collect energy so as to increase
the SNR at its output. Although a matched filter is the optimum receiver for an AWGN
channel, when ISI is present, their performance can be quite poor. Consequently, an
equalizer instead of a matched filter must be employed for the purpose of eliminating
ISI.
2.2.3 The mathed lter bound for band-limited hannel
The matched filter bound (MFB) is an upper bound on the output SNR, or more precisely, the SINR which can be reached provided that all ISI is ideally removed by an
equalizer [24, 38]. For any un-coded system, since MFB delivers the maximum achievable SINR, it also leads to a lower bound on the symbol error rate (SER) [24]. However, it is only a pseudo-bound on the BER, since other receivers such as the maximumlikelihood symbol detection may provide a better result on the BER. Nevertheless, since
it is more convenient to use MFB to obtain a closed form expression for SER [24], the
BER performance provided by the MFB is considered throughout the dissertation as an
30
approximate BER lower bound, with which the BER performances of the considered
equalization schemes can be compared.
The MFB according to [39] is expressed as:
γM F B = d2 ·
Eh
,
Φnn
(2.33)
where Eh is the energy of the channel impulse response, d is the distance between
two adjacent constellation symbols and Φnn = N20 is the double-sided power spectrum
density of the AWGN. If ISI is present, the receiver will generally not achieve the MFB
because of the impact of ISI. Thus, a lower bound on the SER for a unipolar M-ary PAM
transmission scheme in the presence of AWGN can be calculated using the MFB [39]:
√
γM F B
M −1
)Q(
)
Ps ≥ 2(
M
s2
!
d2 Eh
M −1
,
)Q
= 2(
M
4Φnn
(2.34)
where Q(·) is the Q-function. Inserting (2.30):
d2 =
6
· ξa
(M − 1)(2M − 1)
(2.35)
into (2.34), we get:
M −1
Ps ≥ 2(
)Q
M
s
3
ξa Eh
·
2(M − 1)(2M − 1) Φnn
!
(2.36)
.
Finally, the MFB leads to a pseudo lower bound on the BER for a given channel:
2
M −1
Pb ≥
·(
)Q
log2 M
M
s
3
ξa Eh
·
2(M − 1)(2M − 1) Φnn
!
.
(2.37)
(2.37) is used as a theoretical lower bound on the BER throughout the dissertation and
as a figure of merit to judge the capability of an equalizer at suppressing the interference
and noise.
Figure 2.11 shows the theoretical BER lower bound using (2.37) and the simulated
lower bound generated using 108 realizations of a single bit transmission. It can be seen
31
that the simulated and theoretical bounds agree with each other. One might ask for the
difference between this BER lower bound and the BER curve derived for an AWGN
channel. Actually, they are the same. This is because the matched filter collects all the
signal energy and ISI is perfectly removed [24]. However, it is necessary to introduce
the concept of the MFB, since the simulation results should be compared based on the
same type of channel.
−2
10
Simulated BER lower bound
Theoretical BER lower bound
−3
Bit Error Rate
10
−4
10
−5
10
−6
10
−7
10
25
26
Figure 2.11:
27
28
29
SNR [dB]
30
31
Simulation of the mathed lter bound
32
CHAPTER 3
Equalization of the Optial Channel
As pointed out in the last chapter, an increase in the signal-to-noise ratio is unable to
reduce the ISI impact and therefore equalizers are needed. With the purpose of proposing and exploiting diverse well-performing low-complex equalization strategies, Section 3.1 opens with designing the optimal equalization architectures, continues with
approximating the optimal equalizers with a finite length structure, and finally concentrates on the practical digital adaptive equalizers which utilize the least mean square
(LMS), the normalized LMS, the variable step size LMS (VSLMS) and the recursive
least squares (RLS) adaptive algorithms.
In Section 3.2, two transmission strategies are developed for the POF-based optical
channel. One of them is based on the post-equalization technology, and the other one
combines the pre-equalization with the post-equalization method. A detailed design
for the equalizers with respect to the filter length, the tap coefficients, the adaptive
parameters and so forth is expounded. The BER performances of both transmission
strategies are subsequently examined by computer simulations in Section 3.3.
3.1
General Equalization Tehniques
Among all types of equalizers, it is well known that the maximum-likelihood sequence
estimation (MLSE) equalizer is optimal for detecting data sequences under the maximumlikelihood (ML) criterion: it implements the trellis-based Viterbi algorithm to estimate
the most likely transmitted information sequence. However, MLSE is not considered
here for a couple of reasons. First, the complexity of the Viterbi algorithm grows exponentially with the channel order and the size of the symbol alphabet. For the considered
transmission with 8PAM and a channel of order 5, the MLSE computes 85 metrics for
33
34
CHAPTER 3 Equalization of the Optial Channel
each new received sample, which is too complex. Second, by use of channel coding,
it is possible to approach the channel capacity even in the presence of ISI. In this case,
MLSE can be unnecessary [39]. Instead of MLSE, this chapter concentrates on less
complex but good enough sub-optimal equalization techniques, which are the feedforward equalizers (FFE) and the decision feedback equalizers (DFE).
Before deriving the practical adaptive digital equalizers in Section 3.1.3, we first deduce
the optimal equalizer under the zero-forcing (ZF) and the mean-squared-error (MSE)
criterion, respectively.
3.1.1 Optimal equalization arhitetures
The objective of this section is to derive the optimum continues-time receive filter
HR (f ) in Figure 3.1 in the frequency domain, according to different optimization criteria. HR (f ) is then converted into its discrete-time counterpart ZHR (z) in the z-domain.
Without loss of generality, we assume the received signal first passes through a receive
filter HR (f ), and then is sampled by a T-spaced sampler, resulting in the discrete-time
received sequence y ′ [k] and the noise sequence n′ [k] whose double-sided PSD is indicated as Φn′ n′ (ej2πf T ).
Define the cascade of the channel and the receive filter
HT (f ) = Ha (f ) · HR (f )
as the overall transfer function. Its z-domain representation is then:
ZHT (z) = H(z) · ZHR (z),
where
H(z) =
L
c −1
X
h[k]z −k ,
(3.1)
k=0
is the discrete-time channel transfer function using z-transform. For convenience, Figure 3.1 illustrates the relation between Ha (f ), HR (f ) and ZHT (z).
The choice of HR (f ) is considered as follows:
35
n(t), σn2
x[k]
kT
Ha (f )
+
HR (f )
y ′ [k]
n′ [k], Φn′ n′ (ej2πf T )
x[k]
ZHT (z)
+
y ′ [k]
Figure 3.1: The overall hannel transfer funtion HT (f ) is the asade of the
hannel transfer funtion Ha (f ) and the transfer funtion of the reeive lter
HR (f ).
• HR (f ) implemented as the inverse filter to the channel. So HR (f ) = Ha (f )−1
and the overall impulse response hT (t) = F −1{HT (f )} is a delta impulse. However, this strategy is neither power efficient nor required, because only the region of the signal spectrum is of interest instead of the whole receiving spectrum.
Moreover, the inverse operation might lead to a strong noise enhancement at frequencies outside the signal spectrum.
• By contrast, the zero-forcing linear equalizer (ZF-LE) forces hT (t) to a Nyquist
impulse response which has constraints only at the sampling instants. To derive
the optimal ZF-LE HR(ZF-LE)(f ), i.e., to maximize the output signal-to-noise ratio,
the variance of the output noise n′ [k] should be minimized. The reason for taking
SNR rather than BER as the measure for optimization is, that the latter can hardly
be analytically handled for a higher-order modulation scheme [40].
Since the PSD of n′ [k] is:
Φ
n′ n′
j2πf T
(e
∞
µ
Φnn X
|HR (f − )|2 ,
)=
T µ=−∞
T
(3.2)
36
where Φnn = N20 is the two-sided PSD of the AWGN n(t), the variance of n′ [k]
turns out to be:
1
Z2T
σn2 ′ = T
Φn′ n′ (ej2πf T ) df
1
− 2T
(3.3)
1
Z2T X
∞
µ
N0
=
|HR (f − )|2 df.
2
T
µ=−∞
1
− 2T
Minimizing the noise variance σn2 ′ subject to the constraint that the overall transfer
function HT (f ) has a Nyquist characteristic, leads to the optimal ZF-LE. Using
the method of Lagrange multipliers, the result is [40]:
HR(ZF-LE)(f ) =
Ha∗ (f )
1
T
∞
P
µ=−∞
,
|Ha (f − Tµ )|2
(3.4)
The optimal ZF-LE in (3.4) is composed of a matched filter Ha∗ (f ), a T-spaced
sampler and a discrete-time filter
∞
1 X
µ
(
|Ha (f − )|2 )−1 .
T µ=−∞
T
Note that the transfer function of the cascade of the channel Ha (f ), the matched
filter Ha∗ (f ) and the T-spaced sampler is:
j2πf T
Sh (e
∞
1 X
µ
)=
|Ha (f − )|2 ,
T µ=−∞
T
(3.5)
so the discrete-time filter is just the inverse of Sh (ej2πf T ). Therefore, ZF-LE is
able to ideally cancel out the ISI. Obviously, the optimal ZF-LE does not always
exist. It exists only when the folded spectrum Sh (ej2πf T ) has no zeros on the unit
circle.
(3.4) can also be given by the z-domain expression:
(ZF-LE)
ZH
(z)
R
H ∗ (1/z ∗ )
1
H ∗ (1/z ∗ )
=
=
.
=
Sh (z)
|H(z)|2
H(z)
(3.6)
where
Sh (z) = H(z) · H ∗ (1/z ∗ ) = |H(z)|2 .
(3.7)
37
Accordingly, the existence of the optimum ZF-LE can be interpreted in an alternative way. That is, if the channel H(z) is decomposed canonically in the manner
of (the delay is assumed to be zero for simplicity) [41]:
(3.8)
H(z) = H0 · Hmin (z) · Hmax (z) · Hzero(z),
where H0 is a constant, Hmin (z) is the monic, causal, and minimum-phase term,
Hmax (z) is the monic, anti-causal and maximum-phase term, and Hzero(z) represents the term containing zeros on the unit circle. Then (3.6) turns out to be:
(ZF-LE)
ZH
(z) =
R
1
1
=
.
H(z)
H0 · Hmin (z) · Hmax (z) · Hzero(z)
(3.9)
Therefore, it is evident that the ZF-LE exists only if Hzero = 1, i.e., the chan−1
nel has no zeros on the unit circle. Hmin
(z) can be implemented as a causal
−1
minimum-phase filter, while Hmax
(z), if exists, is an all-pole infinite impulse
response (IIR) anti-causal filter that can only be approximated in practice.
• In eliminating ISI, ZF-LE might enhance the noise component at frequencies
where the channel transfer function shows a small gain. An alternative is to
use the minimum-mean-squared-error linear equalizer (MMSE-LE), which minimizes the mean power of the error. The error is defined as the difference between
the input and output of the decision device, and is originated from both ISI and
the colored noise. MMSE-LE has a similar structure as ZF-LE [42]:
HR(MMSE-LE)(f ) =
Ha∗ (f )
1
T
∞
P
µ=−∞
or in the z-domain:
ZH (MMSE-LE) (z) =
R
|Ha (f −
µ 2
)|
T
H ∗ (1/z ∗ )
|H(z)|2 +
2
σn
ξx
.
+
2
σn
ξx
,
(3.10)
(3.11)
The denominator of (3.10) is positive and unequal to zero due to the additional
2
positive term σξxn , which means, unlike ZF-LE, MMSE-LE must exists. Comparing (3.10) with (3.4), it can be seen that both ZF-LE and MMSE-LE contain a
discrete-time matched filter segment H ∗ (1/z ∗ ).
Furthermore, comparing MMSE-LE, MF and ZF-LE, it can be seen that a MMSELE tends to a ZF-LE at high SNRs, while conversely it tends to a MF. The MMSELE combines the MF and the ZF-LE, and outperforms them at all SNRs. Moreover, it should also be pointed out that (3.10) is optimal with respect to the MMSE
38
rather than to the SER. The reason is that (3.11) is biased and a decision device for
an unbiased transmission is no longer optimal for a biased transmission. In this
case, a scaling factor should be multiplied with the output of a biased MMSE-LE
to remove the bias. Although by doing so, the SNR will be decreased by one, the
SER becomes smaller [40].
• A linear equalizer has overall a limited performance, which can be enhanced by
a recursive equalizer (RE). A recursive equalizer contains an additional feedback
filter FB (z) as shown in Figure 3.2. Later, if we introduce a decision device in
front of the recursive part in Figure 3.2, we get a decision feedback equalizer
(DFE). But for the moment, a recursive equalizer without the decision device is
considered.
The optimum feedforward filter FF (z) in a RE also preserves the structure of the
previously discussed optimal front-end, that is, FF (z) represents the cascade of a
MF, a T-spaced sampler and a discrete-time filter WD (z). FF (z) can be written
as:
(3.12)
FF (z) = H ∗(1/z ∗ ) · WD (z).
Requirements for FF (z) are as follows: the noise at the output of FF (z) should
be white, because white noise is desired before the decision. However, the noise
after the matched filter H ∗ (1/z ∗ ) is colored, so the discrete filter WD (z) should
act as a noise whitening filter. Meanwhile, FF (z) should force the channel to
a causal, monic and minimum-phase channel, such that a subsequent feedback
filter FB (z) − 1 can completely remove the post-cursors of the forced channel
H(z) · FF (z).
FF (z) can be designed by applying the minimum phase spectral factorization of
Sh (z) = H(z) · H ∗ (1/z ∗ ) in (3.7). Because Sh (ej2πf T ) is Sh (z) evaluated on
unit circle and has a non-negative real-valued folded spectrum, which satisfies
y[k]
FF (z)
v[k]
y ′ [k]
+
FB (z) − 1
Figure 3.2:
An reursive equalizer omposed of a feedforward lter and a
feedbak lter
39
the conditions according to Appendix A, the monic minimum-phase spectral factorization can be applied to Sh (z) [43, 39], resulting in:
Sh (z) = A2h · G(z) · G∗ (1/z ∗ ),
(3.13)
where Ah is a real-valued constant, G(z) is causal, monic and minimum-phase,
+∞
P
i.e., G(z) = 1 +
g[k]z −k , and G∗ (1/z ∗ ) is anti-causal, monic and maximumk=1
phase [40]. Consequently, by setting
WD (z) = (A2h · G∗ (1/z ∗ ))−1 ,
(3.14)
and by considering (3.7), (3.12) and (3.13), the end-to-end transfer function becomes
H(z) · FF (z) = H(z) · H ∗ (1/z ∗ ) · WD (z)
= Sh (z) · WD (z)
=
A2h · G(z) · G∗ (1/z ∗ )
A2h · G∗ (1/z ∗ )
(3.15)
= G(z).
The choice of WD (z) according to (3.14) also ensures that the noise n′ [k] at the
output of FF (z) is an AWGN with a variance σn2 /A2h . FF (z) is therefore called
the whitened matched filter (WMF), as illustrated by Figure 3.3. Because G(z)
is causal, monic and minimum-phase, the feedback loop of FB (z) can be set
according to:
FB (z) − 1 = G(z) − 1,
(3.16)
for the purpose of completely removing the post-cursors of G(z).
The same result has been shown by Price [20] that without a constraint on the
filter length, the optimal feedforward filter in a zero-forcing decision feedback
equalizer contains the noise whitening filter WD (z) = 1/G∗ (1/z ∗ ).
To summarize, the functionality of the feedforward filter in an optimal zeroforcing recursive equalizer (ZF-RE) can be concluded as whitening the additive
noise and forming an equivalent discrete-time channel with the transfer function
40
n(t), σn2
a)
y(t)
WD (z)
kT
+
1
A2h G⋆ (1/z ⋆ )
ha (−t), Ha (−f )
v[k]
FF (z),WMF
n′ [k], σn2 ′ =
b)
x[k]
2
σn
,
A2h
AWGN
+
G(z)
v[k]
a) The whitened mathed-lter (WMF) onsisting of a MF, a Tspaed sampler and a whitening lter at the reeiver b) The equivalent transmission model by using a WMF at the reeiver
Figure 3.3:
G(z) [41]. Its transfer function is
(ZF-RE)
FF
(z) = H ∗ (1/z ∗ ) · WD (z)
=
H ∗ (1/z ∗ )
.
A2h G∗ (1/z ∗ )
(3.17)
(3.17) can be rewritten as:
(ZF-RE)
FF
(z) =
H ∗ (1/z ∗ ) · G(z)
H ∗ (1/z ∗ ) · G(z)
=
A2h G∗ (1/z ∗ ) · G(z)
Sh (z)
Inserting (3.7) into the above equation, and considering (3.16), we finally get:
(ZF-RE)
FF
(z) =
(ZF-RE)
FB
(z)
G(z)
,
H(z)
(3.18)
= G(z).
• The optimal MMSE-RE can be derived similarly to that for a ZF-RE, only now
the spectral factorization should be done for Sh (z) + σn2 /ξx . The procedure is not
repeated here. Note that since the total error is not Gaussian, MMSE approach is
not equivalent to minimizing the BER. But because of its simplicity, the MMSE
approach is widely used in practice.
41
3.1.2 FIR equalizers
For the reason that the discrete-time channel transfer function for the POF-based optical
link is not necessarily minimum phase, i.e., the discrete-time channel transfer function
H(z) may have zeros outside the unit circle of the z plane, an optimum ZF-LE given
by (3.6) or an optimum ZF-RE given by (3.18) can not be implemented for stability
reasons. But, the optimum equalizers can be approximated by the discrete-time feedforward equalizers (FFE) and decision feedback equalizers (DFE). From a practical
point of view, we restrict ourselves to use the finite impulse response (FIR) equalizers (consisting of transversal filters) for the approximation. The optimal equalizers and
the principles developed in Section 3.1.1 are essential for understanding the theoretical
analysis of the later simulation results.
The FIR equalizers can be either symbol-spaced or fractionally-spaced. Comparing
to a symbol-spaced equalizer (SSE), a fractionally-spaced equalizer (FSE) has several
advantages. First, FSE results in a reduced noise power density in the relevant frequency
spectrum. Due to the oversampling, the noise power appearing at the receiver will then
be spread over a larger frequency range, resulting in a reduced power spectrum density
in the smaller signal spectrum. Second, FSE is less sensitive to the sampling jitter. In
order to relax the requirement of an optimal sampling instant, it is customary to use the
FSE. Third, FSE provides more degrees of freedom and has the potential to perform an
ideal equalization with a FIR structure. For example, often a FSE is able to completely
remove the inter-symbol-interference, whereas SSE is not able to do. Although these
benefits are gained based on an increased complexity, it will be shown later that the
required complexity for a FSE to reach our transmission objective is quite moderate and
because of its better performance, FSE is preferred to SSE. Note that in a DFE design,
the fractionally-spaced structure is only for the feedforward filter, the tap spacing in the
feedback filter remains at T .
3.1.2.1 Design of FIR ZF-FFE
To derive a finite-length zero-forcing feedforward equalizer (ZF-FFE), let us first assume that the discrete-time channel impulse response vector
h = {h[0], h[1], · · · , h[Lc − 1]}T
42
is available at the receiver and it is time-invariant, where Lc is the length of the vector.
Without loss of generality, we also assume that the impulse response of the FIR ZFFFE has a length of nf , or equivalently, nf degrees of freedom. Indicate the coefficient
vector of the ZF-FFE by
w = {w[0], w[1], · · · , w[nf − 1]}T ,
and the cascade of the channel h and the ZF-FFE w by b.
Using the representation of matrix multiplication, b can be simply rewritten as:

b[0]





b[1]

 = H · w,
b=
..

.


b[Lc + nf − 1]
(3.19)
where H is a matrix of dimension (Lc +nf −1)×nf containing the channel information:


h[0] 0
0
···
0



h[1] h[0] 0
···
0



h[2] h[1] h[0]
···
0


H= .
.
.
.
.
.
.
.
.
.
.

 .
.
.
.
.




0
· · · h[Lc − 1] h[Lc − 2]
 0
0
0
···
0
h[Lc − 1]
(3.20)
Clearly, the equalizer output b will have Lc + nf − 1 components if a unit impulse is
sent. The objective of the ZF-FFE with coefficient vector w is to force, in the ideal
case, all components of b except for the main tap to zeros. However, the equalizer w
offers only nf degrees of freedom, implying that at most nf − 1 components of b can
be simultaneously forced to zeros. Thus a symbol-spaced ZF-FFE is generally unable
to remove all ISI at the equalizer output. Nevertheless, a fractionally-spaced ZF-FFE is
able to completely remove the ISI, since now only half the components of b should be
forced to zeros. Therefore, a fractionally-spaced ZF-FFE is a better choice when it is
allowed by the complexity. To give a general formula for solving wZF-FFE , a selection
matrix C of dimension nf × (Lc + nf − 1) is applied to pick out those components of
b which should be forced:
C · H · w ZF-FFE = i,
(3.21)
43
where i of length nf is a vector containing the desired values to which those selected
components of b should be forced. From the above equation, w ZF-FFE can be calculated
as:
w ZF-FFE = (C · H)−1 · i.
(3.22)
It should be noted that when (C · H) is not invertible, it will be replaced by the pseudoinverse (C · H)+ to calculate the coefficients of the ZF-FFE.
3.1.2.2 Design of FIR MMSE-FFE
To derive a FIR MMSE-FFE of length nf for a given discrete-time channel h, we indicate the coefficient vector of the MMSE-FFE by:
w = {w[0], w[1], . . . , w[nf − 1]}T .
The output of the equalizer is then:
y ′[k] = wT y[k],
(3.23)
where y[k] = {y[k], y[k − 1], . . . , y[k − nf + 1]}T is a vector containing the received
samples at time instants k, k − 1, . . . , k − nf + 1. So the error e[k] between the equalizer
output y ′[k] and a desired reference sample r[k] is:
e[k] = r[k] − y ′ [k]
= r[k] − w T y[k].
(3.24)
To minimize the mean-squared-error J = E[|e[k]|2 ], the gradient of J with respect to
w, indicated as ▽w J, is examined. The optimal coefficients wo which leads to the
minimum-mean-squared-error is given by setting ▽w J = −2r yr + 2Ry w o = 0, and
the resulting w o is the coefficient vector of the well known Wiener filter [44]:
wo = R−1
y r yr ,
(3.25)
where:
Ry = E[y[k]y T [k]],
and
ryr = E[y[k]r[k]].
(3.26)
44
Clearly, the computational load to get w o would be quite large for a big nf . A direct
calculation of R−1
y in (3.25) for w o in such a case should be avoided. Instead, iterative
algorithms should be considered which are able to progressively approach the optimal
Wiener filter wo by iteratively updating the coefficients during the transmission. In the
following, the algorithms will be discussed.
3.1.3 Digital adaptive equalizers
Realizing a receiver outlined in the last section has in reality the difficulty that the
considered channel is neither perfectly known nor time-invariant. Besides, the inverse
operation in (3.22) and (3.25) has a high computational complexity if the number nf of
the equalizer coefficients is large. Thereupon, this chapter mainly focuses on the use
of digital adaptive equalizers to overcome the aforementioned problem. And the FIR
equalizers in the previous section will be considered in Chapter 4.
Taking into consideration the complexity, the convergence rate and the steady-state error, the following adaptive algorithms are chosen among various adaptive algorithms:
the least mean square (LMS) algorithm based on the steepest descent method, the normalized LMS (NLMS) algorithm, the variable step size LMS (VSLMS) algorithm and
the recursive least square (RLS) algorithm. Note that these adaptive algorithms can
also be adopted by channel estimation in addition to equalization, as will be shown in
Chapter 4 and Chapter 5.
3.1.3.1 Least mean square (LMS) algorithm
Because of the slowly time-variant nature of the considered optical channel, LMS as
well as its variant NLMS are quite appropriate algorithms for tracking the channel.
LMS is one of the most widely used adaptive algorithms due to its low computational
complexity and ease of implementation.
To avoid calculating R−1
y in (3.25) for the Wiener filter w o , the steepest decent algorithm estimates wo iteratively by taking
µ
▽w[k] J[k]
2
= w[k] + µE[e[k]y[k]].
w[k + 1] = w[k] −
where µ is defined as the positive step size. w[k] = {w0 [k], w1 [k], . . . , wnf −1 [k]}T . is
the coefficient vector of the adaptive equalizer with length nf . It should be noted that
45
w is modified to w[k], indicating that the coefficient vector is able to change at every
time step k.
The LMS algorithm however, simplifies the computation by using an immediately available approximation instead of the mean gradient vector of J[k]:
(3.27)
ˆ
▽
w[k] J[k] = −2e[k]y[k].
The iteration with the LMS algorithm is as follows:
µ ˆ
· ▽w[k] J[k]
2
= w[k] + µe[k]y[k],
w[k + 1] = w[k] −
(3.28)
µ is a parameter that influences the convergence speed and the stability of the algorithm.
For example, a small µ results in a low convergence speed and a small steady-state
average squared error. This can be explained as follows: for a small value of µ, the
steady-state mean square error (MSE) is approximately [39]:
J[∞] ≈ Jo · (1 +
µ
· trace(Ry )),
2
µ small,
(3.29)
where trace(·) denotes the trace of a square matrix and Jo is the minimum MSE attained
by the optimal Wiener solution. Therefore, the steady-state MSE for the LMS algorithm
is always larger than Jo . The difference between them defines the quantity of excess
MSE of the LMS algorithm:
Jex [∞] = J[∞] − Jo
µ
= Jo · · trace(Ry ),
2
µ small.
(3.30)
The above equation shows that the steady-state excess MSE for the LMS algorithm will
be small if µ is small. An excess MSE always exists because a noisy gradient vector
instead of the real mean gradient vector is applied for computing the filter weights. In
this sense, the excess MSE actually describes how well the noisy gradient vector can
approximate the mean gradient vector.
On the contrary, increasing the value of µ can accelerate the convergence speed. However, it increases the excess MSE as well. Besides, a large value of µ might lead to
the instability of the algorithm such that the mean squared error does not converge. To
46
guarantee the convergence of the LMS algorithm, µ should be restricted by [39]:
0<µ<
2
λmax
,
(3.31)
where λmax is the largest eigenvalue of the correlation matrix Ry of the input sample
vector y[k]. Apart from µ, the convergence speed is also affected by the eigenvalue
spread, which is defined as the ratio of the maximum eigenvalue over the minimum
eigenvalue of Ry . For instance, a smaller eigenvalue spread results in a faster convergence and a better steady-state MSE. However, for the considered transmission, the
eigenvalue spread does not really influence the performance, since Ry varies insignificantly with the time. Therefore, we concentrate on the adjustment of µ, in order to find
out a satisfactory tradeoff between the convergence speed and the steady-state MSE.
3.1.3.2 Normalized least mean square (NLMS) algorithm
The LMS algorithm is simple to implement, but a main disadvantage is the slow convergence rate to reach a small steady-state MSE. In contrast, the NLMS algorithm offers
a better tradeoff between the convergence rate and the steady-state MSE at the expense
of a slightly increased computational complexity. That means, NLMS is able to achieve
a similar steady-state MSE with a faster convergence speed than the LMS, or, put it in
another way, it is able to achieve a lower steady-state MSE than the LMS for the same
number of iterations.
NLMS is derived by taking into consideration that the parameters of an adaptive system
should be disturbed in a minimal fashion when the input vector varies. By modifying
the step size into:
µ̃
(3.32)
µ=
,
a0 + y[k]T y[k]
where a0 is a positive constant added to overcome the possible numerical problem when
the energy of the input vector y[k] is close to zero, variations of the input power and
the input vector length can be compensated. Using (3.32), the NLMS algorithm can be
seen as the LMS algorithm with a new data-dependent adaptive step size. Finally, the
update equation of the NLMS algorithm is given by:
w[k + 1] = w[k] +
µ̃
e[k]y[k],
a0 + y[k]T y[k]
(3.33)
where µ̃ should satisfy 0 ≤ µ̃ ≤ 2 to ensure the convergence of the NLMS algorithm in
the mean squared error sense [39].
47
3.1.3.3 Reursive least square (RLS) algorithm
The recursive least square (RLS) algorithm is considered because it has a faster convergence rate and it produces a smaller steady-state MSE, comparing to the LMS and the
NLMS algorithms. The RLS algorithm collects information carried by all input samples from the moment when the algorithm is initialized, the resulting convergence rate
is therefore remarkably faster than that of LMS and NLMS. However, the price to pay
is an increased computational complexity. In order to weaken the influence of the past
errors on the current overall error, RLS algorithm multiplies the old data by a forgetting factor λ. If λ = 1, all previous errors are considered equally for the total error. If
0 ≤ λ < 1, applying the forgetting factor is equivalent to weighting the older errors. If
λ is very small, the past errors play nearly no role in the total error. The RLS algorithm
uses the following equations to update the filter coefficients:
w[k] = w[k − 1] + κ[k]α[k]
κ[k] =
P[k − 1]y[k]
λ + y T [k]P[k − 1]y[k]
(3.34)
T
α[k] = r[k] − w[k − 1] y[k]
P[k] = λ−1 (P[k − 1] − κ[k]y T [k]P[k − 1]),
where λ is the forgetting factor which reduces the influence of the old data, and should
fulfill 0 ≤ λ ≤ 1. To initialize the algorithm, w[0] and P[0] are needed (for example,
w[0] = 0 and P[0] ≥ 100 · σy2 · I nf ×nf ). However, their values are not really important
for a long input vector and for a λ smaller than one, since they will be anyway forgotten
due to the forgetting factor.
3.1.3.4 Comparison of the adaptive algorithms
The decision of the selection of an adaptive algorithm is influenced by several aspects.
One aspect is the convergence rate of the adaptive algorithm towards the steady-state error J[∞]. Basically, the coefficients should converge as fast as possible to the expected
values. In this context, the LMS algorithm offers a relatively slow rate of convergence.
Although its convergence rate can be generally improved by the NLMS algorithm, the
RLS algorithm could converge even faster [45].
Another aspect influencing the decision is the excess MSE Jex [∞] in the steady-state.
The LMS and the NLMS algorithm always produce a non-zero excess MSE which arises
48
from the gradient noise in the algorithms; whereas theoretically RLS algorithm is able
to attain the Wiener solution, which means zero excess MSE in the steady-state.
The third aspect to consider is the computational complexity of the algorithm. Despite
RLS shows superiority over LMS with a faster convergence rate and a smaller steadystate excess MSE, it owns an obvious disadvantage, that is, its complexity increases
approximately with the square of the filter length, whereas the complexity of LMS
increases only linearly [46].
In one word, each algorithm has its own pros and cons. Different algorithms can be
combined properly to find a good compromise between the performance and computational complexity.
3.2
Design of Transmission Systems with Pre- and/or
Post-Equalizers
The core of this chapter is dedicated to developing transmission systems based on equalization techniques for transferring data at 2 ∼ 3 Gbit/s over the MOST150 optical
channel. The design is done with respect to two set-ups: one set-up performs only post
equalization at the receiver, and the other one performs pre-equalization at the transmitter and post- equalization at the receiver. For the latter case, a prefilter or a pre-equalizer
is placed between the mapper output and the channel input. Figure 3.4 shows the complete transmission model, where RS (255,223) coding and 8PAM is used as the error
correction scheme and the modulation scheme, respectively. H(z) is the MOST150
optical channel. The additive white Gaussian noise n[k] is added to the received signal
representing the distortion from an optical receiver.
Possible post-equalizer types are linear FFE and non-linear DFE, which are either
symbol-spaced (SS) or fractionally-spaced (FS). Since the channel is unknown by the
receiver, a direct calculation of the equalizer coefficients as stated in Section 3.1.2 can
not be performed. Instead, the adaptive equalizers, which automatically adjust the
equalizer coefficients and adaptively compensate for variations in the channel characteristics, are utilized. This is feasible because cascading an adaptive equalizer with an
unknown channel causes the adaptive filter to converge to the inverse of the unknown
channel. For instance, if the discrete-time channel transfer function of the unknown
channel is H(z) and the adaptive filter transfer function is W (z), then the error measured between the desired signal and the output signal from the equalizer achieves its
49
source
b′ [m]
RS coding
final BER
b[n]
mapper
BER before RS decoding
a[k]
pre-equalizer
SER
x[k]
H(z)
channel
+
n[k], σn2
y[k]
sink
b̂′ [m]
RS decoding
Figure 3.4:
b̂[n]
demapper
â[k]
post-equalizer
Transmission system blok diagram
minimum as long as W (z) · H(z) = 1 is satisfied, which leads to W (z) = H(z)−1 in
the noise free case.
In this section, the considered adaptive algorithms include the LMS and the RLS algorithms. Parameters of these algorithms, such as the rate of convergence and the computational complexity, are analyzed on the basis that the adaptive equalizer achieves the
desired BER value. The essential concepts for understanding the design in this chapter were introduced in Section 3.1.3. The NLMS and the VSLMS algorithms will be
discussed in later chapters.
3.2.1 Design of adaptive equalizers
3.2.1.1 Struture of the adaptive equalizers
The schematic structure of an adaptive FFE is given in Figure 3.5. The equalizer taps
are spaced at T /K, where T is the symbol interval and K ∈ {1, 2}. The FFE has nf
forward taps, among which one tap is set as the reference tap which defines the delay
introduced by the FFE. When K is 2, the FFE is T /2 fractionally-spaced, otherwise
symbol-spaced. The equalizer operates under two modes: the training mode and the
adaptive mode. In the training mode, the initial set of tap coefficients is determined
via transferring a training sequence which is also known by the receiver; while in the
adaptive mode, the weight setting block adaptively updates the tap weights only on the
basis of the decided symbols.
50
Input
Rate K/T
y[k]
T/K
Weight
Setting
w0
T/K
w1
T/K
T/K
w2
wnf −1
′
y [k]
+
Output
Rate 1/T
Decision
Device
â[k]
e[k]
Figure 3.5:
Training
Sequence
Error
Calculation
Adaptive linear feedforward equalizer (FFE)
The considered adaptive DFE has a structure as illustrated in Figure 3.6. It has nf forward and nb feedback tap coefficients. If K is 1, it is a symbol-spaced DFE, otherwise
fractionally-spaced. Similar as the adaptive FFE, DFE also starts with a training period. During the training period, the error signal consisting of the difference between
the delayed training signal and the equalizer output signal y ′ [k] is minimized by adjusting iteratively the coefficients of the forward and feedback filters. After this period, no
training signal is sent. The equalizer is switched to an adaptive mode where y ′[k] and
the output signal â[k] of the decision device is used to calculate the error signal. It is
therefore essential that the coefficients converge during the training process, such that
the decision-directed adaptive mode is able to work properly.
Input
Rate K/T
y[k]
T/K
w0
T/K
w1
T/K
T/K
w2
Weight
Setting
wnf −1
y ′ [k]
+
wnf +nb −1
wnf +1
Output
Rate 1/T
Decision
Device
w nf
â[k]
T
T
e[k]
Figure 3.6:
T
T
Training
Sequence
Error
Calculation
Adaptive non-linear deision feedbak equalizer (DFE)
51
3.2.1.2 Congurations of the adaptive equalizers
The next step is to configure the adaptive equalizer in terms of: the adaptive algorithm,
the reference tap of the equalizer, the filter length and the adjustment of parameters of
the adaptive algorithms. These points are elaborated in the following:
• The adaptive algorithm for the equalizer:
First of all, it is important that the equalizer coefficients converge properly in the
training period. As outlined in Section 3.1.3, the RLS algorithm converges quite
fast however its complexity grows quadratically with the length of the equalizer.
In contrast, the LMS algorithm converges slower. However, its complexity grows
only linearly with the length of the equalizer. To compare their performances
during the training block, computer simulations are run. Figure 3.7 shows the
corresponding result for the considered MOST150 optical channel, where the
blue and red curve stands for the absolute squared error versus the number of
sent symbols for the LMS and the RLS algorithm, respectively. Comparing both
curves, the RLS algorithm exhibits much lower squared errors as well as a faster
convergence rate in comparison to the LMS algorithm. Based on these aspects, it
is reasonable to apply the RLS algorithm for the training block. And later on, to
avoid the high computational complexity of RLS, the adaptive algorithm can be
switched to the LMS algorithm so as to accelerate the equalization process.
• The position of the reference tap:
The location of the reference tap for an equalizer should exceed at least the channel delay to ensure a proper causal equalization. However, to find the optimal
position is a complex optimization problem, which cannot be solved generally. A
rule of thumb for choosing the position of the reference tap is proposed and analyzed in [47], where it is conjectured that the best delay should be close to half
the time span of the equalizer. However, it is suggested by [42], that one should
try different delays in practice. As a result, simulations are run to find out the
optimal reference tap in relation to the BER performance.
One example is shown on Figure 3.8, where different BERs resulting from various locations of the reference tap are compared at a specific SNR of 41 dB. The
considered length for the feedforward filter in the fractionally-spaced DFE is 25,
29 and 33, respectively. The location of the reference tap is changed from less
than the center tap to the end of the filter. The simulation result reveals that the
reference tap should be placed at least after the central position of the feedforward filter. Consequently, the reference taps in a FFE and in a DFE feedforward
52
3.5
LMS (0.00001)
RLS (0.99)
3
Squared Error
2.5
2
1.5
1
0.5
0
0
200
400
600
800
1000
1200
Number of Samples
1400
1600
1800
Squared error performane using the RLS (forgetting fator: 0.99)
or the LMS algorithm (step size: 0.00001), Lc = 12, nf = 25 and nb = 12
Figure 3.7:
0.02
FS−DFE at 3 Gbit/s, SNR =41 dB, nf = 25
FS−DFE at 3 Gbit/s, SNR =41 dB, n = 29
0.018
f
FS−DFE at 3 Gbit/s, SNR =41 dB, n = 33
f
0.016
Bit Error Rate
0.014
0.012
0.01
0.008
0.006
0.004
0.002
0
10
15
20
25
Position of the Reference Tap
30
35
BER performanes of a FS-DFE versus the position of the referene
tap for various feedforward lter lengths nf , Lc = 12, nb = (nf − 1)/2
Figure 3.8:
53
filter are assigned to the corresponding middle tap or to the last tap for all further
simulations.
• The filter length:
This part aims at assigning an appropriate feedforward filter length nf for the
FFE and the DFE, and assigning the feedback filter length nb for the DFE. “Appropriate” means that nf should be neither too large on account of an increased
complexity nor too small because it might lead to a bad performance. At the same
time, nb should not be a large number, since error propagation in a DFE is heavily
influenced by the length of its feedback filter, i.e., once a decision error is made,
the error stays as long as nb symbol intervals. Ideally, nb should be chosen equal
to the number of post-cursors of the forced discrete-time channel.
Here, nf and nb are first roughly estimated according to the channel length. After
that, they are adjusted by simulation trails for a specific data-rate. For example,
the simulated BERs versus the length of the feedforward filter (FFF) in a FS-DFE
is sketched in Figure 3.9, where nb is equal to one half of nf and SNR = 41 dB
is present at the input of the DFE. Each star “*” in Figure 3.9 represents a BER
value for the FS-DFE for a given nf .
Figure 3.9 reveals that nf = 25 and nb = 12 is sufficient for the 3 Gbit/s transmission with a channel length of 12 (T/2-sampled). Figure 3.10 shows the corresponding details for this set-up, where the forced channel after FFF has a length
of 36, and the feedback filter (FBF) taps are the negations of the post-cursors of
the forced channel. It can also be observed that actually nb = 11 is sufficient for
the FBF, which equals the number of the significant post-cursors of the forced
channel. Figure 3.10 is also a good illustration of the fact that DFE applies two
stages, the first stage FFF provides an end-to-end channel response with postcursors only, which are then canceled by the subsequent second stage FBF.
• Adjusting parameters of the adaptive algorithms:
Usually, there is no general rule for choosing the most appropriate parameters in
an adaptive algorithm, because this is always related to the specific problem. In
practice, computational simulations play an important role and are, in fact, the
most commonly used tool to address the problem [42]. Here, the decisions on the
forgetting factor λ for the RLS algorithm and on the step size µ for the LMS algorithm are also accomplished with the help of simulation trails. Nevertheless, there
is a condition for the step size µ to fulfill: it should guarantee the convergence of
the algorithm by fulfilling (3.31). By observing the mean power and the length
54
0.04
FS−DFE at 3 Gbit/s, SNR =41dB, nb=(nf−1)/2
0.035
Bit Error Rate
0.03
0.025
0.02
0.015
0.01
0.005
0
15
Figure 3.9:
20
25
30
Number of Feedforward Taps (nf)
35
40
The length of the feedforward taps in a FS-DFE inuenes the
BER (Lc = 12)
2
1.5
1
Magnitude
0.5
0
−0.5
−1
Channel
Forced channel after FFF,
n = 25
f
−1.5
Coefficients of FBF,
n = 12
b
−2
0
5
10
15
20
k
25
30
35
40
Coeients of the hannel h[k] (Lc = 12), of the fored disretetime hannel impulse response after the feedforward lter (FFF), and of the
feedbak lter (FBF), respetively.
Figure 3.10:
55
of the received sample sequences, µ should not exceed the maximum value of
0.0001. As long as the condition is fulfilled, the step size µ can be chosen freely.
3
LMS (1e−6)
LMS (1e−5)
2.5
Squared Error
2
1.5
1
0.5
0
0
Figure 3.11:
500
1000
1500
2000
Number of Samples
2500
3000
3500
Squared error performanes with regard to dierent values of the
step size µ, Lc = 12, nf = 25 and nb = 12
As an example, Figure 3.11 shows the squared error curves for µ = 0.00001 and µ =
0.000001 in a training block of length 3500. Since both curves end up with similar
squared errors, both values for µ are feasible. The main difference between the two
curves is that a smaller value of µ brings forth a slower convergence rate. The same
approach used for determining µ is also carried out for determining the forgetting factor
λ, and the simulation results revealed that λ = 0.99 and λ = 0.999 produce similar
performances. Finally, µ is fixed to 0.00001 and λ to 0.99 for all further simulations in
this chapter.
3.2.2 Design of a pre-equalizer
Apart from the post-equalization method, an alternative transmission strategy which
combines a prefilter at the transmitter and a simplified post-equalizer at the receiver is
considered in this chapter. The aim of the prefilter or the so called pre-equalizer is to
broaden the overall channel bandwidth seen by the receiver, resulting in a simplified
post-equalization process. However, a prefilter would at the same time lower the light
56
source modulation depth if the transmit power is fixed, corresponding to reducing the
actual transmit power per symbol. Following this line of thought, it is a natural outcome that the prefilter should be designed with a compromise between the bandwidth
boosting and the loss in the actual transmit power. Besides, a pure pre-equalization is
unfortunately not executable due to the peak power constraint on the transmit signal.
As a result, a fifth-order T/2 fractionally-spaced prefilter has been found being a good
compromise. Its magnitude response is indicated by the green curve on Figure 3.12,
where the blue curve is the channel magnitude response with a 3 dB bandwidth of
135 MHz, and the red curve stands for the magnitude response of the boosted channel
after prefiltering. It can be concluded that the new channel bandwidth seen by the receiver is increased to 214 MHz, and this outcome naturally changes the SNR estimate in
Section 2.1.3.3, because there a bandwidth of 135 MHz is considered for the numerical
calculation. Repeating the steps in Section 2.1.3.3, a new SNR estimate based on the
boosted channel bandwidth is derived, and the new SNR range should be adjusted to
18.4 ∼ 57.6 dB for the joint pre- and post-equalization strategy.
21
Channel
Prefilter
The cascade of
channel and prefilter
18
Normalized Magnitude Response [dB]
15
12
9
6
3
0
new 3dB bandwidth
−3
orig. 3dB bandwidth
−6
−9
−12
−15
−18
−21
0
30
60
90
120
150 180 210
Frequency [MHz]
240
270
300
330
360
The magnitude response of the original hannel (10 m POF), the
prelter and the asade of the hannel and the prelter
Figure 3.12:
3.3
57
Simulation Results
In this section, simulation results will be given for the two transmission strategies illustrated by Figure 3.4. For the joint pre- and post-equalization strategy, a fixed fifth-order
T/2 fractionally-spaced prefilter is placed at the transmitter to boost the relevant channel
bandwidth from 135 MHz to 214 MHz. The additive white noise n[k] is generated at
the receiver, whose power is predefined by a given SNR value. The SNR ranges between 20.4 dB and 59.5 dB for the transmission without a prefilter (B = 135 MHz),
and between 18.4 dB and 57.6 dB for the transmission with a prefilter (B = 214 MHz).
The figure of merit for the system performance is the BER before RS decoding, that is,
the BER of b̂[n] on Figure 3.4. The target BER is set to 10−3 , so the final BER can be
reduced to below 10−9 with the help of RS (255,223) coding.
The RLS and LMS algorithms are implemented to update the tap coefficients of the
adaptive equalizers once per symbol. To initialize the equalizer, the RLS algorithm
which ensures a rapid convergence rate is applied in the training phase. Then the adaptive algorithm is switched to LMS in order to accelerate the simulation speed.
3.3.1 Performanes with post-equalization
Four types of equalizers are investigated in this section: the symbol-spaced FFE (SSFFE), the symbol-spaced DFE (SS-DFE), the fractionally-spaced FFE (FS-FFE) and
the fractionally-spaced DFE (FS-DFE). In addition, the matched filter bound (MFB)
has been simulated as a benchmark for comparison.
Figure 3.13(a) and Figure 3.13(b) shows, the BER before RS decoding for a 2 Gbit/s
and a 3 Gbit/s transmission with merely post-equalizers, respectively.
At 2 Gbit/s, the sampled channel has a length of Lc = 4 (T-sampled) or Lc = 8 (T/2sampled). Correspondingly, the lengths of the feedforward filters are set to nf = 17
for both FS-equalizers and to nf = 9 for both SS-equalizers; while the length of the
feedback filter is set to nb = 7 for FS-DFE and to nb = 3 for SS-DFE, respectively. In
Figure 3.13(a), the target BER is reached by the FS-DFE and the FS-FFE at an SNR
of 31.5 dB and 33 dB, respectively. The SS-DFE also successfully achieves the target
BER, but it requires about 2 dB more SNR than its fractionally-spaced counterpart. On
the contrary, the SS-FFE fails to reach the target BER up to SNR= 34 dB, which is a
natural outcome of the insufficient number of filter taps. Indeed, when the filter length
is doubled (as shown by the green curve), SS-FFE performs similarly to the FS-FFE.
58
0
10
−1
10
−2
Bit Error Rate
10
−3
10
−4
10
MFB
FS−FFE at 2 Gbit/s
FS−DFE at 2 Gbit/s
SS−FFE at 2 Gbit/s
SS−DFE at 2 Gbit/s
SS−FFE at 2 Gbit/s with double length
−5
10
−6
10
27
28
29
30
31
SNR [dB]
32
33
34
(a) BER performanes of the adaptive SS-FFE and SS-DFE (with Lc =
4, nf = 9 and nb = 3), and of the adaptive FS-FFE and FS-DFE (with
Lc = 8, nf = 17 and nb = 7) at 2 Gbit/s
0
10
−1
10
−2
Bit Error Rate
10
−3
10
−4
10
−5
10
FS−FFE at 3 Gbit/s
FS−DFE at 3 Gbit/s
SS−FFE at 3 Gbit/s
SS−DFE at 3 Gbit/s
−6
10
36
37
38
39
40
41
42
SNR [dB]
43
44
45
46
(b) BER performanes of the adaptive SS-FFE and SS-DFE (with Lc =
6, nf = 13 and nb = 5), and of the adaptive FS-FFE and FS-DFE (with
Lc = 12, nf = 25 and nb = 11) at 3 Gbit/s
Figure 3.13:
BER performanes as a funtion of SNR for the transmission
strategy using merely post-equalization
59
Comparing the filter lengths of FS-DFE and SS-DFE, it is straightforward to say that
the 2 dB gain of FS-DFE is obtained through the use of a longer filter length as well
as the fractionally-spaced structure. Finally, based on the preference of low complexity
designs and the fact that SS-DFE and FS-FFE are both able to reach the target BER
with SNRs much lower than the SNR upper-limit, it can be concluded that SS-DFE and
FS-FFE are feasible solutions for the 2 Gbit/s transmission.
In Figure 3.13(b), a significant difference between the FFE and the DFE can be observed
at 3 Gbit/s. Unfortunately, the BERs of both SS- and FS-FFE can no longer be forced
to reach the RS threshold with SNRs under the SNR upper-limit of 59.5 dB. On the
contrary, FS-DFE is able to reach the target BER at a SNR of 41 dB, and 41 dB lies
in the middle of the SNR budget (from 20.4 dB to 59.5 dB). These results describe the
fact that the DFE yields an apparent lower BER than the FFE although they are both
assigned with the same number of feedforward taps.
The simulated FS-DFE has 25 feedforward taps and 11 feedback taps, which were chosen based on a channel length of 12 (T/2-sampled), details about assigning the filter
length see Section 3.2.1.2. The cause for an increased BER compared with the 2 Gbit/s
transmission lies in the stronger ISI and the more severe error propagation effect at
3 Gbit/s. The error propagation is now more problematic because both the tap weights
and the number of feedback taps in DFEs have increased. In the end, it can be concluded that, in order to increase the data-rate from 2 Gbit/s to 3 Gbit/s and achieve BER
= 10−3 , the filter length should be extended by 50 %, and more than 10 dB of SNR is
required.
Besides, a comparison between the DFE and the MFB indicates a fairly large performance loss in the DFE due to the residual ISI which could not be removed and due to
the feedback of incorrect decisions.
3.3.2 Performanes with joint pre- and post-equalization
In the next scenario, a fifth-order T/2 fractionally-spaced prefilter is inserted between
the unipolar MPAM output and the input of the LED. The prefilter broadens the channel
bandwidth from 135 MHz to 214 MHz, as shown in Figure 3.12. At the receiver, a
FS-FFE or a FS-DFE is used.
To boost the channel bandwidth, a prefilter must contain some negative coefficients
which will turn the original all positive transmit symbols partly into negative ones. An
example is illustrated in Figure 3.14(a), where the blue curve stands for the original
60
25
Singal before prefilter
Signal after prefilter
Transmit signal (with DC)
20
Magnitude
15
10
5
0
−5
−10
0
0.05
0.1
0.15
0.2
Time [us]
0.25
0.3
0.35
0.4
(a) Signals before and after the prelter, and the transmit signal at the
hannel input
16
Received signal, DC−contained
Received signal, DC−removed
14
12
Magnitude
10
8
6
4
2
0
−2
100
200
300
400
500
600
Number of Samples
700
800
900
1000
(b) Sampled reeived signals before and after removing the DC
Figure 3.14:
Example: signals at the transmitter and the reeiver when a
prelter is utilized
61
0
10
−1
10
−2
Bit Error Rate
10
SNR upper limit
−3
10
−4
10
−5
10
MFB for unipolar 8PAM
FS−FFE
FS−DFE
−6
10
31
32
33
34
35
SNR [dB]
36
37
38
BER performanes as a funtion of SNR for the transmission
strategy using joint pre- and post-equalization at 3 Gbit/s (Lc = 12, nf = 25
and nb = 11)
Figure 3.15:
unipolar MPAM symbols ranging from [0, 14], and the green curve stands for the output
of the fifth-order T/2 fractionally-spaced prefilter. As can be seen, the magnitude of
the prefilter output is noticeably enlarged and extends into the negative domain. Consequently, a DC must be added to the green curve in order to produce an all positive input
to the LED, which is represented by the red curve.
The corresponding received signal is shown in Figure 3.14(b), where the blue and the
red curve stands for the received signal containing a DC and the signal after removing
the DC, respectively. Comparing both curves, it can be seen that the dynamic range
of the received signal is greatly enlarged due to the DC component introduced at the
transmitter. Clearly, the more the channel bandwidth is boosted by the prefilter, the
more the dynamic range of the received signal will be increased. This phenomenon
forces a modification to the previously deduced power constraint. By simply subtracting
the DC power from the previous SNR upper limit, we get the upper limit for the actual
SNR. Because the DC power is (20 + 20 log(d/2)) dB, where 20 dB is determined
by the prefilter coefficients and d = 2 is the minimum distance between two MPAM
constellation points, the SNR upper limit is reduced from 57.6 dB to 37.6 dB.
The simulated BER curves for a transmission at 3 Gbit/s are shown in Figure 3.15,
where a prefilter and a post-equalizer are used jointly. In the observation window, the
62
BER curve for the prefilter and the FS-FFE combined transmission fails to reach the
target BER, while the curve for the prefilter and the FS-DFE combined transmission
stays beneath the RS correction threshold as long as the SNR is higher than 35.5 dB.
Although this SNR value seems to be very optimistic, we should keep in mind that the
SNR upper limit is now decreased to 37.6 dB, as marked by the green dashed line in
Figure 3.15. Consequently, there are only 2 dB left to the SNR upper limit. Because it
is common to reserve a 3 dB SNR margin for the sake of reliability, in this sense, the
transmission strategy with combined pre- and post-equalization does not yield a fully
satisfactory result. Additional approaches in this case should be applied to increase
the actual SNR upper limit at the receiver, such as decreasing electronic noises at the
receiver or specifying an increased upper power limit for the received optical signal.
In one word, the current overall situation becomes even more critical compared to the
transmission strategy based on the merely post-equalization in Section 3.3.1.
One might now wonder how the situation could become worse when the channel bandwidth is increased. This is actually not surprising. Since if a comparison is done based
on equal transmit power, a linear pre-equalizer will be exactly like a linear equalizer
at the receiver [40]. For example, the prefilter here has increased the transmit power
by about 5 dB. To exclude the effect of power amplification, the curves in Figure 3.15
should by shifted by 5 dB along the SNR axis to the left, which then turns out to be
very close to the FS-FFE and FS-DFE curves in Figure 3.13(b). Since a prefilter can
not truly improve the performance, the power constraint imposed on the received signal
finally worsens the overall situation.
3.3.3 Performanes with regard to the MPAM modulation
order
As is commonly known, a larger modulation order M for the MPAM modulation is
more sensitive to the SNR condition at the receiver. On the other hand, as log2 (M)
bits are carried by each symbol, the required transmission bandwidth is decreased that
perhaps lead to a better performance. With the purpose to find out the most beneficial
modulation order, BER performances as a function of both SNR and M are simulated.
The simulation result is demonstrated in Figure 3.16, where the 4, 8 and 16PAM are
compared for a 3 Gbit/s transmission. It can be observed that 4PAM generates the worst
performance among the three. Increasing M to 8 provides a quite good performance
because it reduces the required transmission bandwidth by 2, so the high-frequency
region in the considered low-pass channel where a high attenuation happens can be
63
0
10
−1
10
−2
Bit Error Rate
10
−3
10
−4
10
FS−FFE, 4PAM
FS−DFE, 4PAM
FS−FFE, 8PAM
FS−DFE, 8PAM
FS−FFE, 16PAM
FS−DFE, 16PAM
−5
10
−6
10
36
37
38
39
40
SNR [dB]
41
42
43
Comparison of BER performanes with regard to the modulation
order at 3 Gbit/s (Lc = 12, nf = 2Lc + 1 and nb = Lc )
Figure 3.16:
avoided. Increasing M to 16 produces unfortunately no further improvement compared
to M = 8. At this point, the gain resulted from the reduced transmission bandwidth
can no longer balance out the increased SNR requirement. Also, when M is beyond
16, increasing the number of constellation points does not bring in any improvement
but negative influences. As a result, 8PAM is seen as the most preferable modulation
scheme for its low complexity and reasonable performance.
3.4
Summary
This chapter introduced the basic principles for designing an equalizer. To be specific, the optimal linear equalizers and non-linear decision feedback equalizers were
first derived analytically. It was then explained that equalizing a minimum-phase channel is relatively straightforward. However, since the considered channel also contains a
maximum-phase term, implementing a perfect equalization is impossible. So the optimal equalizers can be approximated by the FIR equalizers developed under the ZF and
the MMSE criterion, respectively.
The adaptive equalizers were subsequently described by taking into consideration that
an adaptive system is necessary to compensate for the transmission variations. It was
64
shown that the LMS and NLMS algorithms are less complex but a longer time is required for convergence in comparison to the RLS algorithm which is more complex but
converges rapidly.
In the simulation part, performances of two transmission strategies were explored, one
of which uses a post-equalizer and the other strategy combines a pre- and a postequalizer. The BER performances of various adaptive equalizers were then evaluated.
The result illustrates the superiority of a DFE over a FFE when the error propagation
effect is not significant. The results also shows that the initial objective of increasing
the bit-rate to 3 Gbit/s could be met with an adaptive post-DFE.
In the last step, in order to optimize the PAM transmission, the BER was investigated
as a function of the number of levels M of MPAM (M = 4,8,16). 8PAM turns out to be
the most appropriate modulation scheme for the considered optical transmission, both
in terms of performance and complexity.
CHAPTER 4
Tomlinson-Harashima Preoded Systems
4.1
Motivation
In the last chapter, two transmission schemes based on conventional equalizers were
discussed. Although the transmission scheme using DFE has presented a superior performance, the capability of DFE is limited by the error propagation effect, which gets
more severe with an larger number of the feedback taps and/or larger tap weights.
Consider now a more critical situation, where stronger ISI appears due to a higher datarate or a decreased transmission bandwidth. The tap number in a DFE must increase accordingly in order to maintain the same performance. Consequently, DFE will become
more vulnerable to the error propagation in the presence of a stronger ISI. Starting from
some state onward, it could even fail to provide satisfactory results and an error floor
may occur. To prevent the error floor, the error propagation problem in a DFE must be
coped with. This chapter therefore considers a non-linear equalization strategy at the
transmitter - the Tomlinson-Harashima Precoding (THP) - which can be combined with
a linear post-equalizer to handle the aforementioned problem. It has been shown in [48]
and [49] that THP can be applied for fiber-optic communication systems. The use of the
non-linear THP precoding enables the transmitter to share the burden of mitigating ISI
with the receiver. An expectation is that such an arrangement will be more beneficial to
compensate for stronger ISI, and to improve the performance compared to the methods
applied in the last chapter.
65
66
CHAPTER 4 Tomlinson-Harashima Preoded Systems
4.2
Priniples of the Tomlinson-Harashima Preoding for Channels without Pre-Cursors
THP can be regarded as a special form of DFE. The idea of THP is similar to moving the
feedback part of a DFE that cancels out the post-cursors ISI to the transmitter. Because
the past transmitted symbols are known at the transmitter, ISI related with these symbols
can be controlled there.
The prerequisite for implementing THP is that the discrete-time channel impulse response seen by the THP encoder has to be causal, monic and minimum-phase [40],
because THP is only able to remove ISI from the post-cursors. This is equivalent to
saying that the discrete-time channel transfer function seen by the THP encoder must
have a form of
L
X
b[k]z −k
B(z) = 1 +
(4.1)
k=1
(without taking into account the delay), where the post-cursors are located at k =
1, . . . , L. If the channel itself does not fulfill this condition, a feedforward equalizer
(FFE) should be placed at the receiver to generate an overall discrete-time channel impulse response b[k] given by (4.1).
With the purpose to explain the principles of THP, we first assume that the channel
fulfills the aforementioned prerequisite and a bipolar MPAM is used according to Section 2.2.1 with d = 2. A THP precoder consists of a modulo 2M (mod 2M) device and a
feedback loop, as shown in Figure 4.1, where the feedback loop filter is B(z) − 1. Thus,
by replacing the modulo 2M device by a short-circuit, THP is the same as a recursive
pre-equalizer with transfer function 1/B(z), which provides ideal zero forcing equalization to the channel B(z). However, by using the non-linear modulo 2M device, THP
THP
a[k]
+
-
channel
x[k]
mod 2M
B(z)
B(z) − 1
Figure 4.1:
The Tomlinson-Harashima preoder in onjuntion with the hannel
67
Figure 4.2:
The input-output relation of the modulo 2M devie
is able to restrict the amplitude and power of x[k] and to control the performance loss
arising from the transmit power amplification of a linear pre-equalizer. The modulo 2M
device is the key element which enables THP to outperform a linear pre-equalizer. Its
input-output relation has a sawtooth characteristic as depicted in Figure 4.2, provided
that a[k] is a bipolar MPAM modulated signal.
The algorithm of THP can be better explained by its linearized model as shown in
Figure 4.3: for each transmit symbol a[k], a unique integer value d[k] = 2M · Nm ,
where M is the order of MPAM and Nm is an arbitrary integer number, will be added
to or subtracted from a[k], such that x[k] lies in a pre-defined fixed interval, for instance
[−M, M). Obviously, d[k] changes with a[k] at each time instance k. By literally
adjusting Nm , the mod 2M operation guarantees that the amplitude of x[k] and the
transmit power are bounded. Although strictly speaking, THP indeed introduces some
certain amplification to the transmit power, which is however, often small and may be
neglected. Details regarding this point will be discussed in Section 4.5. Finally, from
Figure 4.3 we see that the received signal is v[k] = a[k] + d[k], which is also present at
the transmitter and is just deviating from a[k] by d[k]. Therefore, v[k] is also called the
effective data sequence (EDS).
d[k]
channel
a[k]
+
v[k]
x[k]
+
-
B(z)
B(z) − 1
Figure 4.3:
The linearized THP model
v[k]
68
4.3
Tomlison-Harashima Preoding for Channels
with Pre- and Post-Cursors
The discrete-time channel impulse response for the considered transmission over a plastic optical fiber exhibits pre- and post-cursors with respect to the main value h[L1 ] as
defined in Section 2.2.2. As a natural outcome, the prerequisite of implementing THP
is not fulfilled, so THP must work together with a linear feedforward equalizer which
is placed at the receiver to force the end-to-end discrete-time channel impulse response
seen by the THP to be monic, causal and minimum-phase. Follow this line of thought,
the complete system block diagram at the symbol level is depicted in Figure 4.4, where
H(z) stands for the discrete-time transfer function of the channel given by (3.1), W (z)
is the linear feedforward post-equalizer (FFE), and B(z) = H(z) · W (z) represents
the end-to-end overall discrete-time channel transfer function seen by the THP precoder. W (z) has the function similar to the feedforward filter in a DFE, which forces
B(z) = H(z) · W (z) to have a discrete-time impulse response b[k] that is monic and
contains only post-cursors.
At the transmitter, the information bits are first encoded by a Reed-Solomon (RS) encoder and then mapped into unipolar 8PAM symbols a[k], which are then encoded by
THP to generate the transmit signal x[k]. The amplitude of x[k] is restricted by the THP
to guarantee that there is only minimum amplification of the transmit power. Here, a
modification has been made to the classical THP and bipolar MPAM combined transmission scheme. That is, since we utilize the unipolar MPAM, the output range of the
mod 2M device is shifted from [−M, M) to [0, 2M) accordingly. The reason is that
LED can only take in a positive input current and x[k] should be kept positive. Note
Figure 4.4:
Transmission blok diagram with a THP enoder and an adaptive
reeiver
69
that this modification does not change the principles of THP, but it changes some THP
properties. This point will be expounded in Section 4.5.
By applying the linearized THP model in Figure 4.3 and substituting the original channel H(z) with the new overall channel B(z), the transmission block diagram in Figure 4.4 has an equivalent linearized form as illustrated in Figure 4.5. Note that now the
noise n′ [k] is no longer white but colored by W (z). Assume that the pre-equalization
is done perfectly, then the effective data sequence v[k] = a[k] + d[k] in addition to the
colored noise n′ [k] will be received. Also assume a small or moderate impact from the
colored noise, then the mod 2M device at the receiver can simply repeat the mod 2M
operation to remove the additive sequence d[k], and recover u[k] = a[k] + n′ [k] for low
noise. Ideally, the colored noise n′ [k] can be canceled out by the subsequently decision
device, and â[k] = a[k] will be achieved.
d[k]
a[k]
â[k]
+
n′ [k]
B(z) − 1
v[k]
+
x[k]
u[k]
mod 2M
Figure 4.5:
B(z)
+
v[k] + n′ [k]
Linearized model for the THP-FFE transmission sheme
To initialize the system, the post-cursors of the overall discrete-time channel transfer
function B(z) are required at the THP. To get B(z) = H(z) · W (z), the channel information has to be known. Therefore, channel estimation at the receiver is first carried
out during the start-up using a training block before the data transmission to get the
estimated discrete-time channel transfer function He (z). After that, W (z) will be calculated based on He (z) and then B(z) = He (z) · W (z) · z L1 should be approximated as
close as possible to a causal, monic transfer function, where z L1 is used to compensate
for the channel delay.
At the end of the start-up procedure, the post-cursors of B(z) are fed back from the receiver to the THP to determine the THP coefficients, i.e., the feedback loop of the THP
is set to B(z)−1. By taking into consideration that THP is placed at the transmitter, and
normally it is inconvenient to make the transmitter adaptive due to the extra communications up-link needed, THP coefficients at the transmitter are fixed after the initialization
period, while the feedforward equalizer W (z) at the receiver serves as the adaptive part
70
in the system, whose coefficients are constantly updated once per symbol block in order
to control the transmission error. This method is feasible in our application because
the considered channel is slowly time-varying. Detailed discussions and the design of
W (z) will be introduced in the upcoming section. Note that the pre-cursors of B(z),
if there is any by improper design, can not be handled by THP and consequently, may
severely decrease the performance if they come into play.
4.4
Design of the Feedforward Equalizer
In order to fulfill the constraint of THP in (4.1), one might first think of rescaling h[k]
in (2.28) by factor 1/h[0] to make its very first tap equals one. However, this method
is equivalent to placing a one-tap filter at the receiver, which might heavily enhance the
noise component and result in an unacceptably large loss (e.g., up to 38 dB for the considered transmission). Therefore, it is important to use an additional linear feedforward
equalizer W (z) at the receiver to meet (4.1). Apart from reshaping the channel, W (z)
can also be set as the adaptive part in the transmission system. An adaptive system
often plays an important role in reality, because the channel is never perfectly known
by the receiver, and a mismatching of the THP coefficients may exist due to channel
estimation errors. This mismatch could seriously reduce the transmission quality or
even breakdown the whole transmission link. For this reason, it is necessary for W (z)
being adaptive. Another benefit of using an adaptive W (z) is that the variations, e.g.,
caused by temperature or bending/vibration of the fiber can also be followed and compensated during the transmission. Besides, W (z) is considered as fractionally-spaced
in this chapter, by taking into account that a fractionally-spaced (FS) equalizer is advantageous over a symbol-spaced equalizer as discussed in Section 3.1.2.
4.4.1 During the start-up
The design of W (z) during the start-up is crucial to the system performance, for it
determines the THP coefficients and should not enhance the noise significantly. On
one hand, to keep the noise level low, one should avoid a ZF design, i.e., completely
removing the pre-cursors ISI. On the other hand, W (z) should keep the pre-cursors ISI
as small as possible, since they result in the residual ISI which THP is unable to handle,
and the residual ISI will seriously decrease the performance. Therefore, it is important
to find a trade-off between the noise performance and the residual ISI.
71
For that, a selection diagonal matrix C with dimension (Lc + nf − 1) × (Lc + nf − 1)
is introduced, such that the relevant pre-cursors ISI vector p can be written in the vector
notation as:
p = C · (H · w − i),
where H is the channel matrix defined by (3.20), i is the desired discrete-time channel
impulse response (a unit impulse) with length Lc + nf − 1, w is the coefficient vector
of W (z) with length nf , and Lc is the channel length. Again, the difference between
H · w and i leads to the overall ISI, and p is the pre-cursors ISI that can not be removed
by the THP. Therefore, the diagonal matrix C weights all relevant pre-cursors and the
main tap of H · w − i with 1 and the rest with 0.
Now, we use the MMSE criterion to calculate the FFE coefficient vector w. The goal is
to minimize the total error caused by the pre-cursors p and the colored noise n′ [k] at the
decision device. The optimum FFE coefficient vector w 0 which leads to the minimum
total error is derived together with [50] as the following:
First, indicate the ISI caused by the pre-cursors p as eISI [k], and the order of p as k0 , the
ISI signal eISI [k] is given by:
(4.2)
eISI [k] = x[k] ∗ p[k],
where the THP transmit symbols x[k] are uncorrelated [40, 51, 52] and has a mean value
µx and a variance σx2 .
The mean of eISI [k] is then:
E[eISI [k]] = µx
k0
X
p[k]
(4.3)
k=0
= µx · 1T · C · (H · w − i),
with 1 = (1 · · · 1)T .
And the variance of eISI [k] is:
var[eISI [k]] =
σx2
=
σx2
k0
X
p2 [k]
k=0
· (C · (H · w − i))T · C · (H · w − i)
T
= σx2 · (H · w − i)T · C
| {z· C} ·(H · w − i)
C
=
σx2
T
· (H · w − i) · C · (H · w − i).
(4.4)
72
With the mean and the variance of eISI [k], the power of eISI [k] can be calculated as:
E[e2ISI [k]]
=
µ2x (
k0
X
2
p[k]) +
σx2
k0
X
p2 [k].
(4.5)
k=0
k=0
Secondly, the power of the colored noise n′ [k] = n[k] ∗ w[k] has to be included into the
total error. The power of n′ [k] is:
nf −1
σn2 ′
=
σn2
X
w 2 [k]
(4.6)
k=0
= σn2 · w T w.
Thirdly, by summing up (4.5) and (4.6), the total error or the cost function ε(w[k]) is
given by:
nf −1
k0
k0
X
X
X
2
2
2
2
2
w 2 [k].
p [k] + σn
(4.7)
p[k]) + σx
ε(w[k]) = µx (
k=0
k=0
k=0
Rewrite (4.7) with matrix-vector representation by using (4.3), (4.4) and (4.6), we get:
ε(w) =µ2x · (1T · C · (H · w − i))T · 1T · C · (H · w − i)+
σx2 · (H · w − i)T · C · (H · w − i) + σn2 · wT w
T
=µ2x · (H · w − i)T · C
· 1T · C} ·(H · w − i)+
| · 1{z
D
σx2
· (H · w − i) · C · (H · w − i) + σn2 · wT w
T
=(H · w − i)T · (µ2x · D + σx2 · C) ·(H · w − i) + σn2 · wT w
|
{z
}
(4.8)
B
=(H · w − i) · B · (H · w − i) + σn2 · w T w
T
=w T H T BHw − iT BHw −
T
T
w
{z Bi}
| H
=iT BHw, with B T =B
+iT Bi + σn2 · w T w
=w T H T BHw − 2 · iT BHw + iT Bi + σn2 · w T w.
The optimum FFE coefficient vector w0 should fulfill grad(ε(w0 )) = 0, using (4.8) and
the vector derivative formulas:
grad(wT Aw) = Aw + AT w
(4.9)
73
and
grad(Aw) = AT ,
(4.10)
ε(w 0 ) = 2 · H T BHw 0 − 2 · H T Bi + 2 · σn2 · w 0 = 0.
(4.11)
H T BHw 0 + σn2 · w0 = H T Bi.
(4.12)
we get:
This leads to:
Finally, the optimum FFE coefficient vector w 0 can be calculated as:
w0 = (H T BH + σn2 · I)−1 H T Bi,
(4.13)
where σn2 is the variance of AWGN, I is an identity matrix of dimension (Lc + nf −
1) × (Lc + nf − 1) and B = µ2x · C T · 1 · 1T · C + σx2 · C.
Evaluation of (4.13) requires the channel matrix H. As described in Section 3.1.3.1, the
channel coefficients h[k] can be estimated via the LMS algorithm by sending a training
sequence before the transmission. For that, different values of the step size µ for the
LMS algorithm are compared. In Figure 4.6 [50] where the squared error curves for
three fixed µ values and a decreasing µ value are plotted. The µ which produces the
most precise estimation after the training period should be chosen. Since the gradually
decreased µ value achieves the smallest steady-state squared error and produces the
fastest convergence, it is selected for the LMS algorithm in this chapter to perform the
channel estimation.
Apart from the channel information, (4.13) also needs the noise variance σn2 . By use
of the estimated channel coefficients he [k], the noise variance of the AWGN can be
estimated as:
N −1
1 X
ˆ
2
(4.14)
σn =
(y[k] − (he [k] ∗ at [k]))2 ,
N k=0
where N is the length of the training block, y[k] is the received sequence and at [k] is
the training sequence. After the noise variance is estimated using (4.14), (4.13) can be
evaluated to calculate w 0 .
74
−2
10
µ = 0.5
µ = 0.2
µ = 0.1
Decreasing µ
−3
10
−4
Squared Error
10
−5
10
−6
10
−7
10
−8
10
−9
10
0
1000
Figure 4.6:
2000
3000
4000
5000
Update Number
6000
7000
8000
Convergene performanes with dierent step size µ
At the end of the start-up, w0 will be assigned to the FFE W (z) and the post-cursors of
the channel coefficients b[k] = he [k] ∗ w[k] will be fed back to the THP encoder, where
w[k] are the coefficients of W (z) and the components of w 0 .
Note that it is the MMSE instead of the ZF criterion that is used to calculate the FFE
coefficients. The reason for that is as explained before: since there is a trade-off between
the noise performance and the residual ISI, some pre-cursors ISI is deliberately left to
trade for a smaller noise enhancement caused by the post-cursors ISI.
An example is given in Figure 4.7, where it can be seen that the ZF design in Figure 4.7(a) removes all pre-cursors but produces simultaneously some large post-cursors
which are responsible for a noise enhancement of 20 dB at the receiver. In contrast,
the MMSE-FFE design in Figure 4.7(b) leads to only a 12 dB noise enhancement, by
allowing the existence of a few small pre-cursors.
4.4.2 After the start-up
After initialization, the filter W (z) can operate adaptively or as an non-adaptive MMSEFFE whose coefficients are fixed to the initial values. For the adaptive case, the coefficients of W (z) are calculated utilizing the normalized least mean square (NLMS) algorithm introduced in Section 3.1.3.2. The initial motivation to use the NLMS algorithm
75
Taps at twice the symbol rate
Relevant taps
2.5
big post−cursors,
handled by THP or the
feedback part in a DFE
2
main value
b[k]
1.5
1
zero pre−cursors
0.5
0
40
45
50
55
60
65
70
k
(a) Channel oeients after a ZF-FFE
Relevant taps
2.5
2
post−cursors, handled
by THP or the feedback
part of a DFE
1.5
b[k]
main value
1
pre−cursors,
as small as possible
0.5
0
0
5
10
15
20
k
25
30
35
40
(b) Channel oeients after a MMSE-FFE
Figure 4.7:
Comparison of the ZF and the MMSE design for the FFE, where
b[k] = he [k] ∗ w[k] is the asade of the hannel and the FFE.
(a) b[k] = he [k] ∗ w[k], with w[k] being a ZF-FFE
(b) b[k] = he [k] ∗ w[k], with w[k] being a MMSE-FFE
76
comes from the fact that it makes the adaption process more convenient by automatically adjusting the actual step size according to the input sequence y[k]. The update
equation of the NLMS algorithm is given by [39]:
w[k + 1] = w[k] +
µ̃
e[k]y[k],
y[k]T y[k]
with 0 ≤ µ̃ ≤ 2,
(4.15)
where the error signal e[k] is the difference between the FFE output y[k] ∗ w[k] and the
decided effective data sequence v̂[k], as shown on Figure 4.8.
y[k]
W (z)
e[k]
Figure 4.8:
y[k] ∗ w[k]
v̂[k]
+
Adaptive FFE in the adaptive THP-FFE transmission sheme
It should be noted that in the adaptive mode when no additional training sequence is
sent, v̂[k] is used as the reference signal. As a natural outcome, the adaptive mode
is subjected to wrong decisions. When v̂[k] varies a lot from the actual effective data
sequence v[k], the errors inserted in the adaptive process might lead to an erroneous
behavior in the adaptive mode. In case it happens, the training mode should be repeated.
On the contrary, for a small to moderate symbol error rate, this erroneous behavior is
not significant.
4.5
Disussion of THP Losses
In order to understand the following simulation results, it is important to look into some
properties of the THP. Recall that a THP-FFE is somehow similar to a DFE. However,
unlike a DFE, THP does not suffer from the error propagation but from two certain
losses: the precoding loss and the modulo loss. Many literature such as [40] have
derived these two losses for the bipolar MPAM scheme. Since the unipolar MPAM
is utilized in this dissertation, some properties of the THP as well as the analytical
expression of the THP losses must be adjusted accordingly.
In the following, the THP losses for a unipolar MPAM scheme will be derived by comparing the THP with an ideal DFE without error propagation. For comparison purposes,
the THP losses for a bipolar MPAM scheme are also listed. The transmit signal for the
77
THP transmission scheme is denoted as x[k] and the transmit signal for the DFE transmission scheme is denoted as a[k].
First, THP increases the average transmit power and thus the average received power
compared to the use of DFE. This effect is known as the precoding loss which is defined as the ratio of the received signal power for THP to the received signal power
for DFE [40]. Many literature use the phrase “precoding loss” to describe the transmit power penalty. However, this description is only true for an unbiased transmission
where the ratio between the received signal power for THP and DFE is the same as the
ratio between the transmit signal power for THP and DFE. In the following, we compare
the precoding loss of the biased and the unbiased transmission:
• The precoding loss for an unbiased THP is (a[k] being bipolar MPAM symbols
with d = 2 according to Section 2.2.1. Unbiased means that a[k] exhibits a zero
mean µa = 0.):
Since the transmit signal x[k] for THP is almost independent and identically distributed (i.i.d.) with a uniform distribution function within the region [−M, M)
(this distribution is nearly independent of the channel and becomes more exact
with an increasing value of M), the transmit power for an unbiased THP can be
defined by the variance of x[k] [21]:
σx2 =
M2
;
3
(4.16)
while the transmit power or the variance of the transmit signal a[k] of the system
with unbiased DFE is
M2 − 1
(4.17)
σa2 =
.
3
This leads to the precoding loss for an unbiased THP:
σx2
σa2
M2
= 2
M −1
≈ 0.07 dB for M = 8,
γtransmit power, unbiased =
(4.18)
which is an insignificant number especially for a higher-order M.
• On the other hand, the calculation of the precoding loss for a biased THP is more
complicated (a[k] being unipolar MPAM symbols with d = 2 according to Section 2.2.1. Biased means that a[k] exhibits a non zero mean µa = M − 1.):
78
Probability density funtion of the shifted output of the modied
modulo 2M devie for a unipolar MPAM transmission
Figure 4.9:
Now, the distribution of the THP transmit symbols x[k] is changed to a uniform
distribution between 0 and 2M, as shown in Figure 4.9. As a result, x[k] ∈
[0, 2M) has a non-zero mean value
µx = E[x[k]] = M,
(4.19)
and a mean transmit power ξx
ξx = E[x2 [k]] = σx2 + µ2x
M2
+ (M)2
3
4M 2
.
=
3
=
(4.20)
Similarly, the mean value µa and the mean transmit power ξa of the transmit signal
a[k] for a biased DFE are also changed. Now, the mean value of the unipolar
MPAM symbols a[k] is:
µa = M − 1,
(4.21)
and the mean power is:
ξa = E[a2 [k]] = σa2 + µ2a
M2 − 1
+ (M − 1)2
3
4M 2 − 6M + 2
.
=
3
=
(4.22)
So, the ratio between the transmit power with THP and with DFE turns out to be:
γtransmit power, biased =
ξx
ξa
4M 2
4M 2 − 6M + 2
≈ 0.9 dB for M = 8.
=
(4.23)
79
Unlike the case for an unbiased THP where there is only a 0.07 dB (M=8) difference in the transmit power, a biased THP enhances the average transmit power by
0.9 dB in comparison to a biased DFE.
To calculate the precoding loss in this case at the receiver side, the received signal
power for THP and DFE should be first computed. Let s[k] be the transmit signal
with a mean value µs and a variance σs2 (s[k] = a[k] for DFE or s[k] = x[k] for
THP, note that the modulo outputs x[k] are uncorrelated [40, 51, 52]), the received
signal power without noise can be expressed as:
E[y 2 [k]] = E[(h[0]s[k] + h[1]s[k − 1] + ... + h[Lc − 1]s[k − Lc + 1])2 ]
=
L
c −1 L
c −1
X
X
i=0
j=0
= σs2 ·
L
c −1
X
i=0
h[i]h[j] · E[s[k − i]s[k − j]]
h2 [i] + µ2s ·
L
c −1 L
c −1
X
X
i=0
(4.24)
h[i]h[j].
j=0
If a T/2-fractionally-spaced system is considered, (4.24) can be simplified due to
some special properties of the system. For instance, now E[s[k]s[k − 1]] = 0
holds, since s[k] is an up-sampled version of the transmit symbol, i.e. every
second value of s[k] is zero. Consequently, the received signal power without
noise for a T/2-fractionally-spaced system can be simplified to:
Lc −1
Lc −1 L
LX
c −1
c −1 L
c −1
X
X
1 2 X
1 2 X
2
h[i]h[j]). (4.25)
h [i]+ ·µs ·(
h[i]h[j]+
E[y [k]] = ·σs ·
2
2
i=0
i=0 j=0
i=0 j=0
2
i even j even
i odd j odd
It follows from the foregoing that the precoding loss for a biased THP can be
calculated according to:
γprecoding =
2
E[yDF
E [k]]|s[k] = x[k]
.
2
E[yT HP [k]]|s[k] = a[k]
(4.26)
Unfortunately, this loss depends explicitly on the channel, so a general analytical
expression can not be given.
Figure 4.10:
Congruent signal onstellation of the EDS v[k] for M = 4 and
d=2
80
Secondly, the modulo operation in the THP increases the symbol error rate, which
is known as the modulo loss. Through the addition/subtraction of d[k] in the modulo operation, where d[k] is an integer multiple of 2M, the effective data sequence
v[k] = a[k] + d[k] in THP now has a congruent signal constellation which is a periodical repetition of the original MPAM constellation for a[k]. Figure 4.10 depicts an
example for M = 4, where v[k] = a[k] + d[k] can now take on values out of the congruent signal constellation. The constellation of v[k] leads to a slightly increased symbol
error rate compared to the symbol error rate of a[k], because periodically repeating the
MPAM signal constellation results in no edge constellation points. So the symbol error
rate for THP, which is the symbol error rate of v[k], has the form of:
PS,T HP = 2Q(
d
),
2σn
(4.27)
where d is the minimum distance between two constellation points, σn is the noise
standard deviation, and Q(x) is the Q-function. Because the symbol error rate of the
MPAM symbols a[k] is
PS,M P AM = 2
d
M −1
Q(
),
M
2σn
(4.28)
where the factor MM−1 is an outcome of the fact that the symbol error probabilities for
the edge constellation points are only half of those for the inner points. Therefore, THP
produces a larger symbol error rate, and the increased error rate can be approximated
by the factor:
γmodulo =
=
PS,T HP
PS,M P AM
2Q( 2σdn )
2 MM−1 Q( 2σdn )
M
=
.
M −1
(4.29)
This factor is the modulo loss for THP.
4.6
Evaluation and Simulation Results
In this section, the system in Figure 4.4 is evaluated by simulating the BER before RS
decoding as a function of SNR. The area-of-interest for SNR is 20.4 ∼ 59.5 dB based
81
on the calculation in Section 2.1.3.3. The area-of-interest for BER is 10−4 to 10−3 ,
because the RS (255,223) coding used in the bit level can decrease the final BER to
10−9 and below.
The feedforward equalizer W (z) at the receiver is fractionally-spaced with twice the
symbol rate. In order to convert the discrete-time channel impulse response h[k] into
a monic channel with only post-cursors, coefficients of W (z) are either optimized in
terms of mean squared error or updated by the normalized least mean squares (NLMS)
algorithm if W (z) is set as adaptive.
Simulations are first run to identify the improvement brought by the MMSE THP-FFE
in comparison to the linear FFEs (ZF-FFE and MMSE-FFE) and the MMSE-DFE, all
for 8PAM with d = 2. Then, the robustness and performances of the adaptive THP-FFE
under reduced channel bandwidths are investigated. Finally, THP-FFE is compared to
an ideal DFE without error propagation.
4.6.1 Comparison of THP-FFE, FFE and DFE at 2 Gbit/s
and 3 Gbit/s
The BER performances of the THP-FFE and the post-equalizers considered in Chapter 3
are compared in Figure 4.11 at 3 Gbit/s. The red curve describes the BER performance
of the THP-FFE, whose coefficients are determined via the MMSE criterion (abbreviated as MMSE-THP), while the orange, blue and green curves stand for the BER curve
for the zero-forcing feedforward equalizer (ZF-FFE), the MMSE feedforward equalizer
(MMSE-FFE) and the MMSE decision feedback equalizer (MMSE-DFE), respectively.
First, the result demonstrates that the MMSE-THP outperforms all the post-equalizers
over the considered SNR region, and it reaches the target BER at an SNR of less than 40
dB, indicating that a power margin of nearly 20 dB is reserved according to the power
budget. In addition, a remarkable improvement can be observed by comparing the nonlinear equalizers (THP and DFE) with the linear equalizers (ZF-FFE and MMSE-FFE).
In the BER range form 10−4 to 10−3 , the nonlinear equalizers have 10 dB SNR gain
over the linear ones.
Second, it is easily seen on Figure 4.11 that the MMSE-THP requires 1 ∼ 2 dB less SNR
than the MMSE-DFE for achieving the target BER, and a more significant difference
between them can be observed in the low SNR region. To interpret this, one should
keep in mind of that both THP and DFE have their own losses. In the low SNR region
where noise dominates, the effect of the error propagation in the DFE tends to be more
82
0
10
MMSE−THP
MMSE−DFE
ZF−FFE
MMSE−FFE
−1
Bit Error Rate
10
−2
10
−3
10
−4
10
30
35
40
45
50
55
SNR [dB]
Figure 4.11:
BER performanes as a funtion of SNR at 3 Gbit/s for
frationally-spaed MMSE-THP, MMSE-DFE, ZF-FFE and MMSE-FFE, Lc =
12, nf = 23, nb = 11, the number of THP oeients = 10
0
10
MMSE−THP, 2 Gbit/s
MMSE−DFE, 2 Gbit/s
MMSE−THP, 2.5 Gbit/s
MMSE−DFE, 2.5 Gbit/s
MMSE−THP, 3 Gbit/s
MMSE−DFE, 3 Gbit/s
−1
10
−2
Bit Error Rate
10
−3
10
−4
10
−5
10
−6
10
25
30
35
40
45
50
SNR [dB]
BER performanes as a funtion of SNR at 2 Gbit/s (Lc = 8),
2.5 Gbit/s (Lc = 10) and 3 Gbit/s (Lc = 12) for frationally-spaed MMSE-THP
and MMSE-DFE, nf = 23, nb = 11, the number of THP oeients = 10
Figure 4.12:
83
serious in comparison to the THP losses which are more or less fixed values for a given
system layout. Consequently, it is a natural outcome that there exists an increasing gap
between their BER curves with a decreasing SNR.
A similar phenomenon can be observed in Figure 4.12, where the BER curves for the
MMSE-THP and the MMSE-DFE operating at various data-rates are plotted. In most
cases, THP outperforms DFE, and larger differences between them can be discovered
for lower SNR values and higher data-rates. However at 2 Gbit/s, because DFE is
well capable of getting rid of the ISI, its performance is not much affected by the error
propagation. At high SNR values, the DFE can even exhibit a better result than the
THP. However, with the increasing data-rate, the superiority of THP over DFE becomes
clearer, because THP is less sensitive to severe ISI and DFE is more influenced by the
error propagation effect.
As a conclusion, DFE is recommended due to the simplicity at 2 Gbit/s. However, THP
is more advantageous for transmissions at above 2 Gbit/s.
4.6.2 Eets of a dereased hannel bandwidth
Earlier simulations have revealed the suitability of THP-FFE for high speed transmissions. Now, given the situation that the available transmission bandwidth is permanently
decreased arising from a change in the fiber length or a hardware aging degradation, the
performance of THP-FFE under such a situation should be closely looked into.
For this purpose, Figure 4.13 illustrates the simulated BER performances for THP-FFE
and DFE under three transmission bandwidths: 135 MHz is the original channel bandwidth, and 122 MHz and 114 MHz are the considered decreased channel bandwidths.
Again, the MMSE criterion is used to calculate the equalizer coefficients. In this example, THP-FFE surpasses DFE for all considered bandwidths, and performs remarkably
better than DFE under the smallest transmission bandwidth. This outcome indicates
that the THP-FFE is better at ISI cancellation than the DFE, and it is a better choice for
a smaller transmission bandwidth.
4.6.3 Performane of the adaptive lter
In a practical environment, the optical transmission system might be sensitive to the
change of operating temperature due to the optical devices involved. [53] has reported
that the increased operating temperature makes the transmission bandwidth fall in an
84
0
10
−1
10
−2
Bit Error Rate
10
−3
10
−4
10
MMSE−THP, BW = 135MHz
MMSE−DFE, BW = 135MHz
−5
10
−6
10
26
28
30
32
34
36
38
SNR [dB]
40
42
44
46
48
Comparison of frationally-spaed THP and DFE under dierent hannel bandwidths at 3 Gbit/s, nf = 23, nb = 11, the number of THP
oeients = 10
Figure 4.13:
obvious manner. For instance, in the automotive environment temperature range from
−40◦ C to 105◦ C, the maximum bandwidth decrease can be ∆Bmax ≈ −20 MHz. For
this reason, the robustness of THP-FFE under such a situation is examined in this subsection. Because the operating temperature rises gradually in the transmission process,
the use of an adaptive THP-FFE is considered to compensate for the loss from the
slowly-decreased transmission bandwidth.
Taking into account that the bandwidth can drop at most 20 MHz due to the rising temperature, the performance of the adaptive THP-FFE system is tested for three bandwidth
drops respectively: ∆B = −5 MHz, ∆B = −12 MHz and ∆B = −20 MHz. During
the transmission, ∆B is reached gradually.
The results are depicted in Figure 4.14. The considered SNRs are within the margin of
the SNR specification. The black curve stands for the BER performance of the MMSE
THP-FFE under a room temperature, where the initial 3 dB channel bandwidth is 134
MHz. The colored curves illustrate the performances of the adaptive THP-FFEs, where
the transmission bandwidth decreases progressively during the transmission until ∆B is
reached. For each value of ∆B, two set-ups are compared: an optimum set-up where the
coefficients of the MMSE THP-FFE are re-calculated once there is a bandwidth change
and, an adaptive set-up where the NLMS algorithm is used by the receiver FFE to follow
85
0
10
THP−FFE, B=134 MHz
Optimum, ∆ B=−5 MHz
Adaptive, ∆ B=−5 MHz
−1
Bit Error Rate
10
−2
10
−3
10
−4
10
35
36
37
38
39
40
41
SNR [dB]
42
43
44
45
46
BER performanes of the adaptive frationally-spaed system
with bandwidth redution at 3 Gbit/s, nf = 23, the number of THP oeients
Figure 4.14:
= 10
the bandwidth drop. For the adaptive set-up, THP coefficients at the transmitter remain
fixed after the initialization.
As expected, performances for both set-ups become worse with the decreasing bandwidth. However, comparing the adaptive with the optimum set-up, we notice that for a
small bandwidth decrease of 5 MHz, the difference between them is negligibly small.
In such a situation, a simple adaptive filter at the receiver side is completely sufficient.
For a bandwidth drop of 12 MHz, their performances are still comparable. Only when
the bandwidth is decreased by 20 MHz, the difference turns out to be quite noticeable.
But even so, the adaptive filter performs very stable.
The reason why the adaptive set-up is distinct more from the optimum set-up for a
bigger bandwidth decrease can be well explained by Figure 4.15. The red taps on Figure 4.15(a) represent the relevant coefficients b[k] after the adaptive equalizer. Postcursors of b[k] are assigned to the THP coefficients during the initialization process,
which are then fixed in the adaptive set-up. It can be seen that although channel changes
during the transmission, the adaptive equalizer tries to force the post-cursors of b[k] as
close as possible to the fixed THP coefficients, which are actually no longer optimal
for the varied channel. On the other hand, the optimum MMSE THP-FFE set-up on
Figure 4.15(b) is allowed to renew its THP coefficients according to the current channel
86
condition. Thus, the current post-cursors of b[k], which are actually the optimal THP
coefficients, are now quite different from the initial THP coefficients marked by blue.
This phenomenon is known as the THP mismatch.
Finally, a conclusion can be drawn that the adaptive THP-FFE works properly in tracking a slowly time-varying dispersive channel. And a reboot of the system is unnecessary
within the prescribed range of temperature change.
4.6.4 Evaluation of the THP losses
As discussed in Section 4.5, THP suffers from the modulo and precoding losses whereas
DFE suffers from the error propagation, when compared to an ideal DFE without error
propagation. Unfortunately, it is too complicated to calculate the DFE loss caused by
the error propagation. However, the THP losses can be calculated numerically, and the
performance of THP in comparison to an ideal DFE can be predicted.
The objective of this subsection is to numerically evaluate the THP losses by use of
the formulas in Section 4.5. The theoretical results will then be compared with the
simulated losses which are obtained by comparing the simulated BER curves of a THPFFE and an ideal DFE. Note that an ideal DFE without error propagation can be seen
as an upper performance bound for the THP-FFE, since ideally THP-FFE is equivalent
to a DFE by moving its feedback filter to the transmitter side [40]. Due to this reason,
THP-FFE can never be superior to an ideal DFE without error propagation.
Evaluation of the THP losses is carried out as follows:
• The modulo loss:
When introducing a congruent signal constellation for the THP, the symbol error
probabilities of the edge symbols in a MPAM constellation are now doubled. The
increased symbol error rate from PS,M P AM to PS,T HP can be calculated according
to (4.29). For M = 8, we get:
PS,T HP
PS,M P AM
M
=
M −1
= 1.1429.
γmodulo =
(4.30)
In the logarithmic plot, this factor corresponds to a shift of 10 lg 1.1429 = 0.058
dB upwards.
87
Relevant taps
THP coefficients
3.5
3
2.5
b[k]
2
1.5
1
0.5
0
0
5
10
15
20
25
30
35
40
45
k
(a) The adaptive set-up (with xed THP and an adaptive lter) is used
Relevant taps
THP coefficients of the
adaptive set−up
3.5
3
2.5
b[k]
2
1.5
1
0.5
0
0
5
10
15
20
25
30
35
40
45
k
(b) The optimum MMSE THP-FFE set-up is used
Example at 3 Gbit/s: omparing oeients of b[k] and the initial
THP oeients, ∆B = −20 MHz, nf = 23, length of the estimated hannel
= 20, the number of THP oeients = 10
Figure 4.15:
88
• The precoding loss:
The precoding loss originates from an increased transmit/received power of the
THP-FFE compared to the DFE. As explained in Section 4.5, when biased MPAM
is used, calculation of the precoding loss has to be done at the receiver side, i.e.,
the ratio of the expected received signal power for the THP-FFE to the expected
received signal power for the DEF should be calculated. Using (4.25), (4.26) and
M = 8, we get
γprecoding = 1.2744,
(4.31)
for the considered Gaussian low-pass channel. This power penalty can be regarded as a shift of 10 lg 1.2744 = 1.05 dB to the right on the SNR axis.
Figure 4.16 (a) shows the simulated BER as a function of SNR for the THP-FFE and the
ideal DFE, which is marked in red and black, respectively. It can be seen that an ideal
DFE without error propagation performs always better than the THP-FFE. Then, when
the calculated THP losses are brought into the plot in Figure 4.16 (a), i.e., by adding
the two losses given by (4.30) and (4.31) to the BER curve of the ideal DFE, we get the
black curve in Figure 4.16 (b). Now both BER curves almost overlap with each other, so
the previously deduced THP losses in Section 4.5 are in agreement with the simulated
result. This also confirms our explanation regarding the existence of an increasing gap
between the BER curves of THP-FFE and DFE in Figure 4.11.
4.7
Summary
In this chapter, a transmission strategy that combines a non-linear THP with a linear
FFE for the high-speed short-range fiber-optic transmission system was explored. Performances of THP-FFE were examined and compared to a DFE via computer simulations at 2-3 Gbit/s.
The results show that THP offers no substantial advantage over DFE for a 2 Gbit/s
transmission, since what THP accomplishes is to eliminate the error propagation which
is a minor problem in the absence of severe ISI at a relatively low speed. However, for
a 3 Gbit/s transmission, THP-FFE is able to provide a superior performance over DFE,
benefiting from the immunity to the error propagation.
Other conclusions that can be extracted from this chapter are: first, in contrast to a
conventional linear pre-equalizer, THP is employed with a non-linear device to prevent
the amplification in the transmit power. Thus, THP provides a more energy-effective
89
(a)
0
(b)
0
10
10
THP−FFE
Ideal DFE
−1
−1
10
10
−2
−2
Bit Error Rate
10
10
−3
−3
10
10
−4
−4
10
10
−5
−5
10
10
−6
10
25
THP−FFE
Ideal DFE
+ THP losses
−6
30
35
SNR [dB]
40
45
10
25
30
35
SNR [dB]
40
45
Comparison of THP-FFE and an ideal DFE without error propagation at 3 Gbit/s, nf = 23, nb = 11, the number of THP oeients = 10
Figure 4.16:
solution than a pre-equalizer. Second, it is proven by computer simulations that the
adaptive THP-FFE is robust against moderate changes of the transmission link caused
by temperature variations, hardware degradation and so forth.
On the whole, THP-FFE is a cost-effective, simple and reliable transmission strategy
based on the MOST150 optical physical layer. However, a disadvantage of THP is that
it requires the channel knowledge at the transmitter.
CHAPTER 5
Bidiretional Deision Feedbak Equalization
5.1
Motivation
The previous chapter has described that the error propagation problem in a DFE can be
avoided by using the non-linear THP technique. However THP has the major disadvantage that it requires the precise channel knowledge at the transmitter, just like any
other types of precoding technique. So a feedback communications link is essential for
the THP especially when the channel is time-variant. Now the challenge is, whether a
similar or even better performance can be provided by an alternative method without
using an extra up-link, which processes meanwhile a moderate complexity.
To meet the challenge, this chapter investigates an equalization approach - the bidirectional DFE (BiDFE) technique. The very first idea of BiDFE was independently
proposed by Ariyavisitakul [25] and Suzuki [27]. BiDFE operates in parallel with two
DFEs at the receiver: a forward-mode DFE for equalizing the received sequence, and
a reverse-mode DFE for equalizing the time-reversed replica of the received signal. It
generates some sort of receive diversity artificially: the time-reversal operation applied
in the reverse-mode DFE makes the decision errors propagate in the opposite direction
compared to the forward-mode DFE. Hence, most of the erroneous locations in the two
DFEs are different. This provides a significant advantage over an ordinary DFE in terms
of preventing the first decision error - also known as the primary error. The original DFE
design typically assumes correct decisions for the past transmitted symbols. However,
once the first decision error is made by the receiver, this error is fed back by the feedback
filter in a DFE which will increase the error probabilities of the subsequent decisions.
91
92
CHAPTER 5 Bidiretional Deision Feedbak Equalization
Preventing the primary error is thus a good way to reduce the error propagation. In a
BiDFE mode, primary errors in a forward- and a reverse-mode DFE usually occur at
different positions since decisions are made in opposite directions. Thus, BiDFE has
the potential to perform better than a single DFE.
Various algorithms for BiDFE can be found in different publications. Under the constraint that the performance of a forward-mode DFE and a reverse-mode DFE is different from each other, [26] suggested to use either a forward-mode DFE or a reverse-mode
DFE, depending on which one provides a better performance. Similar ideas are also
suggested by [54] and [55]. Although the term bidirectional DFE was used at that time,
it refers to the selective time-reversal approach, whereas a truly BiDFE was proposed
10 years later by [56], [57] and [22], where decisions from both DFEs are combined
for the arbitrated final decision. In [57], based on a reconstruction technique, an arbitrator performs the symbol-wise decision to select between the outputs of both DFEs
the output, which produces the minimum squared Euclidean distance to the received sequence. This classical BiDFE algorithm is known as the bi-directional arbitrated DFE
(abbreviated as BAD). Then, [56] proposed the linear combining bidirectional DFE
(LC-BiDFE), which employs a weighted linear combination of the soft outputs from
the forward- and the reverse-mode DFE, to minimize the mean squared error assuming
no feedback errors. To reduce the computational complexity of BAD, [58] proposed the
contradictory block arbitration (CBA) algorithm. CBA compares the decisions provided
by both DFEs and later divides them into consistent and contradictory blocks, such that
the arbitration is carried out only for symbols in the contradictory blocks. Facilitating a
similar design of CBA, the trellis-based conflict resolution (TBCR) algorithm was then
reported in [59], where the noiseless received sequences are partially reconstructed and
the squared Euclidean distance for the whole contradictory block is used as a metric for
arbitration. By doing so, [59] has reported that TBCR provides a better performance
and a lower complexity compared to other previously proposed arbitration algorithms.
This chapter first investigates the classical BAD [29] algorithm, which does arbitration
in a window around each conflicting DFE decision and has a rather high computational
load. The TBCR [59] algorithm, which is known for its lower computational complexity
than BAD, is investigated thereafter. However, since both schemes provide only two
reconstructions for the received sequence, we look into the problem of generating more
reconstructions for the received signal based on the consideration that these additional
reconstructions are able to bring forth further improvements for the BER performance.
So finally, a novel trellis-based algorithm, the trellis-based BiDFE (TB-BiDFE) [60] is
proposed and investigated. It is capable of reconstructing more than two reconstructions
for the received sequence by use of a trellis diagram. Unlike TBCR algorithm where a
93
trellis is used only for identifying the conflict event, TB-BiDFE uses the trellis in a real
sense within the decision process, a performance improvement is therefore expected
over all other aforementioned BiDFE algorithms.
5.2
The System Model
Figure 5.1 depicts the considered system block diagram. The considered optical layer
is a cascade of a LED, a 10 m POF and a PD. Here, 8PAM and non-coherent optical
demodulation is applied. The impulse response of the equivalent base-band channel,
measured from the electrical input to output, is indicated as h[k]. Additive white Gaussian noise (AWGN) is an adequate noise model for the POF based transmission link.
Strong inter-symbol-interference (ISI) is caused by insufficient transmission bandwidth.
To ensure a robust transmission, a BER between 10−3 and 10−4 is desired at the output
of the BiDFE. Then, Reed-Solomon (RS) encoding/decoding can be used to decrease
the BER to 10−9 and below.
n[k]
x[k]
h[k]
Figure 5.1:
+
y[k]
x̂[k]
BiDFE
System blok diagram
The BiDFE block on Figure 5.1 is evaluated by three types of BiDFE equalizers, respectively: the symbol-wise arbitrated BAD, the less complex block-wise arbitrated TBCR
and the novel block-wise arbitrated TB-BiDFE. These equalizers share the same architecture with the functional block diagram shown in Figure 5.2, which consists of four
stages:
1. Equalization stage:
Equalization is done in parallel by a forward- and a reverse-mode DFE at the receiver, which is indicated on Figure 5.2 as DF Ef w surrounded by the red box and
DF Erv surrounded by the blue box, respectively. Throughout this chapter, a DFE
always contain a feedforward equalizer (FFE) and a feedback equalizer (FBE)
part. The 8PAM symbols are transmitted in blocks, which are processed independently. Indicate a symbol block with length N as x = {x[k], k = 0, ..., N −1}, the
94
Figure 5.2:
Arhiteture of BiDFE
input of the forward-mode DFE or the received sample sequence y = {y[k], k =
0, ..., Ny − 1} can be expressed as:
y[k] = x[k] ∗ h[k] + n[k],
= b[k] + n[k].
(5.1)
(5.2)
where the length of the received sample sequence y is Ny = N + Lc − 1,
h = {h[k], k = 0, ..., Lc − 1} is the sampled (symbol-spaced) channel impulse
response corresponding to (2.28) and n[k] is the sampled AWGN. At the reversemode DFE, y is first reversed in time into ỹ = {ỹ[k], k = 0, ..., Ny − 1}, where
~ stands for the time-reversal operation. So we have:
ỹ[k] = b̃[k] + ñ[k],
(5.3)
b[k] = b̃[Ny − k − 1],
(5.4)
x[k] = x̃[N − k − 1],
(5.5)
h[k] = h̃[Lc − k − 1].
(5.6)
and
95
Because
b[k] =
L
c −1
X
ν=0
h[ν]x[k − ν]
= h[0]x[k] + h[1]x[k − 1] + · · · + h[Lc − 1]x[k − Lc + 1],
we could derive the following equation by using (5.4), (5.5) and (5.6):
b̃[Ny − k − 1] = h̃[Lc − 1]x̃[N − k − 1] + h̃[Lc − 2]x̃[N − k] + · · ·
+h̃[0]x̃[N − k + Lc − 2]
= h̃[Lc − 1]x̃[N − k − 1] + h̃[Lc − 2]x̃[N − k − 1 + 1] + · · ·
+h̃[0]x̃[N − k − 1 + Lc − 1].
(5.7)
Replace k ′ = N − k − 1 in (5.7) and because Ny = N + Lc − 1, (5.7) turns out
to be:
b̃[k ′ + Lc − 1] = h̃[Lc − 1]x̃[k ′ ] + h̃[Lc − 2]x̃[k ′ + 1] + · · ·
+h̃[0]x̃[k ′ + Lc − 1]
L
c −1
X
=
h̃[ν]x̃[k ′ + Lc − 1 − ν].
(5.8)
ν=0
Again, replace k = k ′ + Lc − 1, we get:
b̃[k] =
L
c −1
X
ν=0
h̃[ν]x̃[k − ν]
= x̃[k] ∗ h̃[k].
(5.9)
Finally, put (5.9) into (5.3), we get:
ỹ[k] = x̃[k] ∗ h̃[k] + ñ[k].
(5.10)
(5.10) well explains the time-reversal operation: reversing the received sequence
is equivalent to reversing the transmitted sequence and the discrete-time channel
impulse response before the convolution [38], as well as the noise vector. Accordingly, taps of the reverse-mode DFE are trained by the time-reversed discretetime channel impulse response. When a channel is asymmetric with respect to
the main tap, coefficients of the two DFEs tend to have different values, which
consequently leads to different decided symbol sequences x̂F = {x̂F [k], k =
0, ..., N − 1} and x̂R = {x̂R [k], k = 0, ..., N − 1}, and brings in the “receiving
96
diversity”. Apparently, the more asymmetric the channel is, the more receiving
diversity can be achieved. On the other hand, if a channel is symmetric with respect to the main tap, BiDFE doesn’t benefit from the channel reversal. However,
since the noise vector in the received sequence is also reversed, which can be seen
as appearing independently at the two DFEs, so in this sense, some diversity gain
is obtained as well.
2. Channel estimation stage:
To acquire the channel reconstruction he = {he [k], k = 0, ..., Le − 1}, which is
required for the upcoming reconstruction stage, maximum likelihood (ML) criterion is used in a training block in order to get a precise estimation start. Afterwards, the least mean squares (LMS) algorithm is used for faster computational
speed and adaptivity to the channel changes. When the channel estimation works
fine, Le is close to Lc . Let us make this assumption throughout the chapter, and
use Lc instead of Le in the following.
3. Reconstruction stage:
In this stage, he is convolved with, depending on the type of BiDFE, either the
complete decided symbol sequences x̂F and x̂R , or some parts of them, in order to
generate different reconstructions for the received sequence y. If BAD or TBCR
is considered, there will be two different reconstructions ŷ F and ŷ R generated,
whereas when the novel TB-BiDFE is considered, there will be more than two
reconstructions. Details regarding this point will be discussed in Section 5.3.3.
4. Arbitration stage:
Arbitration is done in the final step. Different types of BiDFEs are mostly different in here. However, the common idea is to choose from all candidate sequences/symbols, the one that best explains the received sequence, according to
the arbitration criterion.
In this chapter, the three considered BiDFE algorithms share the common stages 1 and
2, but differ at stages 3 and 4. Details are described in the next section.
5.3
Finite-length BiDFE Algorithms
In this section, we review the BAD and TBCR algorithms, generalize the optimization
problem, as well as formulate a novel TB-BiDFE algorithm.
97
5.3.1 Symbol-wise arbitrated bidiretional arbitrated DFE
The comprehensive operation of BAD is first described in [29]. The received sequence
y = {y[k], k = 0, ..., Ny − 1} is processed by stages 1 to 3 in Section 5.2, to get the
decided symbol sequences x̂F = {x̂F [k], k = 0, ..., N − 1} and x̂R = {x̂R [k], k =
0, ..., N − 1} from the forward- and reverse-mode DFE, respectively. Then ŷF [k] =
x̂F [k] ∗ he [k] and ŷR [k] = x̂R [k] ∗ he [k] are reconstructed for k = 0, ..., Ny − 1.
Whenever x̂F [k] = x̂R [k] by comparing x̂F and x̂R , the arbitrated final decision x̂[k]
can be either x̂F [k] or x̂R [k], as this decision is confirmed by both DFEs and is considered reliable. Whenever x̂F [k] 6= x̂R [k] at time K, i.e., x̂F [K] 6= x̂R [K], the squared
Euclidean distances ξi2 between y and ŷ i , for i∈ {F,R}, are calculated within a window
of length 2W + 1 around the conflict time K. Namely,
max{K+W,N +Lc −1}
ξi2 [K+W
K−W ]
=
X
k=min{1,K−W }
|y[k] − ŷi [k]|2 ,
for i ∈ {F,R}
(5.11)
where [K+W
K−W ] identifies a time window from K − W to K + W . The arbitrated final
decision x̂[k] is chosen between the forward- and the reverse-mode DFE decisions, as
the one that produces a smaller ξi2 to the received sequence. Namely, the decision rule
for arbitration is:



x̂ [k] or x̂R [k], if x̂F [k] = x̂R [k]

 F
k+W
k+W
(5.12)
x̂[k] = x̂F [k],
if ξF2 [k−W
] ≤ ξR2 [k−W
]



x̂ [k],
else
R
for k = 0, ..., N − 1.
Clearly, the complexity of BAD is rather high since for each received data-block, two
sequences ŷ F = {ŷF [k], k = 0, ..., N − 1} and ŷ R = {ŷR [k], k = 0, ..., N − 1} are
reconstructed. Furthermore, for each conflict in the decided symbol sequences, the
squared Euclidean distance within a window length has to be calculated.
5.3.2 Blok-wise arbitrated trellis-based onit resolution
For the reason that the symbol-wise arbitrated BAD algorithm yields a high computational complexity, a block-wise arbitrated TBCR algorithm with reduced complexity is
98
k
0
1
2
3
4
5
x̃F [k]
0
1
0
1
0
1
x̃B [k]
0
0
0
1
0
1
Figure 5.3:
Example: onit event and onsistent bloks in the deisions of
the forward- and the reverse-mode DFE
proposed by [59]. The simplification comes from two considerations: first, consecutive identically decided symbol pairs x̂F [k] = x̂R [k] produce the identical reconstructed
symbol pairs ŷF [k] = ŷR [k] and make no contribution to the squared Euclidean distance.
Thus, reconstruction for these symbols is unnecessary, and reconstruction can be limited to the differently reconstructed pairs ŷF [k] 6= ŷR [k]. Second, the differently decided
symbol pairs x̂F [k] 6= x̂R [k] tend to occur in bursts due to the error propagation, which
consequently causes bursts of contradictory reconstructed pairs ŷF [k] 6= ŷR [k]. Instead
of the symbol-wise arbitration, the block-wise arbitration can be adopted for reducing
the computational load. That is, while the arbitrator in BAD calculates and compares
k+W
the squared error ξi2[k−W
] for each inconsistent decision pair x̂F [k] 6= x̂R [k], TBCR cal-
culates and compares the squared error for each inconsistent block ŷF [k] 6= ŷR [k] from
k1 to k2 . Note that ŷF [k] 6= ŷR [k] must appear in a block-wise fashion with a length
longer or equal to the channel length Lc . The reason for that is as follows: whenever
there is an inconsistency in the two DFE decisions, for example, x̂F [K] 6= x̂R [K] at
time K, the corresponding reconstructed pairs ŷF [k] 6= ŷR [k] must appear consecu-
tively in a block-wise fashion from K to K + Lc − 1 due to the convolution delay in the
reconstruction stage.
Therefore, we can divide x̂F and x̂R into conflict events and consistent blocks, which
correspond to the inconsistent and consistent blocks in ŷ F and ŷ R respectively. A conflict event starts from the first contradict symbol pair in x̂F and x̂R , and ends with Lc −1
consecutive consistent pairs. It is used later on for the reconstruction stage. A consistent
block contains consecutively identical symbol pairs in x̂F and x̂R outside the conflict
events, and a consistent block will be taken directly as the arbitrated final decision.
Figure 5.3 gives an example of such a fragmentation for Lc = 3, where the consistent
block is colored in blue and the conflict event is colored in red. It can be clearly seen
that a conflict event starts from the first conflict decision pair, and ends with Lc − 1 = 2
consecutive consistent pairs.
TBCR algorithm can be summarized as [59]:
99
1. Apply stages 1 and 2 to the received sequence y, and get the decided symbol
sequence x̂F from the forward-mode DFE and x̂R from the reverse-mode DFE.
2. Identify the conflict events and consistent blocks in the two DFE decisions x̂F
and x̂R .
3. For symbols in all consistent blocks, output them as the arbitrated final decisions.
4. For symbols in each conflict event, e.g., from k1 to k2 , reconstruct ŷF [k] from k1
to k2 (indicated as ŷF [kk21 ]) and ŷR [k] from k1 to k2 (indicated as ŷR [kk21 ]). ComP2
pute the squared Euclidean distance ξF2 [kk21 ] = kk=k
|y[k] − ŷF [k]|2 and ξR2 [kk21 ] =
1
Pk2
2
k=k1 |y[k] − ŷR [k]| .
5. For symbols in a conflict event from k1 to k2 , the decision output of TBCR is:

x̂ [k2 ],
F k1
k2
x̂[k1 ] =
x̂ [k2 ],
R k1
ξF2 [kk21 ] ≤ ξR2 [kk21 ]
else
(5.13)
5.3.3 The novel trellis-based BiDFE algorithm
TBCR has one significant drawback by observing the decision rule for a conflict event
from k1 to k2 in (5.13). That is, regardless of errors that might be contained in the
forward-mode DFE decision x̂F [kk21 ] and the reverse-mode DFE decision x̂R [kk21 ], either
of them will be chosen as the arbitrated final decision for k1 ≤ k ≤ k2 , as long as
it provides a smaller squared Euclidean distance to the received signal than the other
one. When a conflict event gets longer, it implies that the two DFE decisions have more
contradictions and are more likely to contain errors, so the arbitrated final decision is
also more likely to contain errors. In order to overcome this drawback, we propose a
new algorithm termed the trellis-based BiDFE (TB-BiDFE) algorithm.
Similar to TBCR, TB-BiDFE only does reconstruction for each conflict event from k1 to
P2
k2 , and uses the squared Euclidean distance ξ 2 [kk21 ] = kk=k
|y[k] − ŷ[k]|2 as a figure of
1
merit for the arbitration, i.e., the reconstruction which produces the minimum squared
Euclidean distance to the received sequence is taken as the arbitrated final decision for
k1 ≤ k ≤ k2 . Major difference between TBCR and TB-BiDFE is that for each conflict
event from k1 to k2 , TBCR reconstructs only two reconstructions ŷF [k] = x̂F [k] ∗ he [k]
and ŷR [k] = x̂R [k] ∗ he [k]; whereas TB-BiDFE performs at least two reconstructions:
ŷ[k] = x̂i [k] · he [0] + x̂i [k − 1] · he [1] + · · ·
+x̂i [k − Lc + 1] · he [Lc − 1],
(5.14)
100
Reonstrution and arbitration using a trellis diagram with 3PAM
(symbol alphabet {0, 1, 2}) and a hannel length Lc = 3
Figure 5.4:
with x̂i [k] ∈ {x̂F [k], x̂R [k]}, so x̂F [k] and x̂R [k] can be mixed in (5.14). Comparing
to TBCR, where x̂i [k] is either x̂F [k] or x̂R [k] for all k, TB-BiDFE can reconstruct at
least 2 and at most 2Lc different reconstructions ŷ[k] for k inside a conflict event. These
additional reconstructions provide more diversity to the arbitration, and the drawback of
TBCR can be avoided. It should be noted that the number of different reconstructions
ŷ[k] is only related to the channel length Lc but not to the possible values of x̂i [k]. This
largely simplifies the TB-BiDFE algorithm in comparison to the maximum likelihood
sequence estimation (MLSE) algorithm.
When visualizing (5.14) by a trellis diagram, whose nodes represent 2Lc −1 combinations
of {x̂i [k − 1], x̂i [k − 2], · · · , x̂i [k − Lc + 1]}, and branches stand for x̂i [k], then the
reconstruction progress of different reconstructions ŷ[k] is illustrated by paths on the
trellis diagram.
An example is given in Figure 5.4. There, 3PAM with symbol alphabet {0, 1, 2} and
a channel length Lc = 3 are assumed. First we partition x̂F and x̂R into a consistent
block (K − 2 ≤ k ≤ K − 1, marked in blue) and a conflict event (K ≤ k ≤ K + 5,
marked in red). Symbols belonging to the consistent block are output directly as the
arbitrated final decisions. Symbols from the conflict event are taken for reconstruction.
According to (5.14), we get ŷ[k] = x̂i [k] · he [0] + x̂i [k − 1] · he [1] + x̂i [k − 2] · he [2]. The
corresponding trellis diagram has thus 2Lc −1 = 4 states, which are {x̂F [k − 1], x̂F [k −
2]}, {x̂R [k − 1], x̂F [k − 2]}, {x̂F [k − 1], x̂R [k − 2]} and {x̂R [k − 1], x̂R [k − 2]}. If
x̂F [k] 6= x̂R [k], a node is split into two branches, each stands for x̂i [k] = x̂F [k] (solid
line) or x̂i [k] = x̂R [k] (dashed line). If x̂F [k] = x̂R [k], then x̂i [k] = x̂F [k].
The paths start from state {x̂F [K − 1], x̂F [K − 2]}, since the previous consistent block
makes {x̂i [K − 1], x̂i [K − 2]} = {x̂F [K − 1], x̂F [K − 2]}. For K ≤ k ≤ K + 1,
101
x̂F [k] 6= x̂R [k], each node is split into two for two times. Till k = K + 1, the four
paths on the trellis diagram indicates {x̂[K + 1], x̂[K], x̂[K − 1]} = {0, 1, 0}, {0, 0, 0},
{2, 1, 0} and {2, 0, 0} respectively, which corresponds to four different reconstructions.
For k = K + 2, x̂i [k] = x̂F [k], two paths merge to one node, therefore, we still have
four reconstructions, otherwise, the maximum number of different reconstructions for
the given example, which is 2Lc −1 = 8, will be reached. For K + 4 ≤ k ≤ K + 5,
x̂F [k] = x̂R [k], paths take Lc − 1 steps to merge again.
During the creation of paths (K ≤ k ≤ K+5), the path costs, i.e., the squared Euclidean
distances between the reconstruction and the received sequence at time k are computed
from K to k:
k
X
2 k
ξ [K ] =
|y[i] − ŷ[i]|2 , K ≤ k ≤ K + 5.
i=K
When several paths merge at one node, paths with higher path costs are abandoned
immediately. In the end, the path which has the lowest path cost survives and its corresponding x̂ is the arbitrated final decision for the conflict event. For example, the
survived green path in Figure 5.4 represents that the arbitrated final decision x̂[K+5
K ] for
the conflict event is {1, 2, 0, 1, 0, 1}.
Finally, the whole decision process ends up with 18 branches from K to K + 5, whereas
the MLSE algorithm would have computed 36 = 729 branches for the same situation.
TB-BiDFE algorithm is summarized as:
1. Apply stages 1 and 2 to the received sequence y, and get the decided symbol
sequence x̂F from the forward-mode DFE and x̂R from the reverse-mode DFE.
2. Identify the conflict events and consistent blocks in the two DFE decisions x̂F
and x̂R .
3. For symbols in the consistent blocks, output them as the arbitrated final decisions.
4. For each conflict event from k1 to k2 , use the trellis diagram to reconstruct ŷ[k] =
x̂i [k] ∗ he [k] where x̂i [k] ∈ {x̂F [k], x̂R [k]}. Find out the ŷ[kk21 ] which has the
P2
minimum squared Euclidean distance ξ 2 [kk21 ] = kk=k
|y[k]−ŷ[k]|2 to the received
1
sequence y[kk21 ]. Output the corresponding x̂[kk21 ] as the arbitrated final decisions.
102
5.4
Comparison of Computational Complexities
The computational complexity in the reconstruction and arbitration stage is another figure of merit for comparing the considered BiDFE algorithms. It can be estimated by
calculating the number of additions and multiplications required to process a symbol
block. For TBCR and TB-BiDFE, the computational complexity in the reconstruction and arbitration stage depends especially on the mean number Ncontr of the conflict
events occurred in a symbol block, and the mean length Lcontr of the conflict events.
The complexities of BAD and TBCR in the reconstruction and arbitration stage are
listed in Table 5.1. To estimate the maximum complexity of TB-BiDFE in the reconstruction and arbitration stage, we take into account that x̂F and x̂R consecutively contradict with each other in a conflict event except for the last Lc − 1 symbols. In the
example on Figure 5.4, this is true if x̂F [K + 2] 6= x̂R [K + 2]. With this assumption,
the maximum computational complexity of TB-BiDFE in the reconstruction and arbitration stage can be derived as follows, where “+” denotes number of additions and
“×” denotes number of multiplications:
1. Reconstruction of one branch: +: Lc − 1; ×: Lc
2. Number of the reconstructed paths: 2Lc (Lcontr − 2Lc + 4) − 4, indicate this value
as A;
3. For path metric calculation: +: 2A, ×: A
4. in total we need +: (Lc + 1)A, ×: (Lc + 1)A
The maximum computational complexity of TB-BiDFE in the reconstruction and arbitration stage is also listed in Table 5.1.
Table 5.1:
Computational omplexities of BiDFEs in the reonstrution and
arbitration stage
Algorithm
BAD
TBCR
TB-BiDFE
Algorithm
BAD
TBCR
TB-BiDFE
Number of additions
2N(Lc − 1) + 2Ncontr (4W + 1)
2Ncontr (Lcontr Lc + Lcontr − 1)
Ncontr (2Lc ((Lc + 1)(Lcontr − 2Lc ) + 4Lc ) − 4)
Number of multipliations
2NLc + 2Nc (2W + 1)
2Ncontr Lcontr (Lc + 1)
Ncontr 2Lc ((Lc + 1)(Lcontr − 2Lc ) + 4Lc )
5.5
103
Simulation Results of the BiDFEs
Simulations of BER performances as a function of SNR for the three considered BiDFE
algorithms are presented in this section. The SNR ranges from 20.4 dB to 59.5 dB based
on a given optical link budget [61]. The area-of-interest for BER is 10−3 ∼ 10−4 , then
RS coding can be adopted to decrease the BER to 10−9 and below. The transmission is
on a block-wise basis to ensure that the time-reversal operation is applicable.
All BiDFE algorithms are realized by use of adaptive DFEs, which are either symbolspaced or fractionally-spaced with half the symbol duration. Their tap weights are updated by RLS and LMS algorithms. The RLS algorithm jointly optimizes the feedforward and feedback weights in a training block and ensures rapid tap convergence. The
LMS algorithm is adopted thereafter for a faster simulation speed. Due to the slow fading nature of the optical channel, a small step size is sufficient for tracking the channel.
The channel estimation is also adaptive.
To identify the performance for different schemes, BiDFEs, BAD, TBCR and TBBiDFE are compared with each other, as well as with a theoretical ideal case, which
assumes that an arbitrator has full knowledge of the original transmitted symbol x[k].
Its arbitration rule is:

x̂ [k], if |x̂ [k] − x[k]| ≤ |x̂ [k] − x[k]|
F
F
R
(5.15)
x̂[k] =
x̂ [k], else
R
The following common simulation parameters are used in this section: the modulation
scheme is 8PAM; the channel is given by (2.28) whose 3 dB bandwidth is 135 MHz; the
channel length is Lc ; the symbol block length N is 3200; the number of the feedforward
taps and the feedback taps in a DFE are set to nf and (nf − 1)/2, respectively.
5.5.1 Symbol-spaed BiDFEs
Figure 5.5 and Figure 5.6 demonstrate a 2 Gbit/s and a 3 Gbit/s transmission over the
given channel using symbol-spaced BiDFEs, respectively. In addition to the considered
BiDFE types, the BER curves for the forward- and the reverse-mode DFE are also
plotted. Besides, a lower bound for the BER, indicating the ideal DFE by employing
(5.15), is plotted on both figures as well.
104
−1
10
−2
Bit Error Rate
10
−3
10
−4
10
−5
10
−6
10
29
Figure 5.5:
BAD
TBCR
TB−BiDFE
Ideal
FW−DFE
RV−DFE
29.5
30
30.5
31
31.5
32
SNR [dB]
32.5
33
33.5
34
Symbol-spaed BiDFEs at 2 Gbit/s, Lc = 4, nf = 9, nb = 4
0
10
−1
10
−2
Bit Error Rate
10
−3
10
−4
10
−5
10
−6
10
38
Figure 5.6:
BAD
TBCR
TB−BiDFE
Ideal
FW−DFE
RV−DFE
38.5
39
39.5
40
40.5
41
SNR [dB]
41.5
42
42.5
43
Symbol-spaed BiDFEs at 3 Gbit/s, Lc = 6, nf = 15, nb = 7
105
The results show that BiDFEs remarkably improve the performance of a conventional
DFE. At 2 Gbit/s, BiDFEs achieve 1 dB SNR gain over the DFEs around the target
BER. Moreover, the performance gap between the TB-BiDFE and the ideal DFE is
about 1 dB at SNR of 29 dB, and about 0.2 dB at SNR of 34 dB. This means, at high
SNRs, TB-BiDFE can approach very closely the ideal DFE. At 3 Gbit/s, in comparison
with the reverse-mode DFE, BAD and TBCR require 1.5 dB and TB-BiDFE requires
2 dB less SNR at BER = 10−3 . Again, among the three considered BiDFE algorithms,
TB-BiDFE presents the closest performance to the ideal case. Note now the forwardmode DFE (FW-DFE) performs differently to the reverse-mode DFE (RV-DFE), which
comes from the fact that the channel is asymmetric with respect to the maximum value
after the symbol-spaced sampling.
Finally, it can be seen that the performance loss due to incorrect decisions being fed back
in a conventional DFE is 2 dB, approximately, around the BER under consideration.
5.5.2 Frationally-spaed BiDFEs
Figure 5.7 and Figure 5.8 present the BER curves for a 2 Gbit/s and a 3 Gbit/s transmission with T /2 fractionally-spaced BiDFEs. At 2 Gbit/s, the BER of 10−3 is achieved
for all BiDFEs with a SNR value of about 30 dB. In addition, TB-BiDFE outperforms
BAD by 0.4 dB and TBCR by 0.8 dB at BER = 10−4 . At 3 Gbit/s, the BER of 10−3 is
reached by TB-BiDFE with SNR ≈ 38.2 dB. Compared to the symbol-spaced BiDFEs
in Figure 5.6, which require SNR = 40 ∼ 41.5 dB for the same BER, a gain of about
3 dB is obtained by oversampling.
For BER between 10−3 and 10−4 , TB-BiDFE has a SNR gain of 2 ∼ 2.2 dB over the
conventional DFEs, whereas BAD and TBCR have only 1 ∼ 1.8 dB SNR gain. We
also observe that TB-BiDFE outperforms TBCR by 1 dB for BER = 10−5 at 2 Gbit/s.
At 3 Gbit/s, the 1 dB gain is achieved at BER = 10−6 . Now, the novel TB-BiDFE
with a fractionally-spaced structure outperforms the BAD and TBCR in a more obvious
manner, and the superiority of TB-BiDFE becomes larger with the decreasing BER.
5.5.3 Evaluation of the omputational omplexities in the
reonstrution and arbitration stage
To provide a rough idea for the computational complexity in the reconstruction and arbitration stage of the novel TB-BiDFE algorithm, the complexity is evaluated numerically
106
−1
10
−2
10
−3
Bit Error Rate
10
−4
10
−5
10
−6
10
−7
10
27
Figure 5.7:
BAD
TBCR
TB−BiDFE
Ideal
FW−DFE
RV−DFE
28
29
30
SNR [dB]
31
32
33
Frationally-spaed BiDFEs at 2 Gbit/s, Lc = 8, nf = 17, nb = 8
0
10
−1
10
−2
Bit Error Rate
10
−3
10
−4
10
−5
10
−6
10
35
Figure 5.8:
BAD
TBCR
TB−BiDFE
Ideal
FW−DFE
RV−DFE
36
37
38
SNR [dB]
39
40
41
Frationally-spaed BiDFEs at 3 Gbit/s, Lc = 12, nf = 29,
nb = 14
107
for a 3 Gbit/s transmission with symbol-spaced BiDFEs. The parameters for the simulation are: SNR= 41.5 dB, the channel length Lc = 6, the half arbitration window size
for BAD algorithm W = 10 and the data-block length N = 3200. From the simulation,
we get the average number of contradiction blocks Ncontr = 3.7 and the average length
of a contradiction block Lcontr = 38 within one data-block, in case TBCR or TB-BiDFE
is used. We also get the average number of contradictory symbol pairs Ncontr = 104
for BAD. With these values, the BiDFE complexities in the reconstruction and arbitration stage can be computed according to Table 5.1. Finally, TB-BiDFE turns out to
be similar complex as BAD, and is 25 times more complex than the TBCR. However,
for a even shorter channel length Lc , TBCR and TB-BiDFE show similar complexities.
Besides, the number of states used in TB-BiDFE is 2Lc −1 = 32, whereas it amounts to
8Lc −1 = 32768 in a MLSE equalizer using 8PAM.
5.6
Summary
With theoretical analysis and simulations, advantages and shortages of BAD, TBCR
and TB-BiDFE were investigated in this chapter.
All BiDFEs, whose length is nearly double of the channel length, can effectively eliminate the ISI lasting up to several symbol durations. All BiDFEs present superior performances compared to a conventional DFE. For the target BER between 10−3 and 10−4 ,
BiDFEs provide 1 ∼ 2 dB performance gain over a single DFE at both 2 and 3 Gbit/s.
To improve the performance of BAD and TBCR, a novel trellis-based algorithm – TBBiDFE – was presented. By the use of a trellis diagram, TB-BiDFE is able to generate
more reconstructions for the received sequence than BAD and TBCR. For a 3 Gbit/s
transmission using fractionally-spaced DFEs, TB-BiDFE outperforms BAD and TBCR
by 1 dB at BER = 10−6 while having a moderate complexity.
CHAPTER 6
Conlusions
Looking ahead into the future, an automobile physical layer should be able to support data transmissions at multi-Gbit/s, in order to meet the daily increasing demand
of the in-car infotainment applications. Thus, this dissertation intends to increase the
transmission rate of the up-to-date MOST150 automobile infotainment backbone from
150 Mbit/s to 3 Gbit/s, by means of cost-effective electronic signal processing techniques.
Section 6.1 first summarizes the main discoveries of this dissertation, and weights the
advantages and disadvantages for each proposed transmission strategy with respect to
computational complexity, power consumption and BER performance. Then, some future research directions are pointed out in 6.2.
6.1
Contribution Summary
In order to achieve the target BER of 10−9 for an almost “error free” digital transmission, this dissertation first recommends the commercially widely used Reed-Solomon
(255, 223) code to prevent the use of complex equalizers or modulation schemes. Instead, RS coding is combined with some less complex equalization strategies to achieve
the target BER within the available SNR budget assessed at the receiver.
Second, with the intent to increase the transmission bandwidth efficiency, MPAM is
suggested owing to its simplicity, popularity and suitability for light intensity modulation. Through the comparison of 4, 8 and 16PAM, 8PAM is decided as the most
appropriate modulation scheme. By use of 8PAM, the desired symbol rate is kept down
to 1 Gsymbol/s and the bandwidth efficiency is increased by 3 times.
109
110
CHAPTER 6 Conlusions
The third contribution of this dissertation is the development of various electronic equalizers which are used to combat strong ISI and noise during the high-speed transmission.
To be more specific, four transmission strategies based on different equalization techniques are suggested respectively, in order to provide performance and complexity alternatives.
The first transmission strategy investigates the conventional feedforward and decision
feedback equalizers with either symbol-spaced or fractional-spaced structure. The simulation results show that both sub-optimal equalizers with a fractionally-spaced structure are sufficient for a 2 Gbit/s transmission; whereas at 3 Gbit/s, DFE presents a remarkable improvement over the linear FFE with only a slightly increased filter length.
Therefore, a single post DFE is overall an adequate equalizer when the SNR on the
channel is good and a simple receiver is required.
The second transmission strategy proposes the use of a prefilter at the transmitter to
share the burden of removing ISI with a post-equalizer. The result demonstrates that the
target BER is reached with a noticeable lower value of SNR. Note that unlike in other
transmission strategies, the received signal now contains an extra DC which carries no
information. The DC is introduced by the prefilter and it largely increases the dynamic
range of the received signal. Correspondingly, the power constraint on the received
signal power must be converted into a constraint on the actual signal power. When this
is done, it turns out that the target BER is reached with nearly the maximum available
power budget. That means, the pre- or post-equalization is unfortunately only suitable
for the situation where there is an adequate power reserve.
Design of the third transmission strategy takes into consideration that the structure of
the MOST network is a two-fiber bidirectional optical ring, i.e., each transceiver is integrated with an up- and a down-link. By taking advantage of this feature, the non-linear
Tomlinson-Harashima precoding technique, which is a more energy-efficient technique
than the pre-filtering, is studied. Together with an adaptive FFE at the receiver, the
THP-FFE transmission scheme is able to provide a SNR gain of 1 to 2 dB over the
DFE at 3 Gbit/s. The system is also proven to be robust against moderate transmission
variations or the mismatch of THP coefficients. Critics of THP-FFE comparing to DFE
cite the slightly increased complexity and a necessary additional up-link.
The BiDFE technique is the last investigated transmission strategy. From the simulation
results, it can be seen that BiDFEs have the ability to largely reduce the error propagation effect in a single DFE. Benefiting from that, all investigated BiDFEs successfully
CHAPTER 6 Conlusions
111
achieve a 2 dB SNR gain at 3 Gbit/s over the conventional DFE, while the computational complexity is increased only in a linear fashion. Consequently, BiDFE offers the
largest SNR margin among all considered transmission schemes.
Last but not least, one of the key contributions of this dissertation is also presented in
the last chapter: the novel trellis-based BiDFE (TB-BiDFE). An assessment of the novel
TB-BiDFE regarding the BER performance and complexity shows that TB-BiDFE is
superior to BAD and TBCR, as well as being moderate complex. For the considered
example, TB-BiDFE reaches the target BER with 0.5 dB less SNR than BAD, however
the required computational load is comparable to that of BAD.
To conclude, by applying one of these transmission systems in the physical layer, the
next MOST infotainment backbone may finally reach the 3 Gbit/s transmission rate in
a vehicle; whereby it is able to support not only the traditional automobile infotainment applications like transporting high quality videos, but also new domains such as
supporting the fast in-car Ethernet.
6.2
Future Diretions
The popularity of POF is not just confined to the automobile industry. In many other
fields like access networks for home networking, internet protocol television (IPTV),
aerospace and aircraft [62], the development of POF based optical system for highspeed communications purpose are of great interest to researchers. Because very often
the POF based optical links share many physical characteristics in common, concepts
and solutions presented by this dissertation can certainly be used by a broad variety of
other POF applications, such as those mentioned above, with only minor variations.
Appendix A
Minimum-phase Spetral Fatorization
If a folded spectrum Sh (ej2πf T ) satisfies the Paley-Wiener condition, i.e., if Sh (ej2πf T )
and log Sh (ej2πf T ) are both integrable on the interval (−π/T, π/T ], the spectral factorization of Sh (z) can then be established in the following way:
Sh (z) = A2h · G(z) · G∗ (1/z ∗ ),
(A.1)
where Ah is a real-valued constant, G(z) is causal, monic and minimum-phase, i.e.,
G(z) = 1 +
+∞
X
g[k]z −k ,
k=1
and G∗ (1/z ∗ ) is anti-causal, monic and maximum-phase [40].
Any rational folded spectrum, fortunately, can vanish at only a finite set of points on the
unit circle (at most the number of zeros). Due to the fact that a rational folded spectrum
does not have zero magnitude over a finite band of frequencies, its logarithm is always
integrable. For this reason, the spectral factorization is always possible for a rational
folded spectrum without poles on the unit circle [39].
113
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& military aircraft,” tech. rep., Fiber Optics Subcommittee, 2009.

Digital Pre- and Post-Equalizers for In

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