Seriall oncatenated Conti4nuous Pase Modulation wcrit n HCtdn *

Seriall
oncatenated Conti4nuous Pase Modulation
n
H
wcrit
e
*Ctdn
es
Alexandre Graell i Amat, Charbel Abdel Nour, and Catherine Douillard
GET/ENST Bretagne, Electronics Department, CS 83818 - 29238 Brest Cedex 3, France.
email: {alexandre.graell,charbel.abdelnourcatherine.douillard}@enst-bretagnefr
In spite of the wide literature on concatenated CPM systems,
solutions for practical applications are still to be proposed.
Indeed, most of these previous works are aimed, at optimizing
the outer FEC to improve the power efficiency of a given
from 0.75 to 2.25 bit/s/Hz. A two-step design procedure combining the oute
to approac he power eepcomprehen
ile this approach has provided a deep comprehenEXIT charts analysis and union bound techniques is used to CPM.
optimize the association of the outer code and the CPM. An sion of concatenated CPM schemes, it strongly limits the
exhaustive study of several quaternary and octal CPM schemes consequences of the analysis, since it is limited to particular
is performed.
The proposed concatenated structure offers very low err?r cases, which cannot be in general compared. This prevents
floors (frame error rate below 10-6) and good performance in rom extracting strong conclusions on the performance of
the waterfall region for all spectral efficiencies. A significant im- concatenated CPM systems, and hence it is not suited to
provement with respect to previous concatenated CPM schemes provide solutions for real implementations. On the contrary,
is shown. The envisaged application of the proposed scheme is for practical purposes, the approach should be completely
the return link of broadband satellite communications.
different: given a spectral efficiency, one would be interested
in finding the best association of outer FEC code and CPM
LT
DUC'TION
1I- TROn
modulation.
Since the seminal paper by Aulin et. al. [1], Continuous
The main goal of this paper is to fill the existing gap in the
Phase Modulation (CPM) has attracted an intensive research. literature and give a suitable answer to this question. To the
CPM is a bandwidth and, energy efficient digital modulation best of our knowledge none of the previous works formally
scheme with constant envelope. This characteristic makes it addressed these issues, which are of primary importance for
particularly interesting for band-limited systems employing actual implementation. Our work is strongly motivated, by real
non-linear power amplifiers, such as satellite communication implementation. In particular, this paper focuses on the return
systems.
link of broadband satellite transmissions, where very low error
Shortly after the invention of CPM, convolutional encoded rates (frame error rate below 10-6) are required.
CPM was addressed by many researchers as a simple way to
A serially concatenated CPM structure using simple eximprove power and, bandwidth efficiency while maintaining tended BCH (eBCH) codes as outer code is proposed, tarthe constant envelope 12]. After the breakthrough of Turbo geting a wide choice of spectral efficiencies, ranging from
Codes and iterative decoding techniques, it seemed natural to 0.75 to 2.25 bit/s/Hz. For each spectral efficiency, the outer
apply the same principles to coded CPM. Indeed, CPM is code rate and, CPM parameters are optimized by combining
equivalent to a recursive convolutional encoder (known as the Extrinsic Information Transfer (EXIT) charts and union bound
continuous-phase encoder) followed by a memoryless mapper techniques. Compared to SCCPM in [5,9], lower floors are
[3], hence it can be regarded, as the inner encoder in a serial achieved thanks to the good distance properties of the outer
concatenation. Therefore, iterative decoding/demodulation can block code. The proposed concatenated scheme offers frame
be performed between the outer code and the CPM.
error rates below 10-6 for short block lengths and good
Several papers addressed the concatenation of a Forward Er- convergence behavior for all spectral efficiencies.
ror Correcting Code (FEC) with CPM [4-6]. In [5] a detailed
1. SYSTEm DESCRIPTION
analysis of the serial concatenation of outer convolutional code
and inlner CP:M: (SCC:PM:) is given. Upper bound.s to the error A. Decomposition? of CPM
probabilit and.d.esign criteria are d.erived ................
The analysis inl
.....
We colnsider a C:P:M system with M-arzy symbo:ls wi
[5] was extended in 16] to convo:lutiolna:l codes over rings ..l,...., zl(M-1)} transmitted everysymbol interval Twith
and symrbolL interleaving, improving performance :mainlLy in the energy Es. The CPM signal can be described as [1L]:
stw
E/cs2ft
(,)
o
1
waterfall (WE) region.
Abstract-In this paper, serially concatenated Continuous
Phase Modulation (CPM) is considered. A concatenated structure
consisting of a short extended BCII code as outer code is
proposed, targeting a wide choice of spectral efficiencies, ranging
T
Broadband Satelllite Digitall Transmissionl (BSDT), contract number 19370.
A. Graell i Amat is supported by a M[arie Curie Intra-European Eellowsbip
within the 6th European Communait Eramework Programme.
1-4244-1200-5/07/$25.00 ©)2007 IEEE.
phase shiLft, alnd the sequ.ence of ifI-ar symbolLs vi, respectively, anld (t, w) is the inlformation carring phase, given
by
lKi^nD0/1+XTl,ith'.~ ~ ~ ~Iteraiv cer
__
(t, w)
(2)
wf (t- iT)
4 h
CPM modulator
JIv v
u
w
CPE
MM
I
s(t) AWGN
Q
i=O
--------------------------------------------------------------------------CPMdemodulatoreC
(t) is the phase pulse, which is the integral of a positive
normalized frequency pulse g(t). Parameter h
q/p is a
Filtering
Demapper
ttA
th
th
1
rational number and it is referred as the modulation index.
In [3] it was shown that the transmitted signal does not
change if (t, w) is substituted in (1) by the so-called tilted
«r+T
rnT, w)
L- I
47h
X
2wh
[
[n-L
E
1
w mod
+
(3)
j0
whe referreLdis the phasfullrespuselystengt S hisystems withL
I are referred to as partfal response systems. with
Refeed toasine(RC)
reystem osiHer
LeL >>on1iare
pandSpectral
idr
Cosine
Ri
et
Raies edf
(we
Cy
pulses
g(t)
(SRtC famlisC fLSRe
of
c
with L
{2 3
denotedhas tlRCtad LhasR,the trenspectivelysignalinthenth
tras
ed
thethe
tte signal th th
current
interval tS tilt ps
istauniquely defined-by
dataasymbolaccuthe
interval
M
phase (t, ),
Fig. 1. Transmitter
is
Hard
Iteative recierF
ith-------------------
and receiver for the serial concatenation of extended
BCH code and CPM.
L,,j
T)] mod2 o < <T
wn f(T
r(t)
previous daIta s ymbolsV Wtn-L w, modwp. and theaCcuMlated phase state On
_n W mod p. Therefore, the CPM
modulator can be decomposed into the concatenation of a
continuous-phase encoder (CPE) and a memoryless modulator
At the receiver side iterative decoding/demodulation is
applied. The decoder consists of two Soft-input Soft-output
t(SISO) decoders 'D0 and Di matched to the outer code and
to the CPM modulator, respectively. Note that if h = q/p,
the CPE can be represented by a trellis with p_AIL-l trellis
states. Therefore, APP decoding can be applied, provided that
sufficient statistics are available. In this paper, we assume
optimal detection for the CPM signal. To d.ecod.e the eBCH
code the Chase-Pynd.iah algorithm was used [711 It is a wellknown 5150 suboptimal algorithm for block codes with one
of the best trade-off between complexity and performance,
especially for low error correction capability t (typically 1 or
2) where it is nearly optimal.
III. PERFORMANCE ANALYSIS AND SYSTEM
OPTIMIZATION
(MM) [3].
Most of the previous works on concatenated CPM systems
focus on the optimization of the outer FEC given a particular
B. SeriallyConcatenated System
CPM system. To the best of our knowledge, only [5] provides
We consider the serial concatenated system shown in Fig. 1. some insight into the comparison of CPM system performance
The information data u of length K bit is divided into by fixing bandwidth efficiency. However, the analysis in [5] is
subblocks of k bit and encoded by an (n, k, d,in) eBCH linear far from being exhaustive since only few CPM schemes are
code in systematic form, CO. In this paper, a short double-error- compared, and the study is limited to very low efficiencies.
correcting eBCH code, the (64, 51, 6) eBCH code, will be Indeed, quoting [4], the following issue still remains an open
considered for Co, since it realizes a good compromise between question: "Which is the optimal association of coding and
decoding complexity and code performance. To achieve differ- CPM for a given bandwidth efficiency and decoder complexent code rates, shortening is applied. The proposed structure ity?". In this Section, we try to give a suitable answer to this
may be extended to other systematic block codes or to a family question. With the spectral efficiency (SE) as the reference
of eBCH codes in a straightforward manner. If K is not a parameter, we analyze several concatenated CPM systems and
multiple of k, some few dummy bits are added to the data propose a suitable scheme giving the best performance for a
sequence to adjust its length to the eBCH code. Alternatively, wide range of spectral efficiencies.
the eBCH code encoding the last subblock of information bits
Several criteria could be adopted to select a particular
may be properly shortened to adapt to the information block CPM scheme rather than another one. For instance, the CPM
size K. The two solutions entail only a small rate loss. J parameters could be chosen based on capacity considerations,
codewords of the outer code are concatenated to form the outer which is the classical approach for linear modulations, where,
codeword v of length N = J, which is permuted to v' by in general, best systelms are those achieving higher spectralL
the interleaver lrI workinlg at bit level. The binar sequenace efficiencies for a given signal to noise ratio (4 ,No). fIn
v'is mapped on to the sequence wv of Alt-ar symbols wi, Fig. 2 we repor the capacit curves (expressed in bit/s/Hlz)
and fed to the CPE, Ci. We assulme N to be a lmultiplLe of for severalL quaternary 2RC (Q2RC), quaternary 3RC (Q3RC),
log2 (M), where AlX is the alphabet size of the CPM system. and quaternary 3SRC (Q3SRC) CPM. For instance, for SE
FinallLy, the sequence at the output of the CPE is mapped to 0.9 bit/s/Hz one may choose Q3RC with lmodulation index
form the modulated signal s(t) and transmitted over an AWGN h =2/7, which gives the highest capaciy lHowever, severalL
chanlnel.
CPMV schemes are inl general competitive for a givenl SE.
-V-Q2RC,h=2IS5
Q2RC, h=317
2.25
Q2RC, h=2/7
Q3RC,h=2/7
Q3RC, h=115
015
D
Q3SRC, h=217/5-
709
---
--
Q2RC, h =2/5, Ro
-OP
--
-
--
--
115~~~~~~~~~~~~~~~~~~~~~~~~~~~3R,h2/,R
0.5,EbINo
3SRC,h=16
Q3SRC h
1.
-
10
1-
0~ ~ ~ ~ ~ ~ OA01-0
i
i//z
31 dB
-
03-04-----------05 ------06
----O-----
.7
aaicre
---
--7.
10500511522533544555566577588.599.510
Fi
0.8 EbINO 1.9dB
o evrlCMshms
i.3EI
thegoute FCandit thres (inne CPMs Thereforsee,aCPM schemes.
wIth bettiper capacitymaypserfom worse tan poorerln schiemesn
h
rates we uEXI
A--C
L
----
-----07
ortoCMsseswt
carto
two-se
08----09----
--(C-
1-
----
E
09btsH
aPprahFirtestwitE rel on9 EXITHz
cha analysise toinoptimizeperformance[inItheUWEoandntheny
evrhees
when concatenated,wtnue
inrterMs ofcptiiacity
FEC. In owe deseibondcingteciihnique to d,ensurelowFloors.ngeea
thi paper, insEad
cofe uhsin capacritnmyconsideratieonstoeidel ine
Lometinrather
andanEcompldenoteihnpirgn.heetrni
as
the actual performance mute ntual informateiong(Ifourgolthe Fcntrcde Inyasiemsilar way,
favoreof paiscussdlarCP,ter onsie
cofcathecncatenasyted schemedped,o heasoitino letrfACrmandeih WF
deniote thepior sandsextrinsi MIr for tero
heeoe
Ahe SptecraFEffCiency ofnerCM CPMMchm
CPraes, weuc an two-st canpberound. viastseparaelMont Carlo
in trms f cpaciy wen cncaenatd wih a outr FC.
simeulatioundofnth twoconiustituentscoes for dffeorens vleso
compteatio of then cpowery cospderaltDensit (PSD)e of Lb/No IE(C,
thipaer
TI(Ac)
and.I( dntthpE(C ioTr(ancd. Lb/etrno),
a
CPMo sinlsi
tanskdIn this paculper,feormnsic wee
de ote the EXI fuctodeIns of CimlawandC,
patcolrCMplwex
m tua ifrandtio(I
ofthe compuationamtho n[]aed.onstehautcorreation
Assumin infinItE ientereaes the ror nv exrgensc threhl
M o ea
E. prEdict
adbyplottingtheEXITd
cuvisespfratCMondeCarilth
CPM
ofsinPs
funSpctionHere thebandidthis
Efiie
dfine as 9cc n bad poer, siuametiagram. tocosiuntcds o ifeetvauso
the
equvaen
-20dB
weve in ecthfractionalioty ofSDbandO E(,
copto tono
uI(C) n E(,
j1(C
bN
equvalnt ote -0
d
leelin he racionl ot
o
ind
power function [8]. Since the capacity (when expressed in
bit/channel use) is upper bounded by log2 Al, the SE of coded
CPM can be approximated by
IOP_____2M
SE-.
R
0
l,,T
i//z(
where B, is the normalized double-sided, uncoded bandwidth,
and R0 k/in is the rate of the FEC code. From (4) it arises
that the SE of a concatenated CPM is completely defined by
the CPM parameters and R0. More precisely, given M, L, and
the pulse tpe, the efficiency is defined by the pair {h, R0}.
B. CPM parameters and code rate optimization
In Fig. 3 we plot the EXIT curves for two concatenated
CPM schemes with SE =0.9 bit/s/Hz. The outer code is the
(64
6) eBCH code. Gray Mapping is assumed. The solid
curves
to Q3RC with modulation index h =2/7
and. code rate R= 0.5. It is the best scheme in terms of
capacity, C 0.55 bit/s/Hz. The tunnel between the two
curves opens at Lb/No =3.1 dB, indicating a convergence
threshold around this value. Remarkably, the convergence
threshold can be significantly improved if Q2RC and h =2/5
is used (dashed curves). The outer code rate must now be
raised. to R =0.8 to achieve the same SE. The convergence
threshold is now at Lb/No =1.9 dB, i.e 1.2 dB earlier.
The superior performance of the second CPM scheme may
r51,
correspond,
=
TABLE I
BEST CPM PARAMETERS AND CODE RATE ASsOCIATION
h
SE
ModulatiDn
Threshold
R.
Q2RC
1.4 dB
0.75
3/7 0.7
0.9
1.0
1.25
1.45
1.75
2.00
___
2.25
Q2RC
2/5
2/7
Q3RC
2/7
1/5
1/6
Q2RC
Q3RC
Q3RC
Q3RC
Q3RC
2/7
0.8
0.68
0.69
0.8
0. 76
0.77
Di__
1/7
0.79
Therefore, a concatenated system with a single outer code and
bit interleaving was assumed.
The EF performance of concatenated CPM systems can be
analyzed by using union bound techniques for SCCCs [11I].
The frame error rate (FER), Pw (e), is upper bounded by:
1.8 dB
2.2 dB
3.4 dB
4.2 dB
PW (e
6.7 dB
8.3 dB
9.9 dB
then dictate its rate. Therefore, it may happen that for some
CPM schemes the matching of the two EXIT curves is worse
than for others, leading to poorer convergence thresholds.
It is worth to point out that better matching between the
two curves might be obtained by using other outer codes, such
as codes over rings and symhol interlLeaving 16] or irregular
repeat accumulate (IRA) codes 19]. Unfortunately, such codes
show much higher error floors for the SE of interest, due to a
poor minimum distance. Also, the codes in [6] assume Ml1
which is in not in general a good choice.
EXIT charts have been used to find the hest CPM schemes
in terms of convergence threshold for a wide range of spectral
efficiencies. Q2RC, Q3RC and Q3SRC CPM schemes have
heen analyzed. For Q3SRC we assumed a roll-off factor
1. We also analyzed octal CPM schemes, but the resulting
convergence thresholds were poorer. In Table I we report the
best association of CPM parameters and outer code rate for
for several spectral efficiencies ranging from 0.75 to 2.25
hit/s/Hz. Here, with best we mean the scheme with the hest
convergence threshold yet allowing low floors (below 10-6,
as required for broadband transmission). In all cases Gray
mapping is assumed, since it shows earlier convergence than
Natural mapping (0.3-0.6 dB gain).
From Table I three main conclusions can be drawn:
* For low spectral efficiencies (< 1.2 bit/s/Hz) Q2RC CPM
should be preferred.
* For high spectral efficiencies (>1.2 hit/s/Hz) Q3RC CPM
give the best performance in the WF.
. To achieve good convergence thresholds, a high-rate outer
code is required. This is a crucial issue when selecting
the outer code.
=
C. The choice of the outer code
The choice of a single outer code for the optimization of the
CPM parameters and the cod.e rate in Table I was not casual.
Indeed, several outer codes could be concatenated with CPM,
such as single binary codes (convolutional or block codes),
convolutional codes (CC) over rings or Turho-like codes.
Among them, the choice should he hased on performance
consid.erations in both the W:F and the fEF region. The u.se
of concatenated codes (eg. Turho Codes, LDPC codes) can
he discarded hased on performance in the WF region, since
they entail a penalLty in termrs of decoding conve:rgence. This is
a common hbehavior for three-fold concatenations. Also, CCs
over rings or binar CCs with symbol inter:leaving show high
error floors and they are not well su.ited. for very high-rates.
1
-
<
B
R
(5)
D
where Eb is the energy per information bit and BD the
number of codewords with normalized Euclidean distance D
on the channel. BD can be computed from the output weight
enumerating function (VTEF) of the eBCH code and the inputoutput weight enumerating function (JOWEF) of the CPM [5,
11]. As for standard SCCCs, the interleaving gain is dictated
by the outer code minimum distance dd,
- o-L d
Furthermore, the main term contributing to the EF increases
with do. Therefore, an outer code with high minimum distance
is preferable.
In this paper, we propose a concatenated structure using
shor eBCH codes as outer code. This scheme is inspired hy
the outstanding performance of serial concatenation of eBCH
outer cod.e and, inner accumulator [12]. Here, the CPM can
be regarded as the accumulator of the serially concatenated
structure. The use of eBCH codes has several advantages with
respect to CCs commonly used in the literature [4, 5]. First,
they are excellent codes for high-rates, with much better dmin
than CCs of similar complexit In fact, CCs are well suited
for low rates. However, for very high ones (which are required
according to the previous Section) CCs show a significant
degradation of the EF due to the heavy puncturing of the
code, which leads to poor d,i,. For instance, for rate 0.8, the
16-state and 32-state codes have drni,, 3 and 4, respectively.
Furthermore, the 32-state CC leads to a significant degradation
of the convergence threshold. Second, eBCHl codes show ver
good convergence properties when concatenated with an inner
accumulator [ 12].
To lower the error floors of the codes proposed in literature,
while still achieving good convergence thresholds, we consider
a short double-error-correcting eBCH code, the (64, 51, 6)
-3 is
code, as outer code. A high interleaver gain a
achieved thanks to its large dniii1 Code rate flexibility is
achieved, by shortening the eBCH code. Note that by applying
shortening, the multiplicity of the error events increases, but
dmin does not change, hence the interleaver gain is preserved.
The proposed structure may be easily extended to a family of
eBCH codes to further increase rate-flexibility. Note that the
complexity of the (64, 51, 6) eBCH code, while decoded using
the Chase-Pyndiah algorithmr, is rather low, comparahlLe to a
1L6-state CC in termns of numher of logic gates.
In the paper, we used union hounds [5, 11L] comhuined with
the EXIT charts in the previous section to optimize the
concatenated CPM structure.
=
IV. SIMLULATION RESULTS
fI:n this Section, the performance of the proposed concatenated structure is valid.ated throu.gh simulationl. For all
4-state CC
--
-------------------4-state
16-state--Rieg--CC-[6]-2RC
0- ~~~~~~~~~~~(64,51)eBCH
102
10-2_02RC
0-
h=3/7
~
-------
--------C C----[5]-
-0
::::
4-
state
[5]
(64,51) eBCH
0--
10~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~------
10
0--
10-3
~~~~~
[5]
--
.A
......
--------------
------3 ------4 45 5----
55
-----6 ---6.5
7
7.5
10 3
8-----0
1
2
3-- 4--5-6-7-8-9 10-------------------
11---- 12-------- 13------- 14----
--
Fig-----4---FEperor ane-o-srill-co caente CPM------for------spectral--------efficincy-Fi-5-FE pefrAnce------- of-----serially--------concatenated------- CPM-or-pectal -fficenc
0. and
1.45
bit/s/H z--------0.7
and
2.0
bit/s/H
z.
simulation results we assumed block length K
12 2 proposed tagtn the-retu- link------of-- broadband -satellit
bit,and
ando tatwhn-sin-a-etede
intrleaing.30 ieraionswereconsdere.
wid----2--25bit/s/Hz.--------rane ofspecral
fficencis i
However,
noteB:3SCH code7 a comuniatios.
covered,- from--- 075 ----to
By-combiing-EXI
sim
plestopping
-be----cha-----analysis
---and---union -------bound -------techniques--------- -the-----association------------- ofcriterion,----- ----------------------based -------on - the --syndrom e,----------may--applied,thus--- reducing---- the average-number -o
FEC----code-----andCPM-----parameters-- was----------optimized--------------for------eve--iterations--------- the----InFig.4 we show ~ ~ ~ ~ ~ _tS_performance-------of------the-----proposed----struc-------spectral---efficiency--S
ignificantly---lower--error--floors
--with -respecth
e
turefor spectral- efiie ce
0.9---and--145--bit/s/Hz.---The--best--to--structures----proposed---in--literature---are--achieved---thanks--to--the
a
re---considered------The--set--of--curves----good--distance----prope-----es--of-BCH--codes,----while
---a-limited----lossconfigurations------ of--- Table----onthe left-------- corresond-toQ2RC- 2<,-whil-the-setof-in-covergenc is------------------- shown with------- respect --tosymbol--------- interleaved-------curves on the~~~~~~------2/7.-----------Inbth srutre.Th-rooedsrutreofesfrm-err-ae
rIghtco-rateeson to_Q3C
case thouter----------- code- - ---------is ---0-8.- Very--low-error-floors--below10-----------------for - -shbolock------lengths------and-- good-------------convergence------------------------are observed for~~-------------- the
propose -schem lso,----- note- that---------- if- - an-----thresholds--------- for all
spectral---------------efficiencies.-------------------optimized- interleaver-is used,---------- the
-----error----------floor----------would----------be------pushed-------- -----------ACK LED EM EN TS
down.
For comparison~~~~~~---------purposes--------we-als-repo in-th-figue
Theauthos-woud-lik to-acnowldge-D-Delruell-fro
the performance for the codes in [5] and [6] as well as for NEWTEC for providing the capacity curves of Fig.2.- - - -state C--------- and-------------------\
32-state--------CC .---For----all----curves------bit---interleaving--------isxet ohecd6i6 whr sybo inelevn
REFERENCE
asue,0
is used, The------ codes------ in [6 have- been-punctured-to-achie
C.--E.--Sundberg----Continuous--- Phase--------------------Mod-----------------a------the-[1]-T--Aulin-N.-Rydbec and-------------------desired coderates. To this scope~~~~~~~~~~~~~-------- optim ized--puncturings---have-tion
--CPM)----Pa--and--Part-------E---Trans---Comm--vol. COM--------------------- ---------2------- no------been found.--------The------proposed----structuresinificanty-lower
3---19-225-Ma.-1981
shows--Aulin-an C.-------------------------------S---ndbe----g
-Phaebi-[]- B.--Anderson
T.-------than-----------------with
oltina-o te-coe-nd
floors~~~~~~~~~-----N
ew-------York:--------Plenum---1986.--interleaving,while
marginal or no performance degradation in [3] B. Rimoldi 'A Decomposition Approach to CPM JEFF hans. Jnf The--------------------------------------
----
----------------------------
-------------
the
schemes
con
B.
useofsmbo
5
inelevn.5 For SE 1.45 th gain is large (0.
dB). HowFEverfothese codesillshowncaepatableCP Ef,r duetoa
thiceir
poor
adis45btance z.
49 no 4 pp 67-8 Apr 2001.
Digitl
Mod
latio
12131
MoqispefrandcTMfAuin'erially Concatenated
CPfonpetinuousfPhasenc
011n pp.0190-195s
NvHz 001
[5].
Fg
5
simlario results areobtainedinFg flork
M.72Xiaoosd tandet.iMgAthi SetriallyiCncateate CrontianuousPaselModspecgtra eficecis=6
0.75 and2.0rbit/s/Hz.eTheavn.3
ulationWicthionvolutwionlCdesOvnero RnspeEEra Tracins.commun.
prpoedatconcatenaed structured
cvre.,Mro
[7
Pyndah 5earoptimum Decoding of Prdctode:blockg Turbo
sligHt
loss
sHowsevery loweta
errorfloosingatnh expensedo
[8]rGaLndellsi OnCdedCnontinousdPehaseMoulatioheD disseiationo
Univ.Cof Lundeer Swede 1985.e orevr
thsredcing he avrage umberof itratios.DeptTECcommd her
applid,
such cofiurtin
e[9]aMXfiaoiandcy. AuignfcntIrglarRoepeatrontinuours PihasesModuca
show
yhefomnepoorteF anoosd.atre not
c
waed wouldh likedtomeempasize
siprooedstpinglitrature.ioweve
tha
thy
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