Reduced state adaptive SISO algorithms for serially concatenated
Reduced State Adaptive SISO Algorithms for Serially Concatenated CPM over
Frequency-Selective Fading Channels
Kyuhyuk Chung, Keith M. Chugg, and Jun Heo
Electrical Engineering–Systems Dept., University of Southern California, Los Angeles, CA 90089-2565
Abstract — Iterative detection (ID) has proven to be a near-optimal solution for concatenated Finite State Machines (FSMs) with interleavers
over an additive white Gaussian noise (AWGN) channel. When perfect
channel state information (CSI) is not available at the receiver, an adaptive ID (AID) scheme is required to deal with the unknown, and possibly time-varying parameters. The basic building block for ID or AID is
the soft-input soft-output (SISO) or adaptive SISO (A-SISO) module. The
complexity of SISOs and A-SISOs increases exponentially with constraint
length. In this paper, Reduced State SISO (RS-SISO) algorithms have been
applied for complexity reduction of the A-SISO module. We show that serially concatenated CPM (SCCPM) with AID has turbo-like performance
over fading ISI channels and also RS-A-SISO systems have large iteration
gains. Various design options for RS-A-SISO algorithms are evaluated and
performance and complexity is compared.
I. I NTRODUCTION
Iterative detection (ID) or turbo decoding is a good approximation to the optimal decoder for a concatenated network of Finite State Machines (FSMs) constructed from constituent codes
and intersymbol interference (ISI) channel memory.
In , Trellis Coded CPM (TCCPM) on multipath fading ISI
channels was studied without iterative detection. It is shown
in  that by exploiting the recursive nature of CPM, TCCPM
can have large interleaver gains similar to Serially Concatenated
Convolutional Codes (SCCCs) . Similar independent work
with iterative detection was reported in . None of papers
above considers SCCPM  over fading ISI channels with iterative detection.
Adaptive SISO (A-SISO) modules  were developed as a
natural extension of Soft-Input Soft-Output (SISO) modules 
for use when perfect CSI is not available at the receiver. In ,
various A-SISO algorithms were defined and derived, which
produces a family of suboptimal practical algorithms.
Since the complexity of SISO algorithms increases exponentially with constraint length, Reduced State (forward-backward)
SISO (RS-SISO) algorithms were proposed independently for
complexity reduction at the same time in  .
In this paper, the RS-SISO algorithm is used for complexity reduction of the A-SISO module, namely RS-A-SISO algorithms. RS-A-SISO modules are applied to SCCPM over
frequency-selective fading channels when perfect CSI is not
available and trade-offs between performance and complexity
are considered. RS-SISO algorithms are of paramount importance in A-SISO algorithms because the complexity of A-SISO
algorithms is higher than that of SISO algorithms due to channel estimation. When the input of a joint CPM-ISI trellis is
binary, multiple-step RS-A-SISO algorithms  can be a design option for improved performance with slightly increased
This paper organized as follows. Section II describes the system and channel model. In section III we explain various RSA-SISO algorithms. Section IV presents numerical results and
Section V concludes the paper.
II. S YSTEM D ESCRIPTION AND C HANNEL M ODEL
A block diagram of SCCPM on an ISI and AWGN channel
is shown in Figure 1. After the source bits b n are fed to a convolutional code (CC), the coded bits c m are bit-interleaved and
continuous-phase-modulated. The modulated signals a(t) are
transmitted through an ISI and AWGN channel. The carrier
phase is modulated by a positive normalized frequency pulse
g (t), containing no impulses and being non-zero for L symbol intervals . Its corresponding phase pulse q (t) is the integral of g (t), normalized to q (LT ) = 1=2 with a symbol time
T . The modulation index is assumed to be rational and irreducible, h = K=P , and then the CPM modulator can be viewed
as a continuous-phase encoder (CPE) followed by a memoryless modulator (MM) . For practical purposes, we can assume that W is the bandwidth occupied by the real bandpass
CPM signal. Then the band occupancy of the complex baseband
equivalent lowpass CPM signal is jf j W=2. A tapped-delayline (TDL) model is usually used for a time-varying frequencyselective fading channel The lowpass impulse response for the
f (t; ) =
fi (t)Æ (
where Tc is the channel resolution which satisfies T c 1=W ,
fi (t) = Tc f (t; iTc) is the tap weight coefficient, and (N c +1) is
the length of the channel in terms of T c . Tc is chosen such that
symbol time T = Ns Tc with Ns being an integer. In complex
baseband representation, the received continuous-time signal is
x(t) + n(t)
0-7803-7206-9/01/$17.00 © 2001 IEEE
Fig. 1. System model.
f (t; )a(t
iTc ) + n(t)
where n(t) is the AWGN. Conventionally, the received signal
r(t) is passed through a matched-filter x ( t), followed by a
sampler and a noise whitening filter. But since CPM is a nonlinear modulation, a bank of matched-filters has to be implemented. In  a low-pass filter (LPF) with two-sided bandwidth
W is used, followed by fractionally-spaced sampling every T c .
Then the k th sample of r(t) is
fi (kTc )a(kTc
iTc ) + n~ (kTc )
fi;k ak i + nk
~ (t) is filtered white noise and n k is complex zeromean Gaussian distributed with variance N 0 . The coefficient
ffi;k gNi=0c is a time-varying tap vector at time k and fa k i giN=0c
are fractionally-spaced signal samples during an ISI-channel
length (Nc + 1). The equation (3) can be used as an equivalent Discrete-Time (DT) model for CPM on fading ISI channel
The low-pass-filtered and sampled observations are fed to the
inner joint CPM-ISI A-SISO module . The outer CC SISO
 and the inner A-SISO are connected by an interleaver and a
deinterleaver to exchange soft information iteratively. The inner
A-SISO re-estimates the ISI channel each iteration based on .
III. R EDUCED S TATE A DAPTIVE SISO A LGORITHMS
The SISO module  implements an algorithm performing
an update of the reliability information of the input and the output symbols of a FSM. A-Posteriori Probability (APP) forwardbackward algorithms (FBAs) and Minimum Sequence Metric
(MSM) FBAs  produce soft information that is thresholded
to obtain the Maximum A-posteriori Probability (MAP) Symbol Detection and the MAP Sequence Detection respectively.
By replacing min-sum operations by sum-product operations in
the MSM FBA, one directly obtain the APP FBA. In the following we consider only the MSM FBA. MSM is defined by
MSM(u) = min[
1 (sk )
f(sk+1 ) = ^f(sk ) + [zk
f (sk )
a(tk )]a (tk )
where a(tk ) = fak i gN
i=0 are fractionally-spaced signal samples during an ISI-channel length (N c + 1), is a suitable constant, and the forward channel estimate is initialized by running
the Recursive Least Squares (RLS) algorithm on a training sequence.
RS-SISO algorithms also have two options, namely forward
decision feedback, backward decision feedback (FDF/BDF)
structures and forward decision feedback (FDF) structures 
. In the FDF/BDF structure, survivors are constructed both
in the forward and in the backward direction. In the FDF structure, survivors are constructed only in the forward direction, all
transition metrics calculated in the forward recursion are saved,
and the backward recursion is performed without survivors.
In this paper, the FDF structure is used. RS-SISO with the
FDF structure is consistent with the FA structure in A-SISO.
The forward and backward state metrics F(v k ) and B(vk ) are
defined for the RS-SISO algorithm recursively as
min [F(vk ) + Mk (k )]
min [B(vk+1 ) + Mk (k )]
where z is an observation sequence, u is any quantity derived
from the system input-output sequence pair (a; x(a)), and the
notation a : u means all a consistent with u. The forwardAdd Compare Select(ACS) and backward-ACS recursions in
the MSM algorithm are
MSMk0 (sk+1 )
where MSMkk12 (sk ) denotes soft information of a state at time k
based on observation z kk12 and Mk (tk ) is the transition metric of
When perfect CSI is not available at the receiver, the SISO
module should be able to deal with unknown, and possibly timevarying parameters. SISO modules that jointly estimate the parameters and generate soft information will be referred to as
Adaptive SISO (A-SISO) modules . When we construct ASISO algorithm whether Fixed-Interval (FI) or Fixed-Lag (FL),
two options are available, namely forward adaptive, backward
adaptive (FABA) structures and forward adaptive (FA) structures   . In the first one, adaptive processing is performed both in the forward and in the backward direction. In
the second approach, adaptive processing is only performed in
the forward direction and all transition metrics associated with
the forward adaptive step are saved for the non-adaptive backward recursion. In this paper, FA structure is used assuming
a training sequence only at the beginning of data blocks. The
Least Mean Square (LMS) algorithm is used for forward channel estimation.The LMS algorithm for updating a channel tap
vector ^f(sk ) in the forward recursion is given by
where vk is the truncated trellis state at time k and k =
is the corresponding truncated trellis transition at
time k . After executing both the forward and backward recursions, the soft outputs for a k can be obtained by
(vk ; vk+1 )
min [MSM0 (sk ) + Mk (tk )] (5)
min [MSMk+1 (sk+1 ) + Mk (tk )] (6)
min [F(vk ) + Mk (k )
s : survivor
x : non-survivor
o : channel estimation
Bit error rate
Fig. 2. Standard PSP and multiple-step PSP.
RS-SISO algorithms can be used in A-SISO algorithms to
produce soft-outputs for iterative detection with reduced complexity when perfect CSI is not available. These combined
SISO algorithms will be referred to as RS-A-SISO algorithms.
In RS-A-SISO algorithms the state-survivor paths selected for
updating channel tap vectors are used for calculating transition
metrics. In the following subsections various RS-A-SISO algorithms are described.
A. Multiple-Estimator (ME) RS-A-SISO Algorithm
In the ME-RS-A-SISO each state has a estimated channel
tap vector ^f(vk ) that is updated in the Per-Survivor Processing (PSP) manner . The reduced number of states means
reduced channel-estimation complexity as well as ACS complexity. For the calculation of the truncated trellis transition
metric both the estimated channel tap vector ^f(vk ) and the statesurvivor corresponding to the transition are required. The same
state-survivors are also used in updating channel tap vectors.
B. Single-Estimator (SE) RS-A-SISO Algorithm
In the SE-RS-A-SISO every state has the common estimated
channel tap vector ^fk that is updated employing the decisionfeedback (DF) assumption introduced in . For updating the
estimated channel tap vector ^fk the state with minimum metric
is first chosen, the state-survivor path is traced back as many
steps as tentative delay d, and then ^fk is updated with the truncated trellis transition k d that has the corresponding statesurvivor path. The single estimator structure reduces estimator
complexity more than the multiple estimator structure.
C OMPLEXITY C OMPARISON : S = NUMBER OF STATES , M- ARY INPUT
Number of transitions
E /N [dB]
Fig. 3. Performance of AWGN channel, perfect CSI, and multiple-estimator ASISO (d = 0:002). (For AWGN channel, the 8-state CPM SISO is used.
For perfect CSI and ME-A-SISO, 64 is full state and 16 or 8 are reduced
C. Multiple-Step (MS) RS-A-SISO Algorithm
The A-SISO algorithm based on a trellis in a fading channel is
suboptimal due to a forced-folding of a tree . But it is almost
impossible to use a tree structure A-SISO for detection of large
size sequences so that trellis structures are used in practice. The
MS-RS-A-SISO uses a tree structure during two trellis steps instead of a trellis structure . The channel adaptation scheme
in MS-RS-A-SISO is shown in Figure 2. The channel updates
are executed on all transitions instead of survivor paths in the
first step of each multiple step, which means that non-survivors
in standard trellis-based PSP can be a candidate for a survivor
depending on the decision of the next step. In the second step
of each multiple step, the channel updates are executed only on
survivor paths. Complexity increases slightly with a relatively
large performance gain. Complexity comparison is described in
Table I using the number of transitions and channel updates per
IV. N UMERICAL R ESULTS
In this section we present simulation results for the system
described in Section II. The outer convolutional code of rate 1=2
and constraint-length 4 is used. The code-bits of the convolutional encoder are bit-interleaved using an interleaver with size
1800 = 30 60. Each block of 60 interleaved code-bits with
an additional training sequence is referred as a burst. Bursts
are continuous-phase-modulated with an M = 2, L = 2-RC,
h = K=P = 3=4 partial response CPM scheme (L > 1), which
Bit error rate
Bit error rate
E /N [dB]
E /N [dB]
Fig. 4. Performance of single-estimator A-SISO and multiple-estimator ASISO (d = 0:002). (64 is full state and 16 or 8 are reduced states)
Fig. 5. Performance of multiple-step RS-A-SISO and multiple-estimator RSA-SISO (d = 0:002). (16 or 8 are reduced states)
has the following frequency pulse function
64-state joint trellis. The joint CPM-ISI FSM is recursive because the CPE FSM is recursive. The state of recursive FSM
includes the accumulation (memory) of all the previous input
symbols. When the number of states of the joint CPM-ISI FSM
is reduced, survivors must have the accumulated value. For every Eb =N0 we use fixed values of the step size = 0:011 and
= 0:04 for d = 0:002 and d = 0:01 respectively assuming
the receiver doesn’t estimate signal-to-noise ratio (SNR). The
optimized tentative delays found via simulation in the singleestimator scheme are d = 5 and d = 3 for d = 0:002 and
d = 0:01 respectively. In this section, each system was identified by the number of states in the joint CPM-ISI A-SISO, i.e.,
64 is full-state,
16 or 8 are reduced-state.
This figure also includes a perfect CSI case and an AWGN
channel case for comparison. For the AWGN channel, the 8state CPM-SISO is used. Note that ME-A-SISO algorithms
show steep slopes as an AWGN channel and a perfect CSI channel.
In Figure 4 the comparison of the adaptive iterative detection using SE-RS-A-SISO algorithms and ME-RS-A-SISO algorithms is shown for the fifth iteration. As the number of states
is reduced, the performance of the single estimator schemes becomes closer to that of the multiple estimator schemes. At low
Doppler spread with d = 0:002, the performance of SE-RSA-SISO algorithms is quite close to that of ME-RS-A-SISO
algorithms. It is shown in Figure 6, however, that for more
rapidly varying channels with d = 0:01 the SE-A-SISO with
64 full-states reaches an error floor above a BER 10 2 . At this
Doppler spread d = 0:01 no error floor was observed for the
ME-A-SISO with 64 full-states while the ME-RS-A-SISO with
g (t) =
where T is a symbol time. The CPM signals of each burst are
sent over Nc + 1 = 7 tap equal power Rayleigh fading channels
with maximum normalized Doppler spreads d = fd T = 0:002
and d = 0:01. The taps of the DT channel model are generated under the wide sense stationary, uncorrelated scatter (WSSUS) assumption from an independent Gaussian process with
the Clarke spectrum
Rf (m) = J0 (2d m)
where J0 () is the zero-order Bessel function of the first kind.
The channel resolution T c is chosen as T=2, which means
Ns = 2 samples per symbol 1 . Then the length of the channel
ISI in terms of the symbol time T is L c = bNc =Ns c + 1 = 4. In
Figure 3 the performance of the adaptive iterative detection using ME-A-SISO algorithms is shown for the fifth iteration over
the ISI channel with d = 0:002. If we decompose the CPM
into the CPE and the MM , then the CPE has P M L 1 = 8
states and the MM produces one of P M L = 16 signals. The
ISI channel can be described separately by (P M L )Lc 1 =
16 16 16 = 4096 states, but this number of states is infeasible for SISO algorithms. If we describe the CPM and the
ISI by a joint trellis, only P M L 1 M Lc 1 = 64 states
are needed. A joint CPM-ISI A-SISO is used based on the
1 The normalized double-sided bandwidth of this CPM signal is 1:165 .
This means that 2 samples per symbol are sufficient.
SE-A-SISO (64 states)
Bit error rate
Bit error rate
SCCPM with AID showed turbo-like performance over fading ISI channels. RS-A-SISO algorithms were proposed and
various design options for RS-A-SISO algorithms were evaluated for SCCPM over frequency-selective fading channels. RSA-SISO systems also showed large iteration gains. As the number of states used decreases, the performance of the SE-RS-ASISO algorithms approaches that of the ME-RS-A-SISO algorithms at low Doppler spread. For more rapidly varying channels full-state ME-A-SISO algorithms did not show an error
floor but full-state SE-A-SISO algorithms showed an error floor.
MS-RS-A-SISO algorithms can be a design option to improve
K. M. Chugg, A. Anastasopoulos, and X. Chen, Iterative Detection,
Kluwer Academic Press, 2001.
S. Benedetto, D. Divsalar, G. Montorsi and F. Pollara, “Soft-input softoutput modules for the construction and distributed iterative decoding of
code networks,” ETT, vol. 9, No. 2, Mar./Apr. 1998, pp. 155-172.
S. Benedetto, G. Montorsi, D. Divsalar and F. Pollara, “Serial concate-
Fig. 7. Performance of multiple-step RS-A-SISO and multiple-estimator RSA-SISO (d = 0:01). (16 or 8 are reduced states)
V. C ONCLUSIONS
16 reduced-states hits an error floor below a BER 8 10 4 .
The comparisons of MS-RS-A-SISO algorithms and ME-RSA-SISO algorithms are shown for the fifth iteration in Figure 5
and Figure 7 for d = 0:002 and d = 0:01 respectively. Note
that the gain of MS-RS-A-SISO algorithms over ME-RS-ASISO algorithms increases as the number of states is reduced.
The gains at BER 10 3 are 0.5 dB and 1 dB for a 16 reducedstate and an 8 reduced-state scheme respectively at d = 0:002
and 2 dB for a 16 reduced-state scheme at d = 0:01.
E /N [dB]
Fig. 6. Performance of single-estimator A-SISO and multiple-estimator ASISO (d = 0:01). (64 is full state and 16 or 8 are reduced states)
E /N [dB]
nation of interleaved codes: Performance analysis, design, and iterative
decoding,” IEEE Trans. Inf. Theory, vol. IT-44, May 1998, pp. 909-926.
A. Anastasopoulos and K. M. Chugg, “Adaptive soft-input soft-output algorithms for iterative detection with parametric uncertainty,” IEEE Trans.
Communications, vol. 48, Oct. 2000, pp. 1638-1649.
J. Heo, K. M. Chugg and A. Anastasopoulos, “A comparison of forwardonly and bi-directional fixed-lag adaptive SISOs,” Proceedings of ICC
2000, June 2000, pp. 1660-1664.
K. M. Chugg, “The condition for the applicability of the Viterbi algorithm
with implications for fading channel MLSD,” IEEE Trans. Communications, vol. 46, Sep. 1998, pp. 1112-1116.
P. Moqvist and T. Aulin, “Power and bandwidth efficient serially concatenated CPM with iterative decoding,” Proc. of GLOBECOM 2000, San
Fransico, CA, Nov. 2000.
J.B. Anderson, T. Aulin, and C.-E. Sundberg, Digital Phase Modulation,
New York: Plenum Press, 1986.
L. Yiin, G. L. Stüber, “MLSE and soft-output equalization for trelliscoded continuous phase modulation,” IEEE Trans. Communications, vol.
45, June 1997, pp. 651-659.
K. R. Narayanan and G. L. Stüber, “Performance of trellis coded CPM
with iterative demodulation and decoding,” Proc. of GLOBECOM’99, Rio
de Janeiro, Brazil, Dec. 1999.
X. Chen and K. M. Chugg, “Reduced state soft-in/soft-out for complexity reduction in iterative and non-iterative data detection,” Proceedings of
ICC 2000, June 2000, pp. 6-10.
G. Colavolpe, G. Ferrari, and R. Raheli, “Reduced state BCJR-type algorithms,” Proceedings of ICC 2000, June 2000, pp. 460-464.
K. M. Chugg, “Sequence estimation in the presence of parametric uncertainty,” Ph.D. thesis, University of Southern California, Los Angeles, CA,
T. Aulin and C.-E. Sundberg, “An easy way to calculate power spectra for
digital FM,” IEE Proc. Part F, vol. 130, pp.519-526, Oct. 1983.
R. Raheli, A. Polydoros and C. -K. Tzou, “The Principle of per-survivor
processing: A General approach to approximate and adaptive MLSE,”
IEEE Trans. Commun., vol. 43, March/April 1995, pp. 354-364.
J. P. Seymour and M. P. Fitz, “Near-optimal symbol-by-symbol detection schemes for flat Rayleigh fading,” IEEE Trans. Commun., vol. 43,
March/April 1995, pp. 1525-1533.
B. Rimoldi, “A decomposition approach to CPM,” IEEE Trans. Inf. Theory, vol.34, pp. 260-270, March 1988.