848IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 19, NO. 5, MAY 2001
Reduced-State BCJR-Type Algorithms
Giulio Colavolpe, Associate Member, IEEE, Gianluigi Ferrari, Student Member, IEEE, and
Riccardo Raheli, Member, IEEE
Abstract—In this paper, we propose a technique to reduce the
number of trellis states in BCJR-type algorithms, i.e., algorithms
with a structure similar to that of the well-known algorithm by
Bahl, Cocke, Jelinek, and Raviv (BCJR). This work is inspired by
reduced-state sequence detection (RSSD). The key idea is the con-
struction, during one of the recursions in the reduced-state trellis,
of a “survivor map” to be used in the other recursion. In a more
general setting, two distinct survivor maps could be determined in
the two recursions and used jointly to approximate the a posteriori
probabilities. Three examples of application to iterative decoding
are shown: 1) coherent detection for intersymbol interference (ISI)
channels; 2) noncoherent detection based on an algorithm recently
proposed by the authors; and 3) detection based on linear predic-
tion for Rayleigh fading channels. As in classical RSSD, the pro-
posed algorithm allows significant state-complexity reduction with
limited performance degradation.
Index Terms—Error correcting codes, iterative decoding, soft-
input soft-output, turbo codes.
likelihood that a particular symbol has been transmitted. Soft-
output algorithms – have been considered with renewed
erative decoding of interleaved concatenated codes , . The
Bahl, Cocke, Jelinek, and Raviv (BCJR) algorithm  is based
terion and proves to be optimal for estimating the states or out-
puts of a Markov chain observed in white noise. This algorithm
is rather complex to implement because of the following
1) Problems of numerical representation of very low proba-
2) Use of nonlinear functions;
3) Numerous mixed multiplications and additions
In order to reduce these problems, realizations of this algorithm
in the logarithmic domain have been proposed, which result in
useful simplifications –.
Whenever the overall transmission system can be considered
memory in a trellis diagram. For example, this is the case for a
channel and/or a finite intersymbol interference (ISI) channel.
A frequency nonselective fading channel is not rigorously finite
N CURRENT digital communication systems based on it-
erative detection/decoding, it is necessary to determine the
Manuscript received April 26, 2000; revised November 22, 2000. This work
was supported by Ministero dell’Università e della Ricerca Scientifica e Tec-
nologica (MURST), Italy. This work was presented in part at the IEEE Inter-
national Conference on Communications, (ICC’00), New Orleans, LA, USA,
The authors are with the Dipartimento di Ingegneria dell’Informazione, Uni-
versità di Parma, 43100 Parma, Italy.
Publisher Item Identifier S 0733-8716(01)03902-6.
memory  but it is generally assumed as being so . Once
the overall system memory has been identified, a relevant trellis
diagram can be defined and soft-output algorithms, such as the
BCJR algorithm, can be employed. If the overall memory is
large, the complexity of a BCJR algorithm, even with the cited
logarithmic simplifications, may be unacceptable because of
the trellis size. Various solutions have appeared in the literature
to reduce the complexity of the decoder trellis diagram. The
so-called T-algorithm and M-algorithm reduce the number of
paths which are searched in the trellis diagram . Trellis
splicing based on a confidence criterion may be used to detect
reliable information symbols early on during decoding .
When the channel memory is not finite, as for a “noncoherent
channel” , it is possible to utilize soft-output algorithms
with a structure similar to that of the BCJR algorithm, which
try to partially take into account this memory by means of an
augmented trellis . This approach can be generalized to other
channels with infinite memory by means of suitable algorithms
In this paper, we propose an extension of reduced-state se-
quence detection (RSSD) – to a general BCJR-type al-
gorithm. Further generalizations toward the application of per-
survivor processing (PSP) techniques  in order to dynam-
ically estimate unknown channel parameters are also possible,
as for example in –.
The application of the principle of RSSD to the complexity
applied to a soft-output Viterbi algorithm as in  (see also 
for a recent application). Other applications were brought to the
authors’ attention during the review process –. Most of
the algorithms proposed in these papers deal with ISI channels
of a BCJR algorithm employed as a soft-output equalizer in a
soft-decision feedback is derived from a modified version of
the Lee algorithm  for a minimum-phase channel impulse
response.Itis alsoshown thatthealgorithm thereproposedmay
be given an equivalent formulation by introducing a backward
recursion, making it similar to the BCJR algorithm. In , a
proposed, the T-algorithm is applied to reduce the complexity,
and the possibility of applying RSSD techniques is mentioned.
versions of the fixed-lag soft-output algorithm introduced in 
are proposed based on the application of the T-algorithm and
RSSD, although an explicit introduction of the concept of
survivor is not given. A similar detection strategy is used in
 to detect continuous phase modulations.
0733–8716/01$10.00 © 2001 IEEE
COLAVOLPE et al.: REDUCED-STATE BCJR-TYPE ALGORITHMS849
A reduced-state BCJR algorithm similar to that here pre-
sented and its application to iterative decoding was proposed
independently in  at the same time of the first presentation
of this work .
The paper is organized as follows. In Section II, we give a
general definition of BCJR-type algorithms. In Section III, we
propose a reduced-state BCJR-type algorithm in general terms.
In Section IV, we consider examples of application of the pro-
posed technique for ISI channels, noncoherent channels, and
Rayleigh flat-fading channels. Numerical results are presented
in Section V. Conclusions are drawn in Section VI.
II. BCJR-TYPE ALGORITHMS
An information source emits a sequence of independent and
encoded and transmitted through a channel. We assume that a
proximations of them as in . The calculation of these prob-
abilities is generally based on the observation of a suitable se-
is the observation at the
at the th signaling interval.
be thenumber ofproperly definedreceivertrellisstates,
the channel model and possible approximations. Let
is associated with a transition from state
to state and there are no paralleltransitions. The exten-
andor trellis coded modulations, can be easily ob-
branches, each of them being uniquely identified
by the couple of states it connects. We denote the branch which
mation symbol which drives this transition by
With these definitions, we assume that a good approximation
of the a posteriori probability of symbol
and assume that it can be expressed, in analogy
with a generalized version of the BCJR algorithm ,  as1
is the transmission length
th signaling interval.
, although in generalcould
by and the infor-
can be determined
1As customary, we do not use a specific notation to distinguish between
random quantities and their hypothetical values. Specifically, ??? ? denotes
the a priori probability that the information symbol at the ?th epoch takes on
the hypothetical value ? . A consistent notation is used throughout and the
correct interpretation should be clear from the context.
tions. They depend on the particular trellis branch and their ex-
pressions can be specified for the transmission system under
probabilities of information symbol
, respectively. If the symbols are a priori equiprob-
able, these probabilities reduce to constants. On the contrary,
if the algorithm is used in an iterative decoding process, these
probabilities in general are not constant in successive iterations,
and must be explicitly considered.
essary inorder to correctlyexpress thea posteriori probabilities
in (1). However, an arbitrary normalization of these quantities
would scale the resulting a posteriori probabilities by a factor
which would not affect the correctness of the algorithms under
consideration. As a consequence, we take the notational liberty
of denoting by
a scaled version of the a posteriori
probabilities and referring to them as a posteriori probabilities
for simplicity. Similarly, we refer to
as probability density functions, although strictly speaking they
may be so except for a normalization constant. If necessary, we
may further relax the terminology by referring to probabilities
even if an approximation may be involved.
Similarly to the BCJR algorithm, we assume that we can
compute the probability density functions
by means of forward and backward recursions , , .
as the final state of transition
represent the a priori
and beginning state of
we may write
sibly scaled, probability density functions that, in general,
may depend on two consecutive branches. By
denote the information symbol “lost” in transition
the oldest symbol in the initial state
sum is extended over all the transitions of epoch
in the initial state of branch
. The sum in (3), relative to the
trellis section at epoch
, may be interpreted similarly.
Wemayalso definereliability valuesof eachsymbol
respect to a reference symbol
rithmic likelihood ratios of the following type
andare suitable, pos-
. The couple
. In (2), the
, by considering loga-
850IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 19, NO. 5, MAY 2001
The second term in (4) represents the generated extrinsic infor-
mation, that is the component of the likelihood values to be fed
back in an iterative decoding process (as an example, see ,
). Equations (1)–(3) can be reformulated in the logarithmic
We remark that the algorithm described by (1)–(3) reduces to
the classical BCJR algorithm  in the special case of strictly
finite memory. In this case, the probability density functions
anddepend on transition
and depend on a single transition
The proposed formulation of BCJR-type algorithms general-
izes the classical one  and may be suitable for specific detec-
tion problems characterized by strictly unlimited memory. As
an example, the noncoherent BCJR-type algorithm proposed in
 is obtained by truncating the infinite memory of the consid-
ered channel and falls within theconsidered formulation. Infact
in , the expression of the a posteriori probability involves
a probability density function
transition, the forward and backward recursions involve prob-
ability density functions
cessive transitions, and
on a state). As shown in the Appendix, two alternative formula-
tions of a BCJR-type algorithm may be conceived but none re-
formulation previously described has several motivations. First,
the alternative formulations described in the Appendix involve
augmented trellis diagrams with larger number of states. As this
paper deals with reduced-state algorithms, it would be contra-
dictory to begin with augmented trellis diagrams. Second, the
gorithm when the overall memory is finite, as opposed to the al-
formulation is consistent with that adopted in .
through a single state.
which depends on a single
which depend on two suc-
depend on a transition (not
III. REDUCED-STATE BCJR-TYPE ALGORITHMS
A BCJR-type algorithm formulated as in Section II requires
the calculation of essentially two kinds of probability density
, which depends on a single transition, and
, which may depend on two consecutive transitions. Let us
single information symbol
. By using a vector notation, we
denote this sequence of code symbols by
and refer to as the code (vector) symbol at epoch
general, a single transition can be related to
a one-to-one correspondence between transition
code symbols. The parameter
the channel and/or the detection strategy. As an example, for
an AWGN channel,
in the case of coherent detection,
is the assumed phase memory for noncoherent
detection based on the algorithm proposed in . A single
branch can be related to
information symbols, that is
. The relation between integers
on the coding structure.2We assume that the receiver observes,
are generated by a
is related to
, where the symbol
consider some encoder state variables together with the information symbols in
the definition of ? and ? .
at each epoch, a (vector) sample
the output of the channel. Generalizations to the case of
oversampling or multiple decoder inputs are straightforward.
the length of the discrete-time overall impulse
response, each sample
will depend on
Weconsider a trellis diagramwith a reduced numberof states
and denote its generic state by
ciple of RSSD –, we define the state reduction by as-
suming that a transition
diagram is equivalent to a sequence
formation symbols, with
even with this reduction of states, knowledge of
tion symbols is necessary in order to specify
which is required to compute
, we need to estimate symbols
order to compute the metric. According to RSSD, this estimate
could be obtained by tracking the survivor of each state in the
reduced-state trellis. However, how could we define a survivor
in this case?
A “full-state” BCJR-type algorithm runs first a forward re-
cursion to compute
for each transition
. As shown in (1) and (2),
, instead of a single state
Hence, a “survivor” associated with a single transition has to
be defined. By considering the algorithm in the logarithmic do-
main, the forward recursion of
. Following the prin-
in the reduced trellis
.3The central point is that,
,, and . Hence, for a given
is associated with
as in a classical case.
((2)) can be expressed as
this recursion by the “max-log” approximation as follows 
. We can approximate
In (6), it is intuitive to interpret the term to be maximized as a
“metric” associated with a path ending with transition
a classical Viterbi algorithm.
Considering the reduced-state trellis and assuming we know
the survivors of each transition
survivors can be extended to time instant
ward recursion. We may define by
transitions reaching epoch
, as in
, we now show how the
by using the for-
along the survivor of
. Each transition
, depend on the
uniquely identifies the sequence of information symbols
. Hence, the couple
resent the information symbols in the path history of transition
3More complex techniques based on set partitioning may also be employed
COLAVOLPE et al.: REDUCED-STATE BCJR-TYPE ALGORITHMS851
it is easy to extend the survivors by a step, i.e., to epoch . In
the reduced-state trellis, taking into account paths
For each transition
should be stored.
During the forward recursion, a “full-state” BCJR-type algo-
rithm does not necessarily make a selection between the metrics
of the paths terminating with a given transition
(6) but rather determines a new metric
of these path metrics as in (5). Similarly, in the reduced-state
trellis equation (7) does not necessarily have to be used to com-
but only to determine the survivor associated with
We note that the concept of survivor associated with a transi-
gorithm. The Appendix describes this possible formulation.
In the reduced-state trellis, it is necessary to keep track of the
survivors associated with each transition, at each epoch, in the
forward recursion of
only. In fact, these survivors may be
used in the backward recursion of
in (1).Wewill refertotheensemble ofthesurvivors
generated in the forward recursion as “survivor map.” The basic
structural idea can be further generalized. In fact, it is not nec-
essarily the forward recursion which allows the construction of
a survivor map to be used in the backward recursion. One could
build a survivor map during the backward recursion (run first)
and use it in the forward recursion. Finally, one could build two
distinct survivor maps during the forward and backward recur-
sion, and consider an extended
rest of this section, we will refer for simplicity to the case of a
survivor map built during the forward recursion but generaliza-
Given the above definitions, we can reformulate (1)–(3), ob-
, the branch that maximizes (7)
as shown in
as an average
and in the calculation of
in the combination (1). In the
general reduced-state BCJR-type algorithm.
Forward recursion of the probability density function ? ?? ? for a
general reduced-state BCJR-type algorithm. The metric ? is calculated using
the survivor map previously built in the forward recursion.
Backward recursion of the probability density function ? ?? ? for a
the forward recursion proceeds in the reduced-state trellis ac-
cording to (9). In order to compute
probability density functions
and, for each of them, compute the branch
by considering the symbols associated with the sur-
vivor of transition
. The backward recursion proceeds sim-
ilarly using the survivor map generated during the forward re-
cursion, as shown in Fig. 2.
Two logarithmic simplified versions of a reduced-state
BCJR-type algorithm may be considered. A first one is derived
by applying the max-log approximation ,  to (8)–(10).
mated as follows
are the initial and final states of
, respectively. In Fig. 1, it is shown how
, one should consider
, (9) can be approxi-
and. This version of the algorithm
would be motivated, besides complexity reduction, by the fact
that the forward recursion of
vivors without additional operations. In a second logarithmic
simplified version, the max-log approximation for the forward
and backward recursions is maintained but a “full” combina-
tion is considered in order to compute
allows to determine the sur-
. Hence, we
852IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 19, NO. 5, MAY 2001
may retain (11) for the forward recursion of
sponding one for the backward recursion of
with the use of (8), which in the logarithm domain becomes
and the corre-
This second method, referred to as “hybrid” method, tries to
fully exploit the reduced information carried by
cause of the approximated recursions and will be considered in
the numerical results.
The computational complexity of the forward and backward
recursions is related to the number of branches in the receiver
trellis, which is proportional to the number of states. Hence,
the complexity of the proposed reduced-state BCJR-type algo-
With the introduction of the concept of survivor associated
such as those considered in –, , are possible with
taking into account that a single step in the recursions of a
IV. EXAMPLES OF APPLICATIONS
As examples of application of the above reduced-state
BCJR-type algorithms, we consider the case of coherent
detection for ISI channels (assuming perfect knowledge of the
ISI channel coefficients), noncoherent detection as proposed
in , and detection based on linear prediction for Rayleigh
A. ISI Channels
In the case of an ISI channel, we assume uncoded transmis-
reduces to . The observation samples at the
output of a whitened matched filter, can be written as
. The noise samples
independent Gaussian random variables, with zero mean and
. The distortion on the elementary shaping pulse
makes the samples
each weighted by a different coefficient
the classical BCJR algorithm can be directly applied with
. Hence, the probability density
functions which appear in (1)–(3) reduce to
. By defining the
, it is easy to show that
the reduced-state BCJR algorithm in the case of coherent detection over ISI
Forward recursion of the probability density function ? ?? ? for
tion (1)–(3) can be specialized to this case by letting
(that is, the receiver observes a single sample
has minimum phase –, , an efficient re-
duced-state trellis diagram is obtained by defining a state
in the path history associated with state
denotes proportionality. The general formula-
. Hence, searching
, we may
In Fig. 3, it is shown how the forward recursion proceeds in
the reduced-state trellis in the case of coherent detection over
an ISI channel. Since
depends on a single state
on a transition
, each forward step involves a single transi-
tion. In this case, we associate with each state
chosen among the beginning states of the
. The backward recursion proceeds similarly, by using the
survivors selected during the forward recursion. As one can see
from Fig. 3 in this case, the concept of survivor coincides with
the usual one. With the definition given for state
, in (11) reduces to
for the forward recursion reduces to
. The expression (11)
COLAVOLPE et al.: REDUCED-STATE BCJR-TYPE ALGORITHMS853
the reduced-state BCJR algorithm in the case of coherent detection over ISI
channels. The survivor map is built during this recursion.
Backward recursion of the probability density function ? ?? ? for
and (12) reduces to
If the overall impulse response
efficient definitions of trellis state and transition may be
is maximum phase,4
, respectively. As mentioned in Section III, a new ver-
sion of a reduced-state BCJR-type algorithm can be conceived,
which starts with the backward recursion, builds a “survivor
map” compatible with the new reduced state
forward recursion. Related reverse-time processing structures,
to stateand symbol
formulation is a straightforward extension of that previously
introduced, the only modification being a termination of the
reduced-state trellis necessary to better initialize the backward
recursion. Fig. 4 schematically shows how the survivor map can
be built during the backward recursion. The survivor may be
indicates the final state
This example leads to a more general conclusion on the pos-
sible applications of the proposed technique. Survivor maps can
be built during both the forward and backward recursions, de-
pending on the overall channel impulse response and, conse-
quently, on the structure of the received signal samples. As an
example, a proper use of survivor maps built during both recur-
phase.” A specific investigation of this case was not performed
, and runs the
is discarded. The
= , where
because beyond the scope of the paper. Related aspects are dis-
cussed in .
B. Noncoherent Channels
In the case of noncoherent decoding, we refer to the BCJR-
type algorithm proposed in . We may express this algorithm
in terms of the general formulation given above with
and(hence, the receiver considers a “window” of
consecutive samples at each epoch). Denoting by
coder state, the “decoder state” is defined by
case, the probability density functions
in (1)–(3) specialize as follows
. In this
By defining a reduced state as
can be expressed as
is the transpose conjugate operator.
, a transition in the reduced-state trellis
. Hence, the probability den-
854IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 19, NO. 5, MAY 2001
and (8) specializes as follows
The survivor map is built during the forward recursion and
employed in the backward recursion and to evaluate
Referring to the original formulation proposed in , the exten-
sion of the previously introduced general recursions is not im-
mediate. Noting that
can be expressed as follows
, the forward recursion
Since at epoch
vivor associated with
eration. The term
but it does not affect the survivor selection. Instead of
we propose to utilize
mined through the max operation and replace it with
the survivor of each transition
, we replace
. The choice of the sur-
) is based on the max op-
affects the exact value of
where the expression
ically related quantities. The forward recursion finally reduces
denotes that and are monoton-
Once the survivor map has been determined during the forward
recursion, the probability density function
ward recursion, can be calculated
, used in the back-
C. Fading Channels
We refer to the transmission system considered in  and
the normalized fading rate of the channel. We
assume that each information symbol
The corresponding received signal at the matched filter output
are complex random variables whose quadrature compo-
nents are independent, Gaussian with zero mean, and
samples of a zero mean complex-valued white Gaussian noise
process.Theautocorrelationofthefadingprocess is assumed to
follow the classical isotropic scattering model .
Assuming that the information bits are independent within
each symbol, we can consider
The proposed formulation gives the a posteriori probability of
-PSK information symbols
bit-wise, as in , a posteriori probabilities of bits
to be calculated. To obtain these probabilities (1) can be modi-
fied as follows
corresponds to a group
. If the interleaver works
in practice, linear prediction ,  can be used. The branch
input extrinsic information, this is an approximation.
COLAVOLPE et al.: REDUCED-STATE BCJR-TYPE ALGORITHMS855
whereis the fading gain estimate,
are the prediction coefficients and
of the prediction error, which can be computed as shown in
. We may observe that
, the corresponding trellis consists
states. Assuming that the autocorrelation coefficients
of the channel fading process are known, the optimal predic-
Wiener-Hopf equations . Hence
is the prediction order,
is the variance
. By defining
are obtained by solving the
anddo not depend onand defining
The first term in (35) depends on
neglected, unless one assumes perfect channel estimation.
To reduce the number of trellis states, we proceed as in the
case of an ISI channel. We may define a reduced state
is identical to that used in Section IV-A)
and should not be
V. NUMERICAL RESULTS
The performance of receivers based on reduced-state BCJR-
type algorithms in the three cases considered in Section IV is
assessed by means of computer simulations in terms of bit error
over an ISI channel. Receivers with various levels of complexity are considered
and compared with the full-state receiver (? ? ??). The considered numbers
of iterations are one and six in all cases. The performance in the case of coded
transmission over an AWGN channel, without ISI, is also shown (solid lines
Application of the proposed technique to iterative decoding/detection
rate (BER) versus
per information bit and
spectral density. In any component decoder we consider the
second simplified logarithmic version of the BCJR-type algo-
rithm proposed in Section III (hybrid method).
In the case of coherent detection over an ISI channel, we
consider the scheme of turbo detection in . More precisely,
we consider a binary (
) transmission system character-
ized by a rate 1/2 16-state recursive systematic convolutional
(RSC) encoder with generators
tation), followed by a
nonuniform interleaver. The bits
at the output of the interleaver are sent through the channel by
a binary PSK (BPSK) modulation. The minimum phase dis-
crete-time channel impulse response is identified by the fol-
and. The receiver is based on a
serial concatenation of a detector, which uses the reduced-state
BCJR algorithm with metrics proposed in Section IV-A, and
a decoder which is a SISO module . The extrinsic infor-
mation is used according to the heuristic method proposed in
. By trial and error, we found that a good performance
is obtained when the extrinsic information generated by the
inner detector is weighted (i.e., multiplied) by a parameter
equal to 0.3 and the extrinsic information generated by the
outer decoder is weighted by a parameter equal to 0.5. The
state reduction technique is applied to the inner detector. In
Fig. 5, the performance of the full-state receiver (inner de-
states) is compared with the performance
of the receiver with reduced complexity (inner detector with
= 8, 4, or 2 states). In all cases, we consider one and
six decoding iterations. At six decoding iterations, the perfor-
mance loss for a detector with
, being the received signal energy
being the one-sided noise power
, (octal no-
= 4 states with respect to
856IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 19, NO. 5, MAY 2001
an RSC code. Receivers with various levels of complexity are considered and
compared with a full-state receiver (with ? ? ?) and the coherent receiver.
Application of the proposed technique to noncoherent decoding of
the receiver without state reduction is only 0.75 dB at a BER
, and it reduces to 0.25 dB for a detector with
8 states. For comparison, the performance in the absence of
ISI, that is for coded transmissions over an AWGN channel,
is also shown. In this case, the receiver reduces to the outer
decoder of the RSC code considered above.
16-state RSC binary code with generators
and rate 2/3 obtained by means of puncturing, used as compo-
nent of the turbo code presented in . The modulation format
is BPSK. In Fig. 6 the performance of the noncoherent decoder
using the BCJR-type algorithm with metrics described in Sec-
tion IV-B is assessed for various levels of state reduction and
compared to that of a coherent receiver. With respect to a de-
andstates the performance is appreciably im-
proved at low signal-to-noise ratios (SNRs). For
, the performance loss with respect to the full-state re-
is less than one dB for every SNR, and re-
duces to only 0.5 dB at SNRs above five dB.
We also consider noncoherent decoding of a turbo code
having as component codes those described in the previous
paragraph . The two component encoders are parallelly con-
catenated by a
nonuniform interleaver. At the receiver,
each noncoherent component decoder uses the reduced-state
BCJR-type algorithm described in Section IV-B. In this case
too, the extrinsic information generated by each component
decoder is passed to the other one by following the heuristic
method proposed in . By trial and error, we found that the
best performance is obtained when the extrinsic informations
generated by both the component decoders are weighted by a
parameter equal to 0.3. Fig. 7 shows the performance of the
receiver for various levels of state reduction, specified by the
and, and compares it to that of a coherent re-
ceiver and a noncoherent receiver with
states (full complexity), for
a turbo code. Receivers with various levels of complexity are considered and
compared with a full-state receiver (with ? ? ?) and a coherent receiver. The
considered numbers of iterations are 1, 3, and 6 in all cases.
Application of the proposed technique to noncoherent decoding of
(full number of states). In all cases, the considered numbers
of iterations are 1, 3, and 6. The reduced-state BCJR-type
performance loss of about 3.2 dB at six decoding iterations
with respect to the coherent receiver. For
(), at a BER of a performance gain of about 1.2 dB
is obtained with respect to the full-state case with
and the same number of states (
the use of the proposed state reduction technique makes the
algorithm proposed in  applicable, even for large values of
Finally, we consider transmission of differentially encoded
quaternary PSK (QPSK) signals over a flat-fading channel, as
in . The outer code is a 64-state nonrecursive convolutional
code with generators
an inner differential encoder. In fact, bit interleaving is an ap-
propriate means to combat the effects of fading , . The
tector uses linear prediction and the state reduction technique.
In Fig. 8 the performance of the full-state receiver, with predic-
and 4, respectively, is compared with
the performance of a receiver with various levels of complexity
specified by the couple of parameters (
consider one and six decoding iterations. At six decoding itera-
tions, a detector with
loss of only 0.2 dB at a BER of
). For and
with respectto a full-state receiver with
a BER of
. For comparison, the performance curve in the
case of perfect knowledge of the fading coefficients (coherent
decoding) is also shown.
( ) exhibits a
). We conclude that
and. This code is
). In all cases, we
exhibits a performance
, with respect to a full-state
, the performance gain
is about 0.6 dB at
COLAVOLPE et al.: REDUCED-STATE BCJR-TYPE ALGORITHMS857
through linear prediction, for flat-fading channel. Receivers with various levels
of complexity (in terms of prediction order ? and reduced state parameter ?)
are shown. The considered numbers of iterations are one and six in all cases.
The performance in the case of decoding with perfect knowledge of the fading
coefficients is also shown (solid lines).
Application of the proposed technique to iterative decoding/detection,
A class of BCJR-type algorithms has been defined as an
extension of the well-known BCJR algorithm, whenever the
channel memory is not finite or partially taken into account.
Similarly to the BCJR algorithm, the proposed algorithms run
a forward and a backward recursion. Techniques for state-
complexity reduction for these algorithms havebeen introduced
based on reduced-state sequence detection. The structure of
the proposed BCJR-type algorithms leads to the definition of a
survivor associated with a trellis branch (instead of a state). De-
pending on the structure of the system memory, a survivor map
can be determined during one of the recursions and used in the
other one. Alternatively, two survivor maps can be computed
(one for each recursion) and suitably combined when gener-
ating a posteriori probabilities. A reduced-state BCJR-type
algorithm is well suited for iterative processing applications.
In the considered examples, the proposed reduced-state BCJR-
type algorithms are effective for appreciably limiting the
receiver complexity with minor performance degradation or
improving performance for a given level of complexity.
The considered formulation of a BCJR-type algorithm dif-
fers from that of the classical BCJR algorithm , although it
reduces to the latter when the channel is strictly finite-memory.
In this Appendix, we show that two alternative equivalent for-
mulations of a BCJR-type algorithm exist, but none coincides
exactly with the classical one.
Let us define an augmented “super-state”
correspondence with a transition
this augmented trellis diagram is
in a one-to-one
. The number of states of
times that of the original
one because information symbol
. With these definitions, we may rewrite
recursion (2) as
now appears in the state
transitions ending in
duces to that of the classical BCJR algorithm, except for the
more general use of the probability density function
similar conclusion also holds for the backward recursion. How-
ever, let us consider the “central” combination (1), which gives
a classical BCJR algorithm, if this combination is expressed in
terms of states and transitions in the augmented trellis, the fol-
lowing consequences arise: 1) the probability density functions
anddepend on states relative to the same time instant and
2) the sum must be extended over
(and not over all states). Therefore, even
if this alternative formulation would entail a forward recursion
exhibit differences in this central combination.
central combination may be computed on the original trellis. In
but the two recursions and the central combination must be re-
a single step as involving a single transition. Specifically, in the
forward and backward recursions an augmented trellis must be
times the number of states of the trellis relative to
the central combination.
, respectively, and the sum is extended over all
. The forward recursion (37) now re-
are the beginning and ending states
of the possible super-
References –, , and  were brought to the au-
viewers and W. H. Gerstacker are gratefully acknowledged.
 K. Abend and B. Fritchman, “Statistical detection for communications
channels with intersymbol interference,” Proc. IEEE, vol. 58, pp.
779–785, May 1970.
 L. R. Bahl, J. Cocke, F. Jelinek, and R. Raviv, “Optimal decoding of
linear codes for minimizing symbol error rate,” IEEE Trans. Inform.
Theory, vol. 20, pp. 284–284, Mar. 1974.
 G. Battail, “Pondération des symboles décodé par l’algorithme de
Viterbi,” Ann. Télecommun., Fr., vol. 42, no. 1–2, pp. 31–38, Jan. 1987.
puts and its applications,” in Proc. IEEE Global Telecom. Conf., Dallas,
TX, Nov. 1989, pp. 1680–1686.
 Y. Li, B. Vucetic, and Y. Sato, “Optimum soft-output detection for chan-
nels with intersymbol interference,” IEEE Trans. Inform. Theory, vol.
41, pp. 704–713, May 1995.
 G. Colavolpe, G. Ferrari, and R. Raheli, “Noncoherent iterative (turbo)
decoding,” IEEE Trans. Commun., vol. 48, pp. 1488–1498, Sept. 2000.
 C. Berrou and A. Glavieux, “Near optimum error correcting coding
and decoding: Turbo-codes,” IEEE Trans. Commun., vol. 44, pp.
1261–1271, Oct. 1996.
858 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 19, NO. 5, MAY 2001
 S. Benedetto, D. Divsalar, G. Montorsi, and F. Pollara, “Serial concate-
nation of interleaved codes: Performance analysis, design, and itera-
tive decoding,” IEEE Trans. Inform. Theory, vol. 44, pp. 909–926, May
 P. Robertson, E. Villerbrun, and P. Hoeher, “A comparison of optimal
and sub-optimal MAP decoding algorithms operating in the log
domain,” in Proc. Int. Conf. Communications, Seattle, WA, June 1995,
 S. Benedetto, D. Divsalar, G. Montorsi, and F. Pollara, “Soft-output al-
codes,” in Proc. Int. Conf. Communications, Dallas, TX, June 1996, pp.
 K. M. Chugg and X. Chen, “Efficient architectures for soft-output algo-
rithms,” in Proc. Int. Conf. Communications, Atlanta, GA, June 1998,
 K. M. Chugg, “The condition for applicability of the Viterbi algorithm
with implications for fading channel MLSD,” IEEE Trans. Commun.,
vol. 46, pp. 1112–1116, Sept. 1998.
 X. Yu and S. Pasupathy, “Innovations-based MLSE for Rayleigh
fading channels,” IEEE Trans. Commun., vol. 43, pp. 1534–1544,
 V. Franz and J. B. Anderson, “Concatenated decoding with a reduced-
search BCJR algorithm,” IEEE J. Select. Areas Commun., vol. 16, pp.
186–195, Feb. 1998.
 B. J. Frey and F. R. Kschischang, “Early detection and trellis splicing:
Reduced-complexity iterative decoding,” IEEE J. Select. Areas
Commun., vol. 16, pp. 153–159, Feb. 1998.
 G. Colavolpe and R. Raheli, “Noncoherent sequence detection,” IEEE
Trans. Commun., vol. 47, pp. 1376–1385, Sept. 1999.
 M. V. Eyuboˇ glu and S. U. H. Qureshi, “Reduced-state sequence estima-
vol. 36, pp. 13–20, Jan. 1988.
 A. Duel-Hallen and C. Heegard, “Delayed decision-feedback sequence
estimation,” IEEE Trans. Commun., vol. 37, pp. 428–436, May 1989.
 P. R. Chevillat and E. Eleftheriou, “Decoding of trellis-encoded sig-
nals in the presence of intersymbol interference and noise,” IEEE Trans.
Commun., vol. 37, pp. 669–676, July 1989.
 R. Raheli, A. Polydoros, and C.-K. Tzou, “Per-survivor processing: A
general approach to MLSE in uncertain environments,” IEEE Trans.
Commun., vol. 43, pp. 354–364, Feb. 1995.
 A. Anastasopoulos and A. Polydoros, “Adaptive soft-decision algo-
rithms for mobile fading channels,” Eur. Trans. Telecommun., vol. 9,
pp. 183–190, Mar. 1998.
 A. Anastasopoulos and K. M. Chugg, “Adaptive SISO algorithms for
iterative detection with parametric uncertainty,” in Proc. ICC Commu-
nications Theory Mini-Conf., Vancouver, BC, Canada, June 1999, pp.
 A. Anastasopoulos and K. M. Chugg, “Adaptive iterative detection for
turbocodes withcarrierphase uncertainty,”in Proc.GlobalTelecommu-
nications Conf., Rio de Janeiro, Brazil, Dec. 1999, pp. 2369–2374.
 W. Koch and A. Baier, “Optimum and sub-optimum detection of coded
data disturbed by time-varying intersymbol interference,” in Proc.
Global Telecommunications Conf., San Diego, CA, Dec. 1990, pp.
tions Conf., San Diego, CA, Dec. 1990, pp. 1679–1684.
 S. H. Müller, W. H. Gerstacker, and J. B. Huber, “Reduced-state soft-
output trellis-equalization incorporating soft feedback,” in Proc. Global
Telecommunications Conf., London, U.K., Dec. 1996, pp. 95–100.
 G.-K. Lee, S. B. Gelfand, and M. P. Fitz, “Bayesian decision feedback
techniques for deconvolution,” in Proc. Global Telecommunications
Conf., San Francisco, CA, Dec. 1994, pp. 248–252.
 M. P. Fitz and S. B. Gelfand, “Reduced complexity symbol-by-symbol
soft output algorithms,” in Proc. Global Telecommunications Conf.,
London, U.K., Dec. 1996, pp. 21–25.
 R. Balasubramanian, M. P. Fitz, and J. K. Krogmeier, “Optimal and
suboptimal symbol-by-symbol demodulation of continuous phase mod-
ulated signals,” IEEE Trans. Commun., vol. 46, pp. 1662–1668, Dec.
 L.-N. Lee, “Real-time minimal-bit-error probability decoding of con-
volutional codes,” IEEE Trans. Commun., vol. 22, pp. 146–151, Feb.
 X. Chen and K. M. Chugg, “Reduced-state soft-input/soft-output algo-
rithms for complexity reduction in iterative and noniterative data detec-
 G. Colavolpe, G. Ferrari, and R. Raheli, “Reduced-state BCJR-type al-
gorithms,”in Proc. IEEEInt.Conf.Communications,NewOrleans, LA,
June 2000, pp. 460–464.
 J. Heo, K. M. Chugg, and A. Anastasopoulos, “A comparison of for-
ward-only and bi-directional fixed-lag adaptive SISOs,” in Proc. Int.
Conf. Communications, New Orleans, LA, June 2000, pp. 1660–1664.
 G. D. Forney, Jr., “Maximum-likelihood sequence estimation of digital
sequence in the presence of intersymbol interference,” IEEE Trans. In-
form. Theory, vol. IT-18, pp. 363–378, May 1972.
 C. Douillard, M. Jesequel, C. Berrou, A. Picart, P. Didier, and
A. Glavieux, “Iterative correction of intersymbol interference:
Turbo-equalization,” Eur. Trans. Telecommun., vol. 6, pp. 507–511,
 S. Benedetto, D. Divsalar, G. Montorsi, and F. Pollara, “A soft-input
soft-output APP module for iterative decoding of concatenated codes,”
IEEE Commun. Lett., vol. 1, pp. 22–24, Jan. 1997.
 S. Ariyavisitakul, “A decision feedback equalizer with time-reversal
structure,” IEEE J. Select. Areas Commun., vol. 10, pp. 599–613, Apr.
 N. C. McGinty and R. A. Kennedy, “Reduced-state sequence estimator
with reverse-time structure,” IEEE J. Select. Areas Commun., vol. 45,
pp. 265–268, Mar. 1997.
 P. Hoeher and J. Lodge, “‘Turbo-DPSK’: Iterative differential PSK de-
modulation and channel decoding,” IEEE Trans. Commun., vol. 47, pp.
837–843, June 1999.
 R. Clarke, “A statistical theory of mobile-radio reception,” Bell Syst.
Tech. J., vol. 47, pp. 957–1000, July 1968.
 J. Makhoul, “Linear prediction: A tutorial review,” Proc. IEEE, vol. 63,
pp. 561–580, Apr. 1975.
 D. Makrakis, P. Mathiopoulos, and D. Bouras, “Optimal decoding of
coded PSK and QAM signals in correlated fast fading channels and
AWGN: A combined envelope, multiple differential and coherent detec-
tion approach,” IEEE Trans. Commun., vol. 42, pp. 837–843, Jan. 1994.
 G. Colavolpe, P. Castoldi, and R. Raheli, “Linear predictive receivers
for fading channels,” IEE Electron. Lett., vol. 34, pp. 1289–1290, June
 G. Colavolpe, G. Ferrari, and R. Raheli, “Extrinsic information in turbo
decoding: A unified view,” in Proc. Global Telecommunications Conf.,
Rio de Janeiro, Brazil, Dec. 1999, pp. 505–509.
 G. Caire, G. Taricco, and E. Biglieri, “Bit-interleaved coded modula-
tion,” IEEE Trans. Inform. Theory, vol. 44, pp. 927–946, May 1999.
 X. Li and J. A. Ritcey, “Bit-interleaved coded modulation with iterative
decoding,” IEEE Commun. Lett., vol. 1, pp. 169–171, Nov. 1997.
Giulio Colavolpe (S’97–A’01) was born in Cosenza,
Italy, in 1969. He received the Dr.Ing. degree in
telecommunications engineering (cum laude) from
the University of Pisa, Italy, in 1994 and Ph.D. de-
gree in information technology from the University
of Parma, Italy, in 1998.
From December 1997 to October 1999, he was a
Research Associate with the University of Parma.
Since November 1999, he has been a Research
Professor with the University of Parma. In 2000, he
was a Visiting Scientist with the Institut Eurécom,
Valbonne, France. His main research interests include digital transmission
theory, channel coding, and signal processing.
Gianluigi Ferrari (S’99) was born in Parma, Italy,
in 1974. He received the Dr.Ing. degree (Laurea) in
electrical engineering (cum laude) from the Univer-
sity of Parma, Italy, in 1998. Since November 1998,
he has been working towards the Ph.D. degree at the
University of Parma.
Since July 2000, he has been a Visiting Scholar
with the Communication Sciences Institute, Univer-
sity of Southern California, Los Angeles. His main
Dr. Ferrari is a CNIT member and an AEI young member.
COLAVOLPE et al.: REDUCED-STATE BCJR-TYPE ALGORITHMS 859
Riccardo Raheli (M’87) received the Dr.Ing.
degree (Laurea) in electrical engineering (summa
cum laude) from the University of Pisa, Italy, in
1983, the M. Sci. degree in electrical and computer
engineering from the University of Massachusetts,
Amherst, in 1986, and the Ph.D. degree (perfeziona-
mento) in electrical engineering (summa cum laude)
from the Scuola Superiore di Studi Universitari e
di Perfezionamento (now “S. Anna”), Pisa, Italy, in
Siemens Telecomunicazioni, Cassina de’ Pecchi (Milan), Italy. From 1988 to
1991, he was a Research Professor with the Scuola Superiore di Studi Univer-
sitari e di Perfezionamento S. Anna, Pisa, Italy. In 1990, he was a Visiting As-
sistant with the University of Southern California, Los Angeles. Since 1991, he
has been with the University of Parma, Italy, where he is currently an Associate
Professor of Telecommunications. His scientific interests are in the general area
of statistical communication theory, with special attention toward digital trans-
mission systems, data-sequence detection techniques, digital signal processing
and adaptive algorithms for telecommunications. His research activity has lead
to numerous scientific publications in leading international journals and confer-
ence proceedings and a few industrial patents.
Since 1999, he has served on the Editorial Board of the IEEE TRANSACTIONS
ON COMMUNICATIONS as an Editor for Detection, Equalization, and Coding.