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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.

I. INTRODUCTION

I

likelihood that a particular symbol has been transmitted. Soft-

output algorithms [1]–[6] have been considered with renewed

interest,therecentmostfamousapplicationofthembeingtheit-

erative decoding of interleaved concatenated codes [7], [8]. The

Bahl, Cocke, Jelinek, and Raviv (BCJR) algorithm [2] is based

onasymbol-wisemaximumaposterioriprobability(MAP)cri-

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-

bility values;

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 [9]–[11].

Whenever the overall transmission system can be considered

asfinitememory,thereceiverhastotakeintoaccounttheoverall

memory in a trellis diagram. For example, this is the case for a

convolutionalcodeinanadditivewhiteGaussiannoise(AWGN)

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,

June 2000.

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 [12] but it is generally assumed as being so [13]. 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 [14]. Trellis

splicing based on a confidence criterion may be used to detect

reliable information symbols early on during decoding [15].

When the channel memory is not finite, as for a “noncoherent

channel” [16], 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 [6]. This approach can be generalized to other

channels with infinite memory by means of suitable algorithms

whichwillbereferredtoasBCJR-type.

In this paper, we propose an extension of reduced-state se-

quence detection (RSSD) [17]–[19] to a general BCJR-type al-

gorithm. Further generalizations toward the application of per-

survivor processing (PSP) techniques [20] in order to dynam-

ically estimate unknown channel parameters are also possible,

as for example in [21]–[23].

The application of the principle of RSSD to the complexity

reductionofsoft-inputsoft-output(SISO)algorithmswasprevi-

ouslyaddressed.Asanexample,RSSDmaybestraightforwardly

applied to a soft-output Viterbi algorithm as in [24] (see also [6]

for a recent application). Other applications were brought to the

authors’ attention during the review process [25]–[29]. Most of

the algorithms proposed in these papers deal with ISI channels

andnoneconsidersiterativedecoding.Reference[25],mentions

theuseofRSSDtosimplifytheforwardandbackwardrecursions

of a BCJR algorithm employed as a soft-output equalizer in a

frequency-selectivechannel.In[26],asoft-outputequalizerwith

soft-decision feedback is derived from a modified version of

the Lee algorithm [30] 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 [27], a

familyofBayesianconditionaldecisionfeedbackestimatorsare

proposed, the T-algorithm is applied to reduce the complexity,

and the possibility of applying RSSD techniques is mentioned.

Thisideaisfurtherdevelopedin[28],wherereduced-complexity

versions of the fixed-lag soft-output algorithm introduced in [5]

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

[29] to detect continuous phase modulations.

0733–8716/01$10.00 © 2001 IEEE

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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 [31] at the same time of the first presentation

of this work [32].

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

identicallydistributedinformationsymbols

encoded and transmitted through a channel. We assume that a

soft-outputalgorithmcanbedevisedtocomputetheaposteriori

probabilitiesofthetransmittedinformationsymbolsorgoodap-

proximations of them as in [6]. The calculation of these prob-

abilities is generally based on the observation of a suitable se-

quenceofsamplesattheinputofthereceiver.Wedenotethisse-

quence by

, where

and vector

is the observation at the

Thisvectornotationaccountsforpossiblemultipleobservations

at the th signaling interval.

Let

be thenumber ofproperly definedreceivertrellisstates,

whichdependsonthemodulationformat,thepresenceofcoding,

the channel model and possible approximations. Let

thetrellisstateatepoch .Forsimplicity,weassumethatasingle

information symbol

is associated with a transition from state

to state and there are no paralleltransitions. The exten-

siontomoregeneralcases,suchasconvolutionalcodeswithrate

andor trellis coded modulations, can be easily ob-

tainedbyasuitablenotation.Tosimplifythenotation,wealsoas-

sume that

belongtoany

-aryset.Therefore,atrellissectionischaracter-

ized by

branches, each of them being uniquely identified

by the couple of states it connects. We denote the branch which

connects

to

mation symbol which drives this transition by

With these definitions, we assume that a good approximation

of the a posteriori probability of symbol

onthebasisoftheconsideredobservations.Wedenotethisvalue

by

and assume that it can be expressed, in analogy

with a generalized version of the BCJR algorithm [2], [6] as1

whichispossibly

is the transmission length

th signaling interval.

denote

, although in generalcould

by and the infor-

.

can be determined

(1)

Thesumin(1)isextendedoveralltransitionsofepoch

atedwithinformationsymbol

associ-

.Thequantities,

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.

and

tions. They depend on the particular trellis branch and their ex-

pressions can be specified for the transmission system under

consideration.

and

probabilities of information symbol

branch

, 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.

Apropernormalizationof

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

andprobabilitydensityfunctionsandusingtherelevantnotation

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 [2], [6], [7].

Defining

as the final state of transition

areproportionaltosuitableprobabilitydensityfunc-

represent the a priori

and beginning state of

,,and isnec-

, and

and

we may write

(2)

(3)

where

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

uniquely identifies

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-

we

, i.e.,

. The couple

. In (2), the

that end

, with

, by considering loga-

(4)

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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 [7],

[8]). Equations (1)–(3) can be reformulated in the logarithmic

domain [9]–[11].

We remark that the algorithm described by (1)–(3) reduces to

the classical BCJR algorithm [2] in the special case of strictly

finite memory. In this case, the probability density functions

anddepend on transition

Moreover,

and depend on a single transition

reduce to

and, respectively.

The proposed formulation of BCJR-type algorithms general-

izes the classical one [2] and may be suitable for specific detec-

tion problems characterized by strictly unlimited memory. As

an example, the noncoherent BCJR-type algorithm proposed in

[6] is obtained by truncating the infinite memory of the consid-

ered channel and falls within theconsidered formulation. Infact

in [6], the expression of the a posteriori probability involves

a probability density function

transition, the forward and backward recursions involve prob-

ability density functions

and

cessive transitions, and

and

on a state). As shown in the Appendix, two alternative formula-

tions of a BCJR-type algorithm may be conceived but none re-

ducesexactlytotheclassicalBCJRalgorithm.Thechoiceofthe

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

describedformulationiseasilyreducedtotheclassicalBCJRal-

gorithm when the overall memory is finite, as opposed to the al-

ternativeformulations(seetheAppendix).Finally,thedescribed

formulation is consistent with that adopted in [6].

through a single state.

and

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

functions:

, which depends on a single transition, and

, which may depend on two consecutive transitions. Let us

assume that

code symbols

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

that is

a one-to-one correspondence between transition

sequence of

code symbols. The parameter

the channel and/or the detection strategy. As an example, for

an AWGN channel,

in the case of coherent detection,

whereas

is the assumed phase memory for noncoherent

detection based on the algorithm proposed in [6]. A single

branch can be related to

information symbols, that is

. The relation between integers

informationsymbols)and

(relativetocodesymbols)depends

on the coding structure.2We assume that the receiver observes,

and

are generated by a

. In

code symbols,

denotes

and the

is related to

, where the symbol

(relative to

2Iftheconsideredcodeisrecursive,asitisusuallyforturbocodes,oneshould

consider some encoder state variables together with the information symbols in

the definition of ? and ? [6].

at each epoch, a (vector) sample

the output of the channel. Generalizations to the case of

oversampling or multiple decoder inputs are straightforward.

Denoting by

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 [17]–[19], 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

transition

, 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),

a transition

, 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

at

code symbols

. Following the prin-

in the reduced trellis

of in-

.3The central point is that,

informa-

,

,, and . Hence, for a given

forin

and epochs

is associated with

as in a classical case.

((2)) can be expressed as

(5)

where

this recursion by the “max-log” approximation as follows [9]

. We can approximate

(6)

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

of

transitions reaching epoch

transition

, i.e.,

, as in

, we now show how the

by using the for-

the sequence

along the survivor of

=

. Each transition

andinformationsymbol

, depend on the

, for

,in

, for

transition

uniquely identifies the sequence of information symbols

. Hence, the couple

whererep-

resent the information symbols in the path history of transition

3More complex techniques based on set partitioning may also be employed

[17], [19].

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COLAVOLPE et al.: REDUCED-STATE BCJR-TYPE ALGORITHMS851

.Similarly,wecanextenda coupleoftransitions

inordertogetinformationsymbols.Withthesedefinitions,

it is easy to extend the survivors by a step, i.e., to epoch . In

the reduced-state trellis, taking into account paths

associatedwithtransitionsatepoch,(6)reducesto

(7)

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-

pute

but only to determine the survivor associated with

transition

.

We note that the concept of survivor associated with a transi-

tioncanbereconciledwiththeclassicalonebyproperlydefining

anaugmentedtrellisdiagram,tobeusedintheforwardandback-

wardrecursions,andreformulatingthedescribedBCJR-typeal-

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

theprobability

,byevaluatingtheprobabilitydensity

function

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-

tionsoftheproposedformulationtotheothercasesarepossible.

Given the above definitions, we can reformulate (1)–(3), ob-

taining

, the branch that maximizes (7)

as shown in

as an average

and in the calculation of

in the combination (1). In the

(8)

(9)

(10)

Fig. 1.

general reduced-state BCJR-type algorithm.

Forward recursion of the probability density function ? ?? ? for a

Fig. 2.

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

where

transitions

the forward recursion proceeds in the reduced-state trellis ac-

cording to (9). In order to compute

the

probability density functions

and, for each of them, compute the branch

metric

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 [9], [10] to (8)–(10).

By defining

and

mated as follows

and

and

are the initial and final states of

, respectively. In Fig. 1, it is shown how

, one should consider

such that

, (9) can be approxi-

(11)

Similarlogarithmicextensionsholdfor(8)and(10),bydefining

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

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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-

in combination

(12)

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-

rithmsisapproximately

state case.

With the introduction of the concept of survivor associated

withatransition,furthergeneralizationstoPSPtechniques[20],

such as those considered in [21]–[23], [33], are possible with

taking into account that a single step in the recursions of a

BCJR-typealgorithmcouldinvolvetwoconsecutivetransitions.

andbe-

timeslessthanthatofthefull-

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 [6], and detection based on linear prediction for Rayleigh

flat-fading channels.

A. ISI Channels

In the case of an ISI channel, we assume uncoded transmis-

sion; hence,

reduces to . The observation samples at the

output of a whitened matched filter, can be written as

[34]. The noise samples

independent Gaussian random variables, with zero mean and

variance

. The distortion on the elementary shaping pulse

makes the samples

dependent on

each weighted by a different coefficient

state as

the classical BCJR algorithm can be directly applied with

[2]. Hence, the probability density

functions which appear in (1)–(3) reduce to

are real

information symbols,

. By defining the

, it is easy to show that

(13)

(14)

(15)

(16)

Fig. 3.

the reduced-state BCJR algorithm in the case of coherent detection over ISI

channels.

Forward recursion of the probability density function ? ?? ? for

(17)

(18)

where symbol

tion (1)–(3) can be specialized to this case by letting

(that is, the receiver observes a single sample

epoch).

Iftheimpulseresponseoftheequivalenttime-discretechannel

has minimum phase [17]–[19], [34], an efficient re-

duced-state trellis diagram is obtained by defining a state

,with

tion

in the path history associated with state

determine

=

denotes proportionality. The general formula-

and

at each

,and,consequently,atransi-

. Hence, searching

, we may

andobtain

(19)

(20)

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

in

. 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

tion

, in (11) reduces to

for the forward recursion reduces to

and not

a state,

branches ending

and transi-

. The expression (11)

(21)

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COLAVOLPE et al.: REDUCED-STATE BCJR-TYPE ALGORITHMS853

Fig. 4.

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

(22)

If the overall impulse response

efficient definitions of trellis state and transition may be

is maximum phase,4

and

, 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,

suitedtoimpulseresponseswithenergyconcentratedtowardthe

end,areconsideredin[37],[38].Inthis“specular”versionofthe

previouslyintroducedalgorithm,duringthebackwardrecursion

theinformationsymbol

state

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

denoted by

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-

sionsmayproveusefuliftheoverallimpulseresponseis“mixed

phase.” A specific investigation of this case was not performed

, and runs the

isrelativetothetransitionfrom

is discarded. The

= , where

of transition.

4Forexample,ifacausalwhiteningfilterisselected.

because beyond the scope of the paper. Related aspects are dis-

cussed in [22].

B. Noncoherent Channels

In the case of noncoherent decoding, we refer to the BCJR-

type algorithm proposed in [6]. 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

[6]; hence,

case, the probability density functions

in (1)–(3) specialize as follows

the en-

. In this

appearing, and

I(23)

I

I

(24)

I

I

(25)

where

By defining a reduced state as

, with

becomes

sity function

can be expressed as

is the transpose conjugate operator.

, a transition in the reduced-state trellis

. Hence, the probability den-

I (26)

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854IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 19, NO. 5, MAY 2001

and (8) specializes as follows

(27)

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 [6], the exten-

sion of the previously introduced general recursions is not im-

mediate. Noting that

on

can be expressed as follows

in (26).

, the forward recursion

(28)

Since at epoch

and

with

vivor associated with

eration. The term

but it does not affect the survivor selection. Instead of

computing

we propose to utilize

mined through the max operation and replace it with

the survivor of each transition

, we replace

is known

. The choice of the sur-

) is based on the max op-

affects the exact value of

(that is,

deter-

I

(29)

where the expression

ically related quantities. The forward recursion finally reduces

to

denotes that and are monoton-

I

I

(30)

Once the survivor map has been determined during the forward

recursion, the probability density function

ward recursion, can be calculated

, used in the back-

I

I

(31)

C. Fading Channels

Weconsiderthetransmissionofdifferentiallyencoded

phaseshiftkeying(

-PSK)overaRayleighflat-fadingchannel.

We refer to the transmission system considered in [39] and

denote by

the normalized fading rate of the channel. We

assume that each information symbol

of

bits, i.e.,

symbol

isobtainedbyGray-mappingthebits

Thedifferentiallyencoded

-arysymbols

rule

,with

The corresponding received signal at the matched filter output

canbewrittenas

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 [40].

Assuming that the information bits are independent within

each symbol, we can consider

The proposed formulation gives the a posteriori probability of

the

-PSK information symbols

bit-wise, as in [39], a posteriori probabilities of bits

to be calculated. To obtain these probabilities (1) can be modi-

fied as follows

-ary

corresponds to a group

. The-PSK

.

aredefinedbythe

(actsasareferencesymbol).

,wherethechannelcoefficients

are

.5

. If the interleaver works

have

(32)

for

in practice, linear prediction [41], [42] can be used. The branch

metric

becomes [39]

.Sincethechannelcoefficients areunknown

(33)

5Inthecaseofaniterativedecodingprocess,where???

input extrinsic information, this is an approximation.

?arederivedfrom

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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

[43]. We may observe that

, the corresponding trellis consists

of

states. Assuming that the autocorrelation coefficients

of the channel fading process are known, the optimal predic-

tion coefficients

,

Wiener-Hopf equations [39]. Hence

is the prediction order,

is the variance

. By defining

are obtained by solving the

(34)

Noting that

anddo not depend onand defining

we have

(35)

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

,with

is identical to that used in Section IV-A)

and should not be

,obtaining(theformalization

(36)

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

Fig. 5.

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

with circles).

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 [35]. 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-

lowing coefficients:

and[35]. 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 [36]. The extrinsic infor-

mation is used according to the heuristic method proposed in

[44]. 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-

tector with

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

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856IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 19, NO. 5, MAY 2001

Fig. 6.

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

of

, 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.

Inthecaseofnoncoherentdecoding,wefirstconsiderasingle

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 [7]. 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-

coder with

and

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-

ceiver with

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 [7]. 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 [44]. 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

values of

and, and compares it to that of a coherent re-

ceiver and a noncoherent receiver with

=

,

states (full complexity), for

and

and

Fig. 7.

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

algorithm with

and

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 [6] applicable, even for large values of

phase memory

.

Finally, we consider transmission of differentially encoded

quaternary PSK (QPSK) signals over a flat-fading channel, as

in [39]. The outer code is a 64-state nonrecursive convolutional

code with generators

concatenated,througha

an inner differential encoder. In fact, bit interleaving is an ap-

propriate means to combat the effects of fading [45], [46]. The

normalizedfadingrateis

tector uses linear prediction and the state reduction technique.

In Fig. 8 the performance of the full-state receiver, with predic-

tion order

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

and

loss of only 0.2 dB at a BER of

one (

). 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

and

). We conclude that

and. This code is

nonuniformbitinterleaver,with

.Thedifferentialinnerde-

). In all cases, we

exhibits a performance

, with respect to a full-state

, the performance gain

is about 0.6 dB at

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COLAVOLPE et al.: REDUCED-STATE BCJR-TYPE ALGORITHMS857

Fig. 8.

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,

VI. CONCLUSION

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.

APPENDIX

The considered formulation of a BCJR-type algorithm dif-

fers from that of the classical BCJR algorithm [2], 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

definition.Inthisaugmentedtrellis,wemayidentifyatransition

as

. With these definitions, we may rewrite

recursion (2) as

now appears in the state

(37)

where

of transition

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

theaposterioriprobabilityofaninformationsymbol

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

states

atepoch ,thatisoverallstatescharacterizedbythe

considered symbol

(and not over all states). Therefore, even

if this alternative formulation would entail a forward recursion

inaformsimilartothatofausualBCJRalgorithm,itwouldstill

exhibit differences in this central combination.

Inasecondalternativeformulation,forwardandbackwardre-

cursionsmaybecomputedontheaugmentedtrellis,whereasthe

central combination may be computed on the original trellis. In

thiscase,theBCJR-typealgorithmissimilartoaclassicalBCJR

but the two recursions and the central combination must be re-

ferredtotwodifferenttrellisrepresentationsinordertoconsider

a single step as involving a single transition. Specifically, in the

forward and backward recursions an augmented trellis must be

used with

times the number of states of the trellis relative to

the central combination.

and

, respectively, and the sum is extended over all

. The forward recursion (37) now re-

are the beginning and ending states

. A

.Unlike

of the possible super-

ACKNOWLEDGMENT

References [24]–[29], [31], and [33] were brought to the au-

thors’attentionduringthereviewprocess.Threeanonymousre-

viewers and W. H. Gerstacker are gratefully acknowledged.

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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

researchinterestsincludedigitaltransmissionandde-

tectiontheory,channelcoding,anditerativedecoding

techniques.

Dr. Ferrari is a CNIT member and an AEI young member.

Page 12

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

1987.

From1986to1988,hewasaProjectEngineerwith

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.