Low-Complexity Soft Decoding of Huffman Codes and Iterative Joint Source Channel Decoding

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 60, NO. 6, JUNE 2012 1669

Low-Complexity Soft Decoding of Huffman Codesand Iterative Joint Source Channel Decoding

Amin Zribi, Ramesh Pyndiah, Senior Member, IEEE, Sonia Zaibi, Frédéric Guilloud, Member, IEEE,and Ammar Bouallègue, Member, IEEE

Abstract—Most source coding standards (voice, audio, imageand video) use Variable-Length Codes (VLCs) for compression.However, the VLC decoder is very sensitive to transmission errorsin the compressed bit-stream. Previous contributions, using a trel-lis description of the VLC codewords to perform soft decoding,have been proposed. Significant improvements are achieved bythis approach when compared with prefix decoding. Nevertheless,for realistic VLCs, the complexity of the trellis technique becomesintractable. In this paper, we propose a soft-input VLC decodingmethod using an a priori knowledge of the lengths of the source-symbol sequence and the compressed bit-stream with MaximumA Posteriori (MAP) sequence estimation. Performance in the caseof transmission over an Additive White Gaussian Noise (AWGN)channel is evaluated. Simulation results show that the proposeddecoding algorithm leads to significant performance gain incomparison with the prefix VLC decoding besides exhibitingvery low complexity. A new VLC decoding method generatingadditional information regarding the reliability of the bits of thecompressed bit-stream is also proposed. We consider the serialconcatenation of a VLC with two types of channel code andperform iterative decoding. Results show that, when concatenatedwith a recursive systematic convolutional code (RSCC), iterativedecoding provides remarkable error correction performance.In fact, a gain of about 2.3 dB is achieved, in the case oftransmission over an AWGN channel, with respect to tandemdecoding. Second, we consider a concatenation with a low-densityparity-check (LDPC) code and it is shown that iterative jointsource/channel decoding outperforms tandem decoding and anadditional coding gain of 0.25 dB is achieved.

Index Terms—Communication system performance, sourcecoding, variable-length codes, error correction coding, iterativemethods.

I. INTRODUCTION

NEXT generation communication systems are faced withthe competing goals of providing high quality of service

for multimedia applications while minimizing power andbandwidth consumption. It is clear that both source codingand channel coding are important components to achieve

Paper approved by J. Kliewer, the Editor for Iterative Methods and Cross-Layer Design of the IEEE Communications Society. Manuscript received June1, 2010; revised January 30, August 25, and November 21, 2011, and January23, 2012.

A. Zribi, S. Zaibi, and A. Bouallègue are with the Communications Systems(SysCom) Laboratory, National Engineering School of Tunis, BP 37, LeBelvedere, 1002, Tunis, Tunisia (e-mail: [email protected]; {sonia.zaibi,ammar.bouallegue}@enit.rnu.tn).

R. Pyndiah and F. Guilloud are with the Signal and Communica-tions Department, Institut Telecom, Telecom Bretagne, UMR CNRS 6285Lab-STICC, Technopole Brest Iroise, CS 83818, 29238 Brest Cedex3, Université Européenne de Bretagne, France (e-mail: {ramesh.pyndiah,frederic.guilloud}@telecom-bretagne.eu).

Digital Object Identifier 10.1109/TCOMM.2012.041212.100330

these goals. Shannon’s separation theorem [1] states that thesetwo components can be optimized separately if unconstrainedblock lengths and unconstrained coding and decoding com-plexities are allowed. In most practical applications, it isimpossible to fulfill these requirements. Joint Source/Channel(JSC) coding and decoding techniques have emerged as a prag-matic approach leading to a convergence of the multimediaand digital communication worlds.

Variable-Length Codes (VLCs) are widely used in state-of-the-art image, video, and audio compression schemes. Al-though they provide a reduction in the data rate by exploitingthe redundancy in the source-symbol sequence, the VLC-encoded data are very sensitive to transmission errors (errorpropagation phenomenon). Especially when the compressedbit-stream has to be transmitted over a noisy channel, efficientcoding or decoding techniques have to be considered toguarantee a good source-symbol sequence reconstruction.

The first contributions, reported in [3] as robust entropycoding techniques, add some redundancy in the entropy cod-ing stage to help detect the erroneous bits or prevent errorpropagation. Since a single bit error in the VLC-encodedstream can lead to the loss of synchronization, the authorsin [4] introduced a class of VLCs with good synchronizationproperties. In [5], variable length bit-streams from individualblocks are distributed into slots of equal size to improvetheir synchronization. This technique was called EREC forError Resilient Entropy Coding. Reversible VLCs (RVLCs)were proposed in [6] to increase the compressed bit-streamrobustness against the channel errors. Additional work focusedon exploiting the residual redundancy in the compressed bit-stream to improve the decoding performance [7]–[9], [21].

Recent work exploits the VLC properties to improve itsdecoding efficiency. These techniques consider trellises thatare built using the structure of the VLC codebook to applywell-known algorithms used for channel decoding such asViterbi and List Viterbi [10]–[13]. In [10], [11], the authorsused channel soft outputs to perform the MAP decoding on atrellis whose states are indexed by the VLC-encoded sequencelength. The same decoding strategy was applied in [13] on bit-synchronized trellises describing both a Markov source andthe VLC encoder. Other contributions assume that a prioriinformation such as the number of source symbols or thesize of the encoded bit-stream is available at the decoderto improve the decoding performance. In [14] and [17],bit- and symbol-synchronized trellises were presented, wherethe compressed bit-stream and the source-symbol sequence

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1670 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 60, NO. 6, JUNE 2012

lengths are supposed to be known at the receiver. Decodingis performed using a Viterbi-like algorithm where the branchmetrics consider both the source and channel statistics. How-ever, the state reduction techniques used in these algorithmsdegrade the decoding performance [15].

Later contributions proposed to exploit the efficiency of theturbo decoding by integrating the VLCs in an iterative decod-ing process [21]–[24]. While Bauer and Hagenauer [22], [23]proposed new VLC trellis representations for Soft-Input Soft-Output (SISO) VLC decoding, Guyader et al. [24] introduceda Bayesian network-based decoding approach. Both decodingschemes considered a VLC concatenated with a convolutionalcode, and substantial gains were obtained with iterative decod-ing. Iterative decoding of the serial concatenation of a VLCwith a turbo code and a Low-Density Parity-Check (LDPC)code were respectively studied in [25], [26] and [27]. Allthe described techniques rely on a specific trellis constructionand eventually combined with pruning, resulting in variousefficiencies. However, in the proposed trellises, the statesnumber increases with the source-symbol sequence length Land the source alphabet size U . Thus, the decoding complexitybecomes intractable for large values of L and U .

Finally, some contributions considered the JSC decoding ofan LDPC channel code in the case of image transmission. In[28], the authors considered rate-compatible LDPC codes tostudy unequal error protection on the compressed JPEG2000bit-stream. Extra information provided by the error-resiliencemode of the JPEG2000 encoder was delivered to the LDPCdecoder and iterative decoding was performed in [29].

In this paper, the Maximum A Posteriori (MAP) sequenceestimation criterion is employed for the maximum likelihooddecoding of the VLCs in the case of transmission over anoisy channel. We suppose that the source-symbol sequencelength L and the corresponding compressed bit-stream size lare known at the decoder side. The VLC decoding is basedon a search of the MAP sequence among the length-validsequences (bit-streams of length l decoding L symbols). Theoriginal contribution of our work lies in the adaptation of theChase decoder [31] for soft input VLC decoding. In fact, weproposed a Chase-like VLC decoder with reduced complexityand taking into account the bit-streams’ length validity. Sucha condition was considered in [14], [15], [17], but the authorsused a trellis representation to apply the VLC approximate softinput decoding. The VLC decoding algorithm proposed in thispaper does not use state machine representation for the VLCswhich makes it more simple to implement. The feasibility ofthe proposed approach is demonstrated in different contextsand the results are compared with [14] and [15]. As asecond contribution, we propose an iterative JSC decodingscheme based on a VLC concatenated with a channel code.The VLC is used to perform source compression and thechannel code performs error correction. After the soft-inputsoft-output channel decoding, the VLC Chase-like decoderis modified to identify reliable and non-reliable compressedbits and this extra information is provided to the channeldecoder at the next iteration. We consider two types of channelcodes : a recursive systematic convolutional code (RSCC)and a capacity-approaching low-density parity-check code(LDPC). The source/channel decoding is iterated several times

Fig. 1. Block diagram of the studied transmission system.

to improve the error-rate performance.The paper is organized as follows. In Section II, the system

model and the MAP sequence decoding metric are brieflyreviewed. In Section III, a new low-complexity VLC decodingalgorithm is described and its performance are validated indifferent configurations including JPEG image transmission.The results are also compared with those of previous solutionsusing a VLC trellis-decoding algorithms [14], [15] in terms ofcomplexity and error correcting performance. In Section IV,we propose a simple method to achieve a low-complexityVLC decoding with bit reliability definition. The latter isserially concatenated with a SISO channel decoder to performiterative JSC decoding. Results are investigated in the case of aconcatenation with a RSCC, then with an LDPC code. Finally,Section V gives some conclusions and suggests directions forfuture work.

II. SYSTEM OVERVIEW

The communication system model considered in this paperis presented in Fig. 1 and consists of a VLC encoder, a trans-mission channel, and a VLC decoder. We suppose a sourcegenerating symbols sk from the set Ω = {0, 1, . . . , U − 1}with the probabilities pu, u ∈ Ω. The VLC table (of size U )is supposed to be tree-structured and will be determined usingthe Huffman construction method [2]. Let s = (s1, s2, . . . , sL)denote the source-symbol sequence generated with a non-uniform distribution pΩ. The compressed bit-stream b =(b1, b2, . . . , bl) of length l = f(s) is then transmitted overan AWGN channel, which delivers r = (r1, r2, . . . , rl). Whenusing a BPSK modulation, the channel output is given byrj = hj + nj , j = 1, . . . , l where hj =

√Es(1 − 2.bj) is the

modulated value of bj , Es = Rc ·Eb with Eb being the energyper information bit, and Rc the channel coding rate which isequal to one if no channel code is used. nj is a Gaussiannoise sample with zero mean and variance σ2. A standardVLC decoder performs a hard decision on the componentsof r and uses the prefix-free condition to deliver the decodedsequence s.

As mentioned in the introduction, the VLC decoding algo-rithm presented in this paper is based on an estimation of theMAP sequence which minimizes the source Packet Error Rate(PER). The MAP rule corresponds to the search for the bestsequence s satisfying:

P (s = si|r) ≥ P (sk|r), ∀k �= i (1)

Hence, for a compressed bit-stream bk = (bk1 , bk2 , . . . , b

kl )

obtained after the VLC encoding of the source-symbol se-quence sk = (sk1 , s

k2 , . . . , s

kL), the decoding is based on the

maximization of the metric

P (sk|r) = P (r|sk)P (sk)P (r)

=P (r|bk)P (sk)

P (r)(2)

ZRIBI et al.: LOW-COMPLEXITY SOFT DECODING OF HUFFMAN CODES AND ITERATIVE JOINT SOURCE CHANNEL DECODING 1671

The metric includes the channel-transition probability, the apriori source probability, and the term P (r). The last term canbe ignored since it is constant for all the candidate sequences.The other terms can be calculated using the channel modeland the a priori source model. Regarding the source symbols,for independent source symbols, we have P (sk) =

∏i p(s

ki ),

whereas for a first order Markov source we get P (sk) =∏i p(s

ki |ski−1).

Taking the logarithm of (2), we can reformulate the problemas a maximization of the following metric:

log(P (sk|r)) = log(P (r|bk)) + log(P (sk))− log(P (r)) (3)

Since P (r) is constant for all candidates, in the case oftransmission across an AWGN channel with BPSK signalling,the soft decoding metric can be written as

Mk =

l∑j=1

(− (rj − hk

j )2

2σ2

)+ log(P (sk)), (4)

where hkj =

√Es · (1− 2 · bkj ) is the modulated value of bkj .

The exhaustive decoding approach evaluates the metricMk for all possible pairs (sk, bk) and selects the one withthe highest metric. Based on the a priori knowledge of theencoding VLC table, such a decoding scheme requires thecomputation and the comparison of 2l sequence-metrics: all bi-nary sequences bk of length l. However, decoding these 2l bit-streams results in variable-length source-symbol sequences,and we are sure that the decoded sequences of length differentfrom L are in error. Thus, an extra information regarding thelength of the source-symbol sequence L can help the VLCdecoder to detect erroneous sequences among the 2l possiblebit-streams.

Length-constrained VLC decoding was already consideredin previous contributions like [14], [15], [17]. In fact, the extrainformation about L can reduce the size of the search space,and, consequently, increase the minimum Hamming distancebetween all possible candidate sequences. In the following,we propose a new VLC decoding algorithm that restrictsthe search space to the most probable length-valid streamsbased on the soft information delivered by the channel. Wealso assume that the length information is transmitted to thedecoder with no error. In practice, this can be guaranteedby the use of a channel code as in [19] or possibly byincluding this information in the frame header when a specificcompression standard is used.

III. LOW-COMPLEXITY SOFT DECODING OF HUFFMAN

CODES

A. The Chase-like VLC Decoder

As mentioned previously, much work has been done on thesoft input decoding of Huffman codes. In general, the pro-posed techniques rely on a statistical estimation of the trans-mitted source-symbol sequence based on noisy observationsdelivered by the channel. For soft input decoding, we have tospecify two constraints: the estimator and the search space.When the described MAP sequence estimator is considered, alength-constrained decoder selects the compressed bit-stream

having the best metric Mk (given by (4)) among all length-valid sequences in the search space. However, evaluatingall these possible bit-streams is infeasible in practice andnumerous solutions proposed suboptimal decoding algorithmsbased on a trellis description of the code.

In this section, we propose a Chase-type algorithm [31] toachieve a low-complexity suboptimal soft-input decoding forthe VLCs. The Chase algorithm was originally proposed forsoft-input block-code decoding. The main idea of this algo-rithm is to limit the search space to only the Q most probablesequences, based on soft inputs, to reduce the complexity.

A classical decoding scheme consists in applying a VLCprefix decoding to binary sequence y = (y1, . . . , yl) obtainedby a hard detection on the received values r. The soft-input decoder uses the extra information given by r calledreliabilities. The reliability of a component yj is defined usingthe Log Likelihood Ratio (LLR) of the transmitted bit bj . Foran AWGN channel with BPSK signaling and equal a prioriprobabilities, we can write:

|Λ(bj)| =∣∣∣∣log P (bj = 0|rj)

P (bj = 1|rj)∣∣∣∣ = 2

σ2· |rj | (5)

For a stationary channel, the LLR can be normalized andthe reliability of yj is given by λj = |rj |. Then, the Chase-like decoding search space is based on binary sequenceswith different bit combinations at the q least reliable posi-tions. Such a decoder has been validated by the authors in[40] for soft arithmetic decoding and showed good perfor-mance/complexity tradeoff. The proposed length-constrainedChase-like decoding of the VLCs is as follows:

1) Evaluate the hard decision vector y and the correspond-ing reliability vector Λ(y) = (λ1, . . ., λl),

2) determine the positions of the q least reliable binaryelements of y based on Λ(y),

3) form test patterns ti, 0 < i ≤ Q with Q = 2q: all thel-elements binary vectors with elements tij , 0 < j ≤ lhaving a weight less than or equal to q with “1”s inonly the least reliable positions and “0”s in the otherpositions (all zero test pattern represents the case whereno transmission errors have occured),

4) form test sequences zi, 0 < i ≤ Q with zij = yj ⊕ tij ,for 0 < j ≤ l,

5) decode all the test sequences zi using the standard VLCdecoding. If a sequence zi decodes exactly L symbols,we compute its metric using (4) and append it to thesubset of the competitor valid code sequences Ψ.

6) Finally, the decoded bit-stream corresponds to the se-quence having the best metric Mk in the subset Ψ.

B. Simulation Results

In this section, we present the simulation results of theproposed Chase-like VLC decoder in terms of the PER in thecase of transmission over the AWGN channel using the BPSKmodulation. Results are compared with the prefix-based VLCdecoding and the effects of the chosen code and the patterntest weight q of the Chase-like decoder are detailed.

Let us consider a memoryless source with U = 5 symbols.Table I gives the probabilities and the VLC codewords as-signed to each symbol for the three codes C1, C2, and C3. We

1672 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 60, NO. 6, JUNE 2012

Fig. 2. PER versus EbN0

for the Chase-like VLC decoding in the case oftransmission over an AWGN channel with BPSK modulation. Results aregiven for the codes C1, C2, and C3 with different values of q.

notify that the described source and the codes were introducedin [37], then used in [38] as a reference for performance com-parison when length-constrained soft input VLC decoding isapplied. The presented codes have the same average length andconsequently provide the same PER performance for prefixdecoding. The investigated scheme considers the transmissionof source-symbol sequences of L = 128 symbols and theobtained results are depicted in Fig. 2.

Although codes C1, C2, and C3 have the same average lengthof l = 2.2 bits per symbol, they deliver different results withChase-like VLC decoding. In fact, it can be seen that the codeC1 is a particularly poor code in terms of soft input decoding.However, applying Chase-like decoding to the code C2 yieldsa performance gain of about 0.7 dB with respect to prefixdecoding. The code C3 outperforms by about 0.5 dB the codeC2, and results in an overall improvement of more than 1.2dB compared to prefix decoding. These results confirm theconclusions made in [38] where the authors proposed criteriato select the best VLC code for length-constrained soft inputdecoding. In fact, they defined:

• P (ΔS = 0) as the probability that an error induces a hardresynchronization. In this case, the error would have noeffect on the decoded sequence length.

• H(ΔS) as the amount of information delivered by thelength constraint.

In Table I, we report, for each code, the calculated valuesof P (ΔS = 0) and H(ΔS) which are, according to [38],the best criteria for choosing the best code for soft inputdecoding. For a fixed length l, if P (ΔS = 0) is small, thenumber of discarded sequences due to the length constraintswill increase and, consequently, the probability of correctdecisions will increase. This observation is confirmed by thesimulation results presented in Fig. 2. In fact, the code C3 hasthe smallest P (ΔS = 0) and the highest entropy H(ΔS), andis also the code which presents the best performance gain bycomparison to C1 and C2. The worst results are obtained withC1 which presents the highest P (ΔS = 0) and the smallest

TABLE IDIFFERENT VLCS FOR A MEMORYLESS SOURCE WITH U = 5 SYMBOLS

AND ENTROPY=2.12 BIT/SYMBOL.

Symbol u pu C1 C2 C3

0 0.4 01 0 01 0.2 00 10 112 0.2 11 110 1013 0.1 100 1110 10004 0.1 101 1111 1001

l 2.2 2.2 2.2P (ΔS = 0) 0.675 0.305 0.131

H(ΔS) 1.39 2.40 4.38

TABLE IICODE CA . A SYMMETRIC REVERSIBLE VLC [6].

Symbol u pu CA

0 0.33 001 0.3 112 0.18 0103 0.1 1014 0.09 0110

H(ΔS).In the second experiment, we propose to investigate the

effect of the test pattern weight q on the Chase-like VLCdecoding performance. We present, in Fig. 2, simulationresults obtained for the best VLC code C3 with q = 1, 2, and4 bits. We can see that with q = 2 we have, for a PER=0.1,an improvement of about 1 dB over the classical prefix VLCdecoding. Increasing q to 4 results in a small additional gainfor small to medium values of Eb

N0, but causes a significant

increase in complexity. Thus, for this experiment, the bestperformance/complexity tradeoff is obtained with q = 2 bits.

Nowadays, Reversible VLCs (RVLCs) have emerged as analternative to Huffman codes that allow decoding in bothdirections, which can be exploited by the decoder to improveits performance. They were first introduced by Takishima etal. in [6] and are widely considered in video and imagecompression standards. In this experiment, we evaluate theefficiency of the soft-input Chase-like decoder with such acode and compare results to an optimal MAP decoding schemetaking into account the length constraints. We assume a BPSKmodulation and an AWGN channel transmission with source-symbol sequences of length L = 100.

Fig. 3 illustrates the Levenshtein symbol error rate (SER)results for the symmetric RVLC which is given in [6] andpresented in Table II. We recall that the Levenshtein distanceis defined as the minimal number of insertions, deletions,or substitutions that transform one sequence into the other.Results of the instantaneous VLC decoder, the Chase-likedecoder with q = 4 bits, and the MAP decoder are depicted.We note that the simulated MAP decoder was presented in[11], and applies the Viterbi algorithm on a specific trellistaking into account the length information. More details canbe found in [11].

According to the obtained results, the Chase-like VLCdecoder offers almost a 3.0 dB gain in terms of Eb

N0when

compared to prefix decoding. This remarkable gain demon-strates that RVLCs are good candidates for Chase-like decod-

ZRIBI et al.: LOW-COMPLEXITY SOFT DECODING OF HUFFMAN CODES AND ITERATIVE JOINT SOURCE CHANNEL DECODING 1673

Fig. 3. Levenshtein SER versus EbN0

for the Chase-like VLC decoding in thecase of transmission over an AWGN channel with BPSK modulation. Resultsare given for the RVLC CA and compared to MAP decoding presented in[11].

ing. Compared to MAP decoding, the suboptimal Chase-likedecoder offers similar results with increasing Eb

N0values. Thus,

we can conjecture that the Chase-like VLC decoder is veryefficient and asymptotically optimal.

C. Comparison With VLC Decoding Using a Trellis Repre-sentation

Now, we propose to compare the performance of thedescribed Chase-like VLC decoder with trellis-based MAPdecoding using state reduction algorithms proposed in [14]and [15]. We recall that these techniques are based on thebit-constrained directed graph of [14] taking into account thelength constraints. Moreover, at each bit step, the states aregrouped into sets each representing the encoding of a symbolfrom the alphabet (we have U sets at each bit step). In [14],Park and Miller proposed an optimal MAP decoding methodand an efficient approximation by directed graph search. Themain problem with the optimal trellis decoding is the hugenumber of states in the graph when the encoded sequencelength grows. It was proven in [15] that the number of statesgrows linearly with the value of the trellis bit step. In orderto reduce the complexity, at each bit step, Park and Millerproposed to prune all the states but one in each of the Usets. In [15], the authors proposed two approximate methodscalled AMAP1 and AMAP2 with different state reductionoperations. At each bit step, AMAP1 keeps only one statefor every possible symbol number. The corresponding savedsequence should be the best in the sense of partial a posterioriprobability for all graph states at this specific bit step. Thesecond method proposed in [15], denoted AMAP2 keeps onlythe best T paths in the sense of partial a posteriori probabilityfor each trellis bit step.

In order to make a fair comparison, we consider in thisexperiment a first-order Gauss-Markov source with correlationcoefficient ρ = 0.9. We applied a three-level scalar-quantizer

on the obtained source and the corresponding probabilitymatrices were estimated using a long training sequence. Thesource sequence s is the quantization index-sequence outputby the described scheme. The resulting alphabet consists ofU = 3 symbols with probabilities {0.5, 0.25, 0.25}. Thecorresponding Huffman code is {0, 10, 11} giving l = 1.5 bitper symbol. The experiments are based on BPSK modulationand transmission over an AWGN channel with source-symbolsequences of length L = 100.

Table III presents the performance results in terms of PERcorresponding to several cases: prefix VLC decoding, approx-imate decoding of [14], the AMAP1 method, and the AMAP2VLC decoding with T = 3 [15]. We propose to compare thesereferences with the Chase-like VLC decoding. In the same ta-ble, we propose a comparison of the computational complexityof the simulated decoders. The computational complexity of adecoder was defined in [16] as γ =

∑li=1 γi =

∑li=1 Gi · Ji

where γi is the overall complexity at a bit depth i determinedwith Gi and Ji which are respectively the number of reservedstates and the number of each state projected branches atthe same bit depth. Results show that the Chase-like decoderimproves the decoding performance when compared with theclassical decoding, and outperforms the approximate methodof [14]. By reference to the decoding methods of [15], wecan see that for medium to high values of Eb

N0, the Chase-like

decoding performance are almost the same as AMAP1 andAMAP2. We recall that it was demonstrated in [15] that theAMAP2 presents nearly the same performance as the optimaltrellis-based soft decoding, which confirm the fact that theChase-like VLC decoder is asymptotically optimal.

For a fixed value of q, the computational complexity ofthe Chase-like decoder is constant. In fact, it has an averagetotal complexity of l · 2q VLC prefix decoding and metriccomputation operations. In Table III, we used three differentq values for Chase-like VLC decoding according to thechannel conditions. When the channel is noisier, we need moretest patterns for Chase-like VLC decoding to reach the bestperformance. We can see that the computational complexityis less than the other trellis-based decoders at medium to highSignal to Noise Ratio (SNR). We also notify that increasing thealphabet size U induces more states per trellis bit step and,consequently larger complexity for trellis based algorithms.Furthermore, additional computations are needed for the trellisconstruction since for a given L, the trellis depends on l.Indeed, for every l value, we need to build a new trellis, whichis unacceptable in practical applications. Thus, we can con-clude that, compared to the described decoding algorithms, theChase-like decoder presents the best performance/complexitytradeoff at medium to high SNR. Moreover, the Chase-likedecoder can be easily extended to adaptive versions of VLCs.The trellis-based decoders are difficult to apply in the case ofadaptive VLC since the latter is described by a time-variablecode, which means variable trellis for the same source-symbolsequence.

D. JPEG Image Transmission Application

Now, we propose to evaluate the improvement achieved bythe Chase-like VLC decoder on the visual quality of JPEG

1674 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 60, NO. 6, JUNE 2012

TABLE IIICOMPARISON OF DIFFERENT SOFT INPUT VLC DECODING SCHEMES IN TERMS OF PER AND COMPUTATIONAL COMPLEXITY γ .

EbN0

= 4 dB EbN0

= 6 dB EbN0

= 8 dBVLC decoder PER Complexity γ PER Complexity γ PER Complexity γ

prefix decoding 0.83 150 0.29 150 0.028 150Chase-like decoding 0.73 (q = 4) 2400 0.2 (q = 3) 1200 0.015 (q = 2) 600Park & Miller [14] 0.73 1350 0.27 1350 0.027 1350

AMAP1 [15] 0.67 16725 0.2 16725 0.016 16725AMAP2 T = 3 [15] 0.63 1350 0.2 1350 0.015 1350

Fig. 4. Average PSNR versus EbN0

for classical and Chase-like VLC decodingin the case of an AWGN channel transmission with BPSK modulation, andJPEG image compression.

image transmission. We used the baseline configuration of theJPEG standard. The input image is partitioned into 8 × 8sub-blocks followed by the 2D Discrete Cosine Transform(DCT) for each sub-block. The number of quantization levelsof the DCT coefficients controls the achieved compressionratio (we used the quantization table proposed in the standardfor luminance components [39]). The quantized coefficientsare then zigzag scanned. The coefficients are partitioned intoDC and AC coefficients which are respectively compressedusing a DPCM scheme and run-length encoding (RLE). Theresulting stream is then Huffman encoded to deliver thefinal JPEG compressed bit-stream. We used the VLC tablepresented in [39] with an alphabet of U = 162 symbols.Applied to the grayscale test image lenna 512× 512 (8 bpp),the JPEG outputs 0.64 bpp and a PSNR= 35.78 dB. Weconsider the transmission over an AWGN channel and evaluateperformance in terms of the average PSNR over 500 imagetransmissions.

Fig. 4 shows that the use of the proposed Chase-like length-constrained decoding with JPEG image transmission enablesremarkable gains. With a single additional test sequence(q = 1), we can achieve more than 5 dB of gain in PSNR forEb

N0= 7 dB. Increasing the number Q = 2q of test sequences

improves decoding performance for a relatively noisy channel.In fact, for Eb

N0= 5 dB, increasing Q from 2 to 32 gives

approximately a gain of 4 dB in average PSNR.

IV. ITERATIVE JOINT SOURCE/CHANNEL DECODING OF

HUFFMAN CODES

In the previous section, a new soft input VLC decoderwas presented. We showed the efficiency of the algorithmin terms of performance and complexity. In this section, westudy the case where the communication system uses an ErrorCorrecting Code (ECC) applied to the VLC compressed bit-stream. In fact, if the VLC decoder can generate additionalinformation regarding the reliability of the compressed bit-stream components, iterative JSC decoding can be considered.The latter is based on extrinsic information exchange betweenthe VLC and the channel decoder.

A. System Description and Notations

In the following, we describe a transmission system con-sisting of an outer VLC and an inner channel code. Thestructure of the system is depicted in Fig. 5, where the S/P(resp. P/S) denotes the transition from a serial to a parallelvector (resp. from parallel to serial). A memoryless sourceis considered and generates sequences sh = (sh1 , s

h2 , . . . , s

hL)

of L source symbols. A standard VLC encoder encodesthe source symbol sequences. The resulting compressed bit-streams bh = (bh1 , b

h2 , . . . , b

hlh) are concatenated to form

the binary sequence b = (b1, b2, . . . , bl). The number ofconcatenated compressed bit-streams will be denoted H . Se-quence b is then permuted by a random interleaver Π and,if required, some dummy bits are introduced to obtain thebinary sequence m of length K . The latter is encoded by thechannel encoder into a codeword x = (x1, x2, . . . , xN ), andtransmitted through an AWGN channel by means of the BPSKmodulation. A tandem decoding scheme involves soft inputchannel decoding using the received data r = (r1, r2, . . . , rN )followed by a VLC prefix decoding. The proposed iterativeJSC decoding is based on the information transfer betweenthe Chase-like VLC decoder and the channel decoder and isdetailed below.

The reliability of a component bj of the decoded sequenceb is defined using the a posteriori LLR of the transmitted bitaccording to

Λ(bj) = logP (bj = 0|rj)P (bj = 1|rj) = log

P (rj |bj = 0) · P (bj = 0)

P (rj |bj = 1) · P (bj = 1).

(6)Often, contributions dealing with channel decoding assume

an AWGN channel and equal a priori probabilities P (bj =1) = P (bj = 0), and use the expression given in equation (5).However, in the case of iterative channel and VLC decoding,all the components of the decoded sequence don’t have the

ZRIBI et al.: LOW-COMPLEXITY SOFT DECODING OF HUFFMAN CODES AND ITERATIVE JOINT SOURCE CHANNEL DECODING 1675

Fig. 5. General block diagram of the proposed concatenated transmissionscheme with iterative JSC decoding.

Fig. 6. PER versus EbN0

for the tandem decoding, and the Chase-like VLCdecoding (q=4 bits) with two configurations in the case of transmission overan AWGN channel with BPSK modulation. Results correspond to iterativedecoding performance from the first to the fifth iteration.

same reliability and consequently the a priori term can berefined.

We have seen in the previous section that the proposedChase-like VLC decoder calculates the J (J ≤ Q) most likelylength-valid compressed bit-streams b

i. The decoded VLC

sequence b corresponds to the best sequence, among the Jcandidates, according to the metric given in equation (4). Itis clear that the positions j where we have the same bit forall possible candidate sequences (bi1j = bi2j ∀i1 �= i2) are themost reliable positions. In the following, we assume such bitsas reliably decoded, and assign to them a constant extrinsicinformation wj = β · (1 − 2.bj). We notify that a similarreliability definition was proposed in [32] for SISO decodingof linear block codes and that the value of β is determinedexperimentally. All the other bits are supposed non reliable andwe assign to them an extrinsic information equal to zero. It’sworth noticing that in some cases, especially with a relativelynoisy channel, the Chase-like VLC decoder is not able to finda length-valid codeword among the Q test sequences. In thiscase, we propose the decoding rule b = y and to assign to allits components an extrinsic information equal to 0.

B. Concatenated VLC and RSCC Iterative Decoding

We consider the system sketched in Fig. 5 with an innerRecursive Systematic Convolutional Code (RSCC). At thereceiver side, the latter is decoded using a Soft Input SoftOutput (SISO) BCJR [33] optimal decoder that provides theouter VLC decoder with soft values for the estimate of thevariable-length encoded sequences. The output of the proposedChase-like VLC decoder is a hard decision of the sourcesymbols sh on the one hand, and a bit reliability value onthe other hand. These reliability values are fed back to theRSC decoder as a priori information so as to improve itsperformance at the next iteration.

The experimental setup assumes the use of the source givenin Table II with the corresponding code CA. The sourcegenerates source-symbol sequences of length L, encoded bythe VLC. The resulting binary streams are grouped into Hpackets to form the bit-stream x. We note that no dummybits are required and we have K = l. The utilized RSCChas a memory M = 4, a generator polynomial G(D) =(1, 1 +D2 +D3/1 +D4), and a rate Rc = 0.5.

The reference system considers a Huffman code, and tan-dem decoding with classical VLC decoding and the soft inputViterbi algorithm is used for RSC decoding. The Chase-likeVLC decoder uses test patterns of maximum weight q = 4 andthe best experimental value of β is β = 1.0. Experimentalresults in terms of PER as a function of Eb

N0in the case of

transmission over an AWGN channel with BPSK modulationare depicted in Fig. 6. The presented curves correspondrespectively to tandem decoding, and Chase-like decodingwith five JSC iterations and for two different configurations.The first one assumes H = 8 source-symbol sequences withL = 50, and the second has H = 4 and L = 100 symbols.Results show that the iterative JSC decoding based on theChase-like algorithm provides a gain of about 1.7 dB withthe first configuration (L = 100), and 2.3 dB with the secondone with respect to tandem decoding. Results also show thatthe proposed iterative system presents a performance improve-ment along iterations until it converges to a fixed error rate.Moreover, we can see that when the number of concatenatedsequences H decreases, we have faster convergence behaviorand worst results. The configuration with H = 8 source-symbol sequences is 0.75 dB better than the one with H = 4.However, for a fixed q, the overall Chase-like VLC decodingcomplexity increases when H is bigger.

Now, the best configuration with H = 8 sequences andL = 50 symbols is used. We compare the iterative JSCdecoding performance in the case of the use of the Chase-likealgorithm to trellis-based SISO VLC decoding methods. Wealso assume transmission over a Rayleigh channel by meansof BPSK signalling.

In Fig. 7, the simulation results in terms of PER as afunction of Eb

N0are illustrated for the JSC iterative scheme at

the fifth iteration. The SISO reference decoders were presentedrespectively in [23] and [20] and consider different types oftrellis. In fact, authors of [23] proposed a new symbol-clocktrellis representation for VLC that enables the application ofthe BCJR algorithm [33] and provides a posteriori probabilityof the transmitted source-symbols. These probabilities were

1676 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 60, NO. 6, JUNE 2012

Fig. 7. PER versus EbN0

for the tandem decoding, the Chase-like VLCdecoding (q=4 bits), and the trellis-based SISO VLC decoding methodspresented respectively in [23] and [20] for the fifth JSC iteration in the caseof transmission over a Rayleigh channel with BPSK modulation.

Fig. 8. Bipartite graph with message exchange for the proposed JSC iterativedecoding.

used to generate an estimate value for the bit reliability ofthe compressed bit-stream, that enables iterative JSC decodingwhen the VLC is serially concatenated. Moreover, the decoderpresented in [20] applies the BCJR algorithm on a VLC bit-clock trellis first presented in [30]. The advantage of themethod presented in [20] is that it provides directly thereliability of the compressed bit-stream needed for iterativedecoding in the case of a serial concatenation.

Results show that the iterative JSC decoding using theChase-like VLC decoder provides a gain of about 4.0 dB atthe fifth iteration with respect to tandem decoding. Comparedto the SISO decoder [23], we can see that a slight performancegain is achieved. The use of the BCJR soft-output algorithmon a bit-clock VLC trellis [20] induces the best results and weobserve a gain of about 0.3 dB compared to the Chase-likedecoding performance. However, we point out that the Chase-like decoder has a much lower complexity than the decodersproposed in [23] and [20]. In fact, the latter apply optimaldecoding on a trellis with a number of states increasinglinearly with the binary sequence length.

C. Concatenated VLC and LDPC Iterative Decoding

Low-density parity-check codes, introduced by Gallager in1960 [34], have regained attention as a capacity-approachingchannel coding alternative to the turbo codes [35]. Recently,they have been adopted by the Digital Video Broadcastingby Satellite (DVB-S2) standard [36]. In this paragraph, weconsider the serial concatenation of the VLC with an LDPCcode and iterative decoding is performed to improve thesystem’s performance.

The considered system is similar to the one depicted inFig. 5, but no interleaver is required since interleaving isimplicitly incorporated in the LDPC structure. Unlike theconcatenation with the RSCC, the value of K is specific toa given LDPC. Thus, the number of concatenated variable-length bit-streams H is variable and some dummy bits areneeded to provide exactly K bits at the input of the LDPCencoder.

An LDPC code of length N and rate RLDPC can berepresented using a Tanner graph with N variable nodes vnrepresenting the N bits. Each variable node is connected tosome of the M parity-check nodes cm. Let Fm,n denotethe message from a check node cm to a variable node vn.Similarly, let Tn,m denote the message from a variable nodevn to a check node cm.

The bipartite graph of the JSC scheme and the messagesinvolved in the iterative decoding are depicted in Fig. 8. Theupper part of the figure is the Tanner graph representing theLDPC code. The nodes VLCDh in the lower part denotethe VLC Chase-like decoding stages with the proposed com-pressed bits reliability definition.

The iterative JSC decoding starts with the initialization tozero of the a priori information delivered by the VLC decoder(wn = 0 for 1 ≤ n ≤ N ). The check to variable messagesFm,n, 1 ≤ n ≤ N , are also initialized to 0. Then, the iterativedecoding proceeds as follows:

1) LDPC decoding step: apply ItLDPC iterations of theBelief Propagation (BP) algorithm:

a) BP horizontal step: each variable node vn com-putes the output messages Tn,m according to:

Tn,m = In + Fn − Fm,n + wn, (7)

where Fn =∑

m′∈M(n) Fm′,n is the overallinformation associated with each variable nodevn, M(n) is the set of checks connected to vn,In = 2

σ2 · rn is the intrinsic information deliveredby the channel output, and wn is the correctionterm given by the VLC Chase-like decoder.

b) BP vertical step: every check node cm has tocompute the output messages Fm,n using the inputmessages Tn,m [34] according to the equation

Fm,n = 2 · tanh−1

⎛⎝ ∏

n′∈N (m)\ntanh

Tn′,m

2

⎞⎠ ,

(8)where N (m) \ n is the set of variable nodesconnected to check cm with bit vn excluded.

c) Horizontal and vertical decoding steps are iterateduntil a valid LDPC codeword is obtained or the

ZRIBI et al.: LOW-COMPLEXITY SOFT DECODING OF HUFFMAN CODES AND ITERATIVE JOINT SOURCE CHANNEL DECODING 1677

Fig. 9. PER, BER, and Levenshtein SER versus EbN0

for the concatenated Chase-like VLC (q = 4 bits) and LDPC iterative decoding scheme in the case oftransmission over an AWGN channel with BPSK modulation.

maximum number of iterations ItLDPC is reached.Then, messages qn, 1 ≤ n ≤ N , containing theoverall information about the bit bn are updated:

qn = Fn + In (9)

2) Chase-like VLC decoding step:

a) The H sub-streams qh of respective lengths lh,extracted from q, are Chase-like decoded one-by-one and the extrinsic information wh is updated asdescribed above.

b) Messages wh are concatenated and the resultingsequence w is then fed back to the LDPC variablenodes for further iterations.

The algorithm is repeated as from step 1 until a stoppingcriterion is reached. In the system we are dealing with,both the LDPC and the VLC decoders are able to detecta valid sequence. In fact, the LDPC decoder can detect avalid codeword based on its syndrome. The VLC decodercan detect an erroneous bit-stream if the decoded source-symbol sequence is not length-valid. We propose to stop thedecoding process if both detections are verified: we have avalid LDPC codeword, and the corresponding H bit-streamsare VLC length-valid.

The experimental setup assumes the use of the quaternarysource and the VLC of the previous sub-section (see Table II)source symbol sequences of length L = 100. The LDPC codeused is one of those proposed in the DVB-S2 standard [36],which has a rate RLDPC = 0.88 with K = 14400 bits andN = 16200 bits. The average degree of the variable nodes(resp. check nodes) is 2.99 (resp. 26.99).

We considered two decoding scenarios. The first scenarioinvolves tandem decoding with binary messages passing be-tween the LDPC iterative decoder (using a maximum of 250iterations) and the prefix VLC decoder. The second is based oniterative decoding as described in the previous subsection with

a maximum number of ItLDPC iterations for the BP-LDPCdecoding and It joint source/channel decoding iterations.Simulation results of the PER, the BER, and the LevenshteinSER as a function of Eb

N0are presented in Fig. 9. Three

configurations of the iterative system are considered withIt = 2, 5, and 10 and the respective values ItLDPC = 125,50, and 25. For all these configurations, the Chase-like VLCdecoder uses test patterns with a maximum weight q = 4 (bestperformance/complexity tradeoff) and a value of the extrinsicinformation β = 0.3 which was optimized experimentally toachieve the best performance.

We note that compared to tandem decoding, the extrinsicinformation exchange between the Chase-like VLC decoderand the BP-LDPC decoder results in a remarkable reductionof the PER and the BER for the three configurations. Indeed,with It = 2 JSC decoding iterations, the number of erroneouspackets is almost divided by a factor of 200 for Eb

N0= 3.6

dB when compared to tandem decoding. Additional gainsare possible with It = 5, and we have an overall gain ofabout 0.25 dB with respect to the classical decoding. Theiterative decoding performance gain is due to the Chase-likeVLC decoding error correction capability and to the extrinsicinformation available for each sub-packet at the input of theLDPC decoder (inter-packet correction). We also observe thatincreasing It to 10 results in a degradation of performance.Thus, the best performance/complexity tradeoff is obtainedwith It = 5 and ItLDPC = 50. It’s worth noticing that asimilar concatenated LDPC and VLC scheme was consideredin [27] with iterative decoding. However, authors applied theoptimal BCJR algorithm on a bit-clock trellis for SISO VLCdecoding and simulation results showed an improvement of0.3 dB across tandem decoding. The performance differencebetween the Chase-like VLC decoder and the optimal scheme[27] is small and can be neglected if we consider complexityconstraints.

1678 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 60, NO. 6, JUNE 2012

By using the error detection capabilities of the LDPC andthe Chase-like VLC decoders, the average complexity of theiterative decoding scheme is negligible at low PER. For aPER< 10−3 at the channel output, the iterative scheme isapplied to less than 0.1% of the received LDPC codewords.Thus, for a maximum of It = 5 iterations, the averagecomplexity of the iterative scheme is increased by less than0.4%. Compared to tandem decoding, the use of the iterativedecoding presented in [27] induces a huge complexity increasesince authors applied a BCJR forward-backward operationafter every LDPC BP iteration.

V. CONCLUSION

In this paper, we have proposed a novel low-complexitysoft-input decoder for VLCs. This algorithm is derived fromthe Chase decoding algorithm initially developed for blockcodes and has been tested for transmission over an AWGNchannel in the case of BPSK signaling. The decoding solutionproposed here shows numerous significant advantages. First,the Chase-like decoding of the VLCs achieves good errorcorrection performance with very low complexity, especiallyfor medium to high values of signal-to-noise ratio. Second,the proposed decoder can be slightly modified to deliverthe reliability of the decoded bits. The Chase-like decoderwas concatenated with a SISO channel decoder and iterativedecoding was performed. On one hand, we have shown that,when a RSCC is used as channel code, the proposed decodingalgorithm achieves a gain of about 2.3 dB compared to tandemdecoding. On the other hand, an additional coding gain of 0.25dB has been demonstrated in the case of the concatenationwith the high performance LDPC code, compared to tandemdecoding, with a reasonable complexity increase.

Ongoing work includes the generalization of the proposedtechnique to context-based adaptive entropy codes and theapplication to image and video compression standards. Theextension of the Chase-like decoder to the context-basedadaptive VLCs is quite easy contrary to the trellis-baseddecoders. Moreover, additional work will focus on improvingthe reliability at the output of the VLC Chase-like decoder.

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Amin Zribi received the engineering degree in in-formation and communication technologies in 2005,and the M.Sc degree in communication technologiesin 2006, both from the National Engineering Schoolof Tunis, Tunisia. In 2010, he received the Ph.D de-gree in information and communication technologiesand sciences from Telecom Bretagne, Brest, France.In September 2010 he joined the Higher Instituteof Communication Technologies (IsetCom), wherehe is now an Assistant Professor. Since January2012, he has been a associate researcher at Telecom

Bretagne and his research interests are in the fields of multimedia signalprocessing and communications. In particular, his expertise includes imageand video coding, data compression, and joint source channel coding.

Ramesh Pyndiah qualified in telecommunicationseng. from Telecom Bretagne in 1985. In 1994,he received his Ph.D. in electronics engineeringfrom the Université de Bretagne Occidentale, andin 1999 he earned the HDR (Habilitation à Dirigerdes Recherches) from Université de Rennes I. From1985 to 1990, he was a Research Engineer atthe Philips Research Laboratory (LEP) in France,where he was involved in the design of MonolithicMicrowave Integrated Circuits (MMIC) for digitalradio links. In 1991 he worked for one year at

TRT (Philips Telecommunications Division, France) as Development ProjectManager for MMIC components. In October 1991, he joined the Signal& Communications Department of Telecom Bretagne where among othershe invented and developed block turbo codes (turbo product codes). Since1998 he is the Head of the Signal & Communications Department. He hasco-authored more than 20 journal papers and 90 conference papers in theareas of channel coding and digital communications. He has also contributedto two book chapters and 16 patents. He was co-editor of special journalissues on turbo techniques (Eurasip and Annals of Telecom), TPC memberof several major conferences on communications (ICC, Globecom, ECWT,etc.) and executive committee member of ICC’04, ISTC (97,00,03, and 10),PIMRC’08, and WCNC’12. He was involved in the evaluation of several EUprojects, and has chaired the IEEE ComSoc France chapter since 2000. Heis recipient of the 2001 Blondel Medal (SEE). His current research interestsare: modulation, channel coding (turbo codes), joint source channel codingand decoding, cooperative wireless coding, and MIMO techniques.

Sonia Zaibi was born in Tunis, Tunisia, in 1974.She received the degree in telecommunications en-gineering (with honors) from the National Schoolof Engineers of Tunis, in 1999, and the M.Sc. andPh.D. degrees in electronic engineering from theUniversity of Bretagne Occidentale, Brest, France,in 2000 and 2004, respectively. During her M.Sc.and Ph.D. studies, she was at the department ofsignal and communications of Telecom Bretagne,Brest, France. Since 2004, she has been with theNational School of Engineers of Tunis, where she

is currently an Assistant Professor and a member of the SysCom laboratory.Her research interests include information theory, joint source-channel coding,error control coding, data compression, and digital communications.

Frédéric Guilloud received the Eng. degree inelectronics from the Ecole Nationale Superieure del’Electronique et de ses Applications (ENSEA) in1998, and is a former student of the Ecole NormaleSuperieure de Cachan (ENS Cachan). He receivedthe Ph.D. degree in 2004 from Telecom ParisTech insignal and communications, entitled Generic Archi-tecture for LDPC Codes Decoding. He is ProfesseurAgrégé in electronics and power electronics since1999, and Associate Professor at Telecom Bretagnein the Signal and Communications Department since

2004. His research and teaching activities focus on error correcting codes anddigital communications.

Ammar Bouallègue was born in Tunis, Tunisia, in 1945. He receivedthe electrical engineering and engineer doctor degrees from ENSERG ofGrenoble, France, in 1971 and 1976, respectively, and the Ph.D. degreefrom ENSEEIHT, INP of Toulouse, France, in 1984. In 1976, he joined theengineering school of Tunis (ENIT), Tunisia. From 1984 to 1992, he was thehead of the Electrical Department, ENIT, and from 1993 to 1995, he wasDirector of Telecommunication for the High School of Tunis, Tunisia. He isthe head of the Communication Systems (SysCom) Laboratory at the NationalEngineering School of Tunis. His research interests include passive and activemicrowave structures, signal coding theory, and digital communications.