[IEEE 2014 21st International Conference on Telecommunications (ICT) - Lisbon, Portugal (2014.5.4-2014.5.7)] 2014 21st International Conference on Telecommunications (ICT) - Reduced-complexity

Reduced-Complexity Iterative Decoder forWavelet-Coded Systems in Flat Fading Channels

Luiz Gonzaga de Queiroz Silveira JuniorDepartment of Communications EngineeringFederal University of Rio Grande do Norte

Natal, RN, BrazilEmail: [email protected]

Abstract—This paper presents a quantization scheme for log-likelihood ratios in order to reduce the computational complexityof an iterative decoder recently proposed for wavelet-codedcommunication systems. In order to reduce the computationaleffort, the real-valued LLRs of encoded bits are quantized intothree levels of reliability, with a threshold that is determinedby minimizing the mean squared error between the originalLLRs and the quantized ones. As result, a reduced-complexityiterative decoder is obtained. Performance evaluations of thisiterative decoder were carried out by computer simulation andshow that it is an effective means for reducing the computationalload of wavelet-coded systems. Therefore, this new approach foriterative decoding may lead to new alternatives for exploitingthe potential of wavelet coding for digital communications overwireless channels.

Keywords—wavelet coding, iterative decoding, Rayleigh fadingchannels, flat fading, time diversity.

I. INTRODUCTION

The performance of several wireless communications sys-tems of interest nowadays is severely limited by flat multipathfading [1]. In order to mitigate the impairments produced bythose channels, many techniques have been recently proposed,such as the use of new associations of channel coding tech-niques and receiver strategies.

The wavelet coding has been proposed by Tzannes in [2].In this approach, the wavelet encoder multiplies successivesource bits by distinct rows of a wavelet coefficient matrix(WCM), called wavelet codewords, to encode the informationbits. The resulting symbols, called wavelet symbols, are non-equiprobable, multilevel and depends on several informationbits. Since the wavelet symbols are non-equiprobable, modula-tion schemes play a significant role in the overall performanceof wavelet-coded systems [3], [4], [5], [6], [7].

Because of the orthogonality properties of the WCM rows,the spread information of each bit may be collected in thedecoder by proper correlation with the wavelet matrix usedfor encoding. The simplicity of correlative decoding is one ofthe main advantages of the wavelet coding technique [2], [8].

Since its advent, wavelet coding has been recognized asa promising technique to overcome the harmful effects ofmultipath fading on performance of wireless communicationsystems [2], [9], [10], [11], [12].

Recently, additional investigations on the use of waveletchannel coding to combat time-varying flat fading effects

were carried out in order to provide some important issuesnot addressed in [2], such as improvements in the spectralefficiency of wavelet-coded systems [13] and in the realizationof iterative decoding scheme [14], [15].

A strategy to exchange soft information iteratively betweena BCJR decoder and a soft-input soft-output (SISO) waveletdecoder was proposed in [14] using a soft-decision demo-dulation technique evaluated in [11]. The results there pre-sented showed the effectiveness of the proposed decodingtechnique for performance improvement of wavelet-coded sys-tems. Nonetheless, many fundamental questions concerning itscomputational complexity remained open.

The present work aims to develop an efficient new mecha-nism to update the a priori probability distribution of waveletsymbols for iterative decoding porposes and thereby derivea new decoder with lower computational cost than the oneproposed in [14].

The performance of this novel iterative decoder was eva-luated over Rayleigh flat fading conditions through computersimulations. The results show the effectiveness of the proposeddecoding technique.

The remainder of this paper is organized as follows. SectionII presents an overview of the wavelet coding processing. Theiterative wavelet decoding system proposed in [14] is reviewedin section III. Section IV is devoted to present the proposedmethod for updating the probability distribution of waveletsymbols. Simulation results are presented in Section V. Finally,Section VI brings the conclusions of this work.

II. WAVELET CODING

Wavelet encoding uses the rows of wavelet coefficientmatrix (WCM) to encode the information bits. A WCM oforder m and genus g has dimension m×mg, and it is denotedby

A =

⎛⎜⎜⎜⎜⎝

a00, . . . , a

0mg−1

a10, . . . , a

1mg−1

......

am−10 , . . . , a

m−1mg−1

⎞⎟⎟⎟⎟⎠ , (1)

whose entries can belong to the field of complex or realnumbers.

A WCM is called flat if all its entries have a same absolutevalue. A flat real matrix A = (aj

k), where ajk ∈ {+1,−1}, is

said to be a wavelet coefficient matrix of order m and genus

2014 21st International Conference on Telecommunications (ICT)

978-1-4799-5141-3/14/$31.00 ©2014 IEEE 374

g if it satisfies the modified wavelet scaling conditions givenbelow[2], [8]:

mg−1∑k=0

aj

k = m√

gδ0,j , 0 ≤ j ≤ m− 1 (2)

mg−1∑k=0

aj

[k+ml]aj′

[k+ml′] = mgδj,j′δl,l′ , 0 ≤ j, j′ ≤ m− 1,

0 ≤ l, l′ ≤ g − 1 (3)

where δj,j′ is the Kronecker delta and the notation [k + ml]stands for k + ml modulo mg.

Equation (3) states that the rows of an order m WCM aremutually orthogonal at shifts of length lm, and also states thateach row is orthogonal to a copy of itself shifted by lm, forany 0 < l ≤ g − 1. These properties are the basis of waveletchannel coding.

x n

Conv.

S/P

ypm+qm−1

ypm+qj ypm+q

ypm+q0

x (p+1)m−1

x pm+j

x pm

0

j

m−1

WCM

WCM

WCM

Source

(a) Encoder overview.

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ajm a

j

2m aj

(g−1)m0aj

ja gm−1

ja 3m−1

ja 2m−1

jam−1

jam+q aj

(g−1)m+qa jq

a j

2m+q

jy(p+1)m−1

jypm+q

jypm

jypm+q

x pm+j

D−1 D−1 D −1

D−1 D−1 D −1

D D D−1 −1 −1

(b) Detailed view of the WCMj block.

Fig. 1. Wavelet encoder based on m × mg WCM. The WCMj block isdefined by the jth row of the WCM.

An overview of the wavelet encoder originally proposed byTzannes is illustrated in Figure 1(a), where the source outputis a bit sequence {xn}, with xn ∈ {−1, +1}. This one isinitially converted to m parallel sequences which are encodedby filter banks WCMj, j = 0, 1, . . . , m−1}, giving rise to mparallel sequences of symbols {yj

pm+q, j = 0, 1, . . . , m− 1},here named wavelet sub-symbols. At every time n = pm + q,m wavelet sub-symbols are added on to produce a waveletsymbol yn. Figure 1(b) gives details of the structure of theblock WCMj, which is composed of m FIR filters, each onewith g coefficients taken from the jth WCM row.

The wavelet symbol produced at time n = pm+ q is givenby

ypm+q =

m−1∑j=0

g−1∑l=0

ajlm+qx(p−l)m+j , (4)

and takes value in set Y = {−mg, . . . ,−2k, . . . , 0, . . . , 2k,. . . , mg}, with cardinality equal to mg+1. Hence, the waveletsymbols are multilevel and depends on several informationbits. As an example, Table I shows symbols produced when anWCM of dimension 2×8 is used, where it can be observed theoccurrence of a transient response, which lasts until nTs = 5.

TABLE I. WAVELET SYMBOLS PRODUCED BY WCM 2 × 8

nTs yn

0 a00x0 + a1

0x1

1 a01x0 + a1

1x1

2 a02x0 + a1

2x1 + a00x2 + a1

0x3

3 a03x0 + a1

3x1 + a01x2 + a1

1x3

4 a04x0 + a1

4x1 + a02x2 + a1

2x3 + a00x4 + a1

0x5

5 a05x0 + a1

5x1 + a03x2 + a1

3x3 + a01x4 + a1

1x5

6 a06x0 + a1

6x1 + a04x2 + a1

4x3 + a02x4 + a1

2x5 + a00x6 + a1

0x7

7 a07x0 + a1

7x1 + a05x2 + a1

5x3 + a03x4 + a1

3x5 + a01x6 + a1

1x7

.

.

. · · ·

A. Original Wavelet Decoding Processing

The sequence of information bits {xn} can be recoveredfrom the received sequence by using a bank of m correlatorsmatched to the lines of the WCM [10]. Assuming that thewavelet symbols are correctly received, the output of thecorrelator matched to the row aj at time i = m(g + p)− 1 isgiven by

zji =

mg−1∑k=0

aj

(mg−1)−kyi−k. (5)

By using (3), it may be verified [2] that under imperfectreception of the wavelet symbols zj

i can be used to decideon bit xj+i−(mg−1) by comparison with a threshold set tozero, i.e., the bit will be −1 if zi = −mg, or, +1 if zi =+mg. The simplicity of correlative decoding is one of themain advantages of the wavelet coding technique.

B. Probability Distribution of Wavelet Symbols

The distribution of the output wavelet symbols, yn, can beobtained from the corresponding generating function [16], [17]

Gyn(z) = E

[z

∑mg−1k=0 akxk

]= E

[mg−1∏

0

zakxk

], (6)

where E(·) denotes the expected-value operator. Assuming thatthe random variables {xk} are mutually independent, it followsthat

Gyn(z) = E

[mg−1∏

0

zakxk

]=

mg−1∏0

E [zakxk ] .


375

Hence,

Gyn(z) =

mg−1∏0

[P (xk = −1)z−ak + P (xk = 1)zak

]. (7)

It should be noted that a recursive procedure based on convolu-tion can be used in order to obtain the probability distributionof wavelet symbols (PDWS).

III. ITERATIVE DECODING FOR SERIAL CONCATENATIONWITH WAVELET ENCODER

The communication system model with iterative decodingand wavelet encoding evaluated in this work is illustrated inFigure 2, where source produces independent and equiprobableinformation bits {xn}, xi ∈ {−1, +1}, which are partitionedin blocks {ak} with k bits. These blocks are initially encodedby convolutional encoder, producing blocks {bn} with n bits.Therefore, the code rate, R, of this outer encoder is R = k/n.

The encoded bits are randomly interleaved and sent inblocks to wavelet encoder, which produces blocks of non-equiprobable and correlated wavelet symbols, {yn}, using aMCW of dimension m×mg.

Finally, the wavelet symbols yn are mapped into a PSK-like of unit energy and transmitted by a single antenna overa Rayleigh fading channel. The channel is assumed to be flatand essentially constant during a signalling interval, despitebeing time-varying.

The baseband-equivalent signal received during the n-th signaling interval may be given by r(t) = αns(t) +w(t), nTs ≤ t ≤ (n + 1)Ts, where Ts is the signallinginterval, αn represents the sampled Rayleigh fading, s(t) isthe transmitted signal, and w(t) is a complex white Gaussianprocess with double-sided power density N0/2.

At the first iteration, since no extrinsic information isavailable at the input of the inner decoder, it is assumed thatthe encoded bits are equiprobable. From (6), it follows that thea priori probability distribution of wavelet symbols is given by

Pr(yn = 2k −mg) =

(mgk

)0, 5mg, 0 ≤ k ≤ mg. (8)

The soft-decision demodulation of wavelet-coded modu-lated signals can be performed using the rule [11]

yon =

mg−1∑i=0

yi P (yi|r), (9)

where P (yi|r) denotes the a posteriori probability of waveletsymbol yi. It is worth to notice that (9) has an interpolationform where wavelet symbols are conveniently weighted by itsprobability estimates, which under iterative decoding will beupdated in each iteration, until numerical convergence criterionbe satisfied.

In order to exploit the turbo principle, blocks are sentto a BCJR decoder containing measures which evaluate thereliability of decisions obtained from SISO wavelet decoder,one soft information per soft output, ck, denoted by Lck

, andexpressed as

Lck=−2μ

σ2c

· ck, (10)

where μ is the data mean and σ2c is the data variance. A

recursive method to estimate the data mean, μ, and variance,σ2

c , from the mean and variance of each block obtained at theoutput of wavelet decoder was derived in [15].

In the end of the first iteration, from extrinsic informationit is straightforward to update the a posteriori probabilitiesfor encoded bits. Hence, by using (7) it is possible to updatethe a posteriori PDWS estimates. This is performed in the“Probability Updater” (PU) block, as shown in Figure 2.

In fact, since there are N observations from the channeland each wavelet symbol has the information of mg coded bits,the PU needs N blocks, each one with mg soft informations,in order to provide the updated distributions necessary to makeN soft-decisions in accordance with (9). Therefore, N updatesof a posteriori PDWS estimates are required for each block ineach iteration step.

Despite its benefits in terms of precision, the mecanismused to update the probability distribution of wavelet symbolshas high computational cost. This cost increases when a higherdiversity gain for the system is required or when N increases.

IV. PROPOSED METHOD FOR UPDATING THEPROBABILITY DISTRIBUTION OF WAVELET SYMBOLS

The above shown method for updating the PDWS esti-mates requires many computational resources because there arecontinuos levels of reliability to evaluate. Therefore, in orderto reduce the computational efforts to a minimum, it shouldbe interest to quantize the LLRs with a reduced number ofquantization levels.

With respect to the extrinsic information, the sign of thesoft information indicates whether the encoded bit is morelikely to be 0 or 1, while its magnitude gives an indication ofhow likely it is that the sign gives the correct bit value.

In this paper we propose to quantize the extrinsic informa-tion according into three levels of reliability, (−D, 0, D), beingthe value of D adapted for every block of soft information.This quantization may be described by the following mappingfunction:

M : R → R

l �→ q, (11)

where l and q are the random variables corresponding to theinput and the output of the quantizer, respectively. Figure 3illustrates such a mapping.

In order to reduce the lost of precision produced by thequantization, the mean squared error between l and q wasminimizing. As result of this task, the decision threshold value,L, was determined as being equal to D/2. Assuming that theD value can represent a measure of dispersion, it follows thatit can be taking as equal to mean of absolute input values,D = E[|l|].

V. NUMERICAL RESULTS

Simulation experiments were carried on in order to investi-gate the performance improvements produced by the proposedtechnique over Rayleigh fading channel. In order to estimatethe SNR value, a minimun amount of 1000 errors has been


376

Reception

Σ

Deinterleaver

−

+

EncoderWavelet

WaveletDecoderDecoder

ProbabilityUpdater

ReliabilityEstimator

Source

BCJR

EncoderConvolutional

Interleaver

Interleaver Modulator Antenna

Channel

Demodulator

{xk} {bn} {cn} {yn} {sn}

{rn}

{yn}{cn}{Lbn}{xk} {Lcn}

{L(bn|yn) }

{Lext(bn|yn) } {Lext(cn|yn) }

{Pcn (S = sn)}

Fig. 2. System Model

D

−D

−L L l

q

Fig. 3. LLR proposed mapping.

observed in the simulation experiments. The wavelet-codedsystem adopted in these experiments was based on a flatinteger WCM with dimension 2 × 8. Figure 4 exhibits the

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−8

042

0102087

066

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−6−4

6

Fig. 4. PSK constellation for a wavelet coding with 2 × 8 WCM.

PSK constellation used in this system, where the signals arelabelled with the values of the corresponding wavelet symbols.This constellations was obtained empirically in [12] by sub-optimum searches in the unitary circle, under the criterion ofminimizing the bit error rate of a non concatenated wavelet-coded system.

Figure 5 illustrates performance results obtained with thenew iterative decoding approach. From this figure, it can beobserved that a performance gain in the Eb/N0 on the orderof 3dB even from first iteration, for values of bit error ratebelow 10−5.

1e-08

1e-07

1e-06

1e-05

1e-04

1e-03

1e-02

1e-01

1e+00

0 1 2 3 4 5 6 7 8 9 10

BE

R

Eb/No (dB)

Iterat. 0

Iterat. 1

Iterat. 2

Iterat. 3

Iterat. 4

Iterat. 5

Fig. 5. Performance evaluation of iterative wavelet decoding system over flatfading channels.

VI. CONCLUSIONS

In this paper, a new mechanism to update the probabilitydistribution of wavelet symbols in the context of iterativedecoding was proposed. Numerical results presented here showthat the proposed method is an effective tool for reducing thecomputational complexity of iterative decoding and suggestthat it is potentially useful as a new means to increase theperformance benefits produced by wavelet coding in flat fadingconditions over wireless channels.

ACKNOWLEDGMENTS

The author would like to thank Professor Ernesto LeitePinto of the Department of Electrical Engineering at theMilitary Institute of Engineering for his insight, advice andcomments on portions of the manuscript. He would also liketo thank the reviewers for their useful comments. The currentresearch is supported by the FAPERN (Fundacao de Apoio aPesquisa do Estado do Rio Grande do Norte) and CNPq.


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REFERENCES

[1] J. G. Proakis, Digital communications, McGraw-Hill, 3rd., New York,1995.

[2] M. A. Tzannes and M. C. Tzannes, Bit-by-bit channel coding usingwavelets, Proceedings of IEEE GLOBECOM’92, pp. 684–688, April,1992.

[3] L. F. Q. Silveira, L. G. Q. Silveira Junior, E.L. Pinto and F.M. de Assis.Analysis and optimization of wavelet-coded communication systems.IEEE Transactions on Wireless Communications, v. 8, p. 563–567,2009.

[4] L. G. Q. Silveira Junior, L. F. Q. Silveira, F.M. de Assis and E.L. Pinto,Signal constellations design based on genetic algorithm for digitaltransmission in wavelet-coded systems over Rayleigh fading channels(in Portuguese), VIII Brazilian Congress on Neural Networks, 2007,Florianpolis-SC, Brazil.

[5] T.R, Ferreira, L. G. Q. Silveira Junior, S.G. Silva and L. F. Q.Silveira, Optimization of psk wavelet-coded systems over Rayleighfading channels (in Portuguese), XXVII Brazilian TelecommunicationsSymposium -SBrT 2009, Blumenau-SC, Brazil.

[6] E. A. SANTOS, E. C. GURJO and F. M. ASSIS, Signal constellationsdesign of wavelet-coded systems (in Portuguese), XXVII BrazilianTelecommunications Symposium –SBrT 2011, Curitiba-PR, Brazil.

[7] E. A. SANTOS, F. M. ASSIS and E. C. GURJO, Improved optimizationalgorithm for constellation mappings of wavelet-coded communication,V International Workshop on Telecommunications, 2013, Santa Rita doSapuca-MG, Brazil.

[8] H. L. Resnikoff and R. O. Wells Jr., Wavelet analysis: the scalablestructure of information, Springer-Verlang, ISBN 0-387-98383-X, 1998

[9] L. F. Q. Silveira and F. M. Assis and Ernesto L. Pinto. Performanceof a psk system with wavelet time diversity over flat Rayleigh fadingchannels. Proceedings of International Microwave and OptoelectronicsConference, IMOC’2003. 2003.

[10] L. F. Q. Silveira, F. M. Assis and E. L. Pinto, A new waveletcoded psk system over time-varying flat fading channels, Fifth IEEEWorkshop on Signal Processing Advances in Wireless Communications- SPAWC’2004, July 2004.

[11] L. G. Q. Silveira Junior, L. F. Q. Silveira, E.L. Pinto and F.M.de Assis. Soft-decision demodulation of wavelet-coded psk signalsover flat Rayleigh fading channels. ITS’2006 – VI InternationalTelecommunications Symposium. September 3-6, 2006, Fortaleza-CE,Brazil.

[12] L. F. Q. Silveira, Wavelet-coded system analysis over time-varying flatfading channels (in Portuguese), Ph.D. Thesis/DEE-UFCG, 2006.

[13] E. A. SANTOS, E. C. GURJO and F. M. ASSIS, Improvement ofspectral efficiency in wavelet-coded systems (in Portuguese), XXVIIBrazilian Telecommunications Symposium –SBrT 2013, Fortaleza-CE,Brazil.

[14] L. G. Q. Silveira Junior, L. F. Q. Silveira, E.L. Pinto and F.M.de Assis. Iterative decoding for serial concatenation with waveletencoding. Proceedings of IMOC’2007 – International Microwave andOptoelectronics Conference, 2007, Salvador-BA,Brazil.

[15] L. G. Q. Silveira Junior, Iterative decoding in wavelet-coded systems(in Portuguese), Ph.D. Thesis/DEE-UFCG, 2006.

[16] A. Papoulis and S.U.Pillai. Probability, Random variables and stochas-tic processes. McGraw-Hill, USA, 2002.

[17] Y. Viniotis Probability and random processes for electrical engineers.McGraw-Hill, Singapore, 1998.


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[IEEE 2014 21st International Conference on Telecommunications (ICT) - Lisbon, Portugal (2014.5.4-2014.5.7)] 2014 21st International Conference on Telecommunications (ICT) - Reduced-complexity