ITERATIVE DECODING OF Q-ARY PRODUCT TURBO CODES IN FFH-CDMA SYSTEMS

Vagner Vale do Nascimento, Jaime Portugheis

State University of CampinasSchool of Electrical and Computer Engineering

Campinas, Sao Paulo, Brazil

Daniel Carvalho da Cunha

University of PernambucoPolytechnical School

Recife, Pernambuco, Brazil

ABSTRACT

This paper presents a study about q-ary product turbocodes applied to FFH-CDMA systems. A method toimplement the q-ary soft decoding is proposed. First, anadaptation of Chase decoding algorithm to satisfy q-arysymbols requirements is developed. The adaptation does notsignificantly increase decoding complexity when comparedto the binary algorithm. Second, a modification of iterative(turbo) decoding allows appropriate feedback of decodedq-ary symbols and their reliabilities in each decoder iteration.Simulation results indicate a good performance/complexitytrade-off of the q-ary turbo coded system when compared toother previous proposed systems.

1. INTRODUCTION

A Fast Frequency Hopping-Code Division Multiple Access(FFH-CDMA) system originally proposed by A. J. Viterbifor multiple-access satellite communication by mobile userswas extended and investigated in [1] and [2] for possibleapplication to mobile telephone service. M -ary FrequencyShift Keying (M-FSK) with L chips per signal intervalwas the considered modulation signal set together withsingle-user noncoherent detection at the receiver. It wasdemonstrated in [3] that this FFH-CDMA system has highercapacity per dimension than a conventional Slow FrequencyHopping-CDMA system in both Additive White GaussianNoise (AWGN) and Rayleigh fading channels. The capacitieswere obtained considering only quantized detector outputs.The results of [3] were extended in [4] by consideringunquantized detector outputs. It was shown that hard limitingof detector outputs can significantly degrade the channelcapacity of these FFH-CDMA systems. A similar conclusionwas obtained based on the evaluation of the channel cutoffrate of these systems. Coding schemes for these systems wereinvestigated in [5], [6], [7] with quantized detector outputs,and in [8], [9] with unquantized outputs. One of the codingschemes described in [8] could work near cutoff rate.

In [6], [7], [9] iterative decoding methods for binary codesare considered and reliability information on each receivedbit is derived. In this paper, a study of coded FFH-CDMA

systems based on q-ary turbo-product codes is described. Inorder to cope with reliability information for a q-ary symbol,modifications of both the Chase decoding algorithm and theturbo decoding are proposed. Simulation results for thestudied systems are obtained and are compared to previousproposed systems. The comparisons indicate a good systemperformance when q-ary turbo decoding is applied. However,they do not indicate that the system can work near cutoff rate.The outline of this paper is organized as follows. We describethe proposed coded FFH-CDMA system in Section 2. InSection 3, an adaptation of the Chase decoding algorithmis proposed to implement q-ary soft decoding. In Section4, a turbo decoding algorithm is defined using the q-aryChase decoding presented in the previous Section. This turbodecoding algorithm is applied to the FFH-CDMA systemconsidered in Section 2. Numerical results are presented inSection 5. To summarize, conclusions are drawn in Section 6.

2. SYSTEM DESCRIPTION

Fig. 1 shows the block diagram of the proposed codedFFH-CDMA system. The channel code is a product codewith rate Rp according to [10]. If its component codesare q-ary codes, there is no need of a converter after theproduct encoder. Iterative decoding is accomplished by amodified turbo decoding algorithm. A detailed descriptionof the uncoded part of the system considered in this articlewas done in Yue's paper [2]. Here, we assume that themodulation signal set is M -FSK, where M == 2K . Themodulator accepts a block, B m , m E {a, 1, ... ,M - 1},of K bits from the source and outputs a sequence of L tones(chips) selected from the M -FSK signal set. The tones hasduration Tc and energy Ec . The sequence is determined bythe frequency hopper in the modulator according to the user'shopping pattern. Throughout this paper we consider onlyrandom hopping patterns.

In [2] it is assumed that the channel can be modeled asM independent frequency selective Rayleigh fading channels.The modulator output is then recovered by the frequencydehopper which is chip-synchronized. Assuming further thatreceiver noise and interference from other users follow the

978-1-4244-2046-9/08/$25.00 ©2008 IEEE 121 SPAWC 2008

Fig. 1. Block diagram of the coded FFH-CDMA system.

frequency dehopper, the conditional probability densities ofenergy detector outputs Rnl are then given by

(7)

(5)P(Yi == mlvi) > P(Yi == klvi) > P(Yi == jIVi),V j t= k t= m.

t. - { Y't - Yi, Y; = Y't (6)~ - 0, Y~ == Yi

Thus, we have a q-ary set with only 2Ld1nin/2J test patterns,that is the same quantity of the binary case.

Similarly to the binary case, the reliability of a receivedq-ary symbol Yi may be generated by using Log-LikelihoodRatio (LLR). However, we have to consider all symbols fromthe alphabet GF (q) to generate the following LLR:

Thus, the sequence ya will have the symbol yi with thesecond greatest a posteriori probability.

Utilizing ya, the sequence y' can assume, alternately,the symbols from the sequences V and Va belonging to theldmin/2J least reliable positions. Therefore, the test patternsymbols are obtained as

P(Yi == mlvi) > P(Yi == jIVi), V j t= m, (4)

where Vi is a sufficient statistic for the ith transmitted symboland j, m E GF(q).

Using an inequality similar to (4), we can get a secondhard-decision sequence ya with symbols yi == k so that

due to the channel information measurement, or reliability,extracted from the received word Y. The complete Chasedecoding methods are detailed in [11].

The Chase algorithms developed in [11] were appliedonly to binary codes. The application of Chase decodingto q-ary codes requires bit-to-symbol conversions that mayaffect the accuracy of the symbol reliability evaluation,mainly in systems using modulation with more than twosignals. Motivated by this restriction, a generalization ofChase decoding in order to deal with q-ary symbols isproposed. It is based on Chase Algorithm II due to its goodcomplexity/performance trade-off. To develop a q-ary Chasedecoding, it is necessary to modify the test pattern generationand the received symbol reliability evaluation. Similar q-aryChase decoding algorithms are described in [12] (see alsoreferences therein).

The Chase Algorithm II defines the test pattern set asthe arrangements, with repetition, of bits 0 and 1 in theldmin /2J least reliable positions. Instead of arranging bitso and 1, one could arrange the elements of GF (q) inthe ldmin /2J least reliable positions. However, this newq-ary test pattern set would have qLd1nin /2J elements andwould increase significantly the decoding complexity whencompared to the binary method.

The hard-decision symbol Yi == m is obtained at thereceiver by using the following inequality:

(3)

(2)

Kr/ == U LM Rp .

3. Q-ARV CHASE DECODING

( )

-1d - No _ ~ Eb

- E c - KRp No

J" B(k; J, /k) [ Rnl ]

p(Rnzlm) = ~k+d+Onm exp k+d+8nm (1)

where /k == 1/M, 6nm is the Kronecker delta, U == J + 1 is

the number of simultaneous users, B(k; J, p,) = (~) .p,k .

(1 - /-l)J-k and d is the noise-to-signal ratio per chip:

Chase proposed in [11] a class of decoding algorithms usingchannel information measurement. These algorithms arebased on the following description. Consider the receivedhard-decision sequence Y == (Y1,"" Yi, ... ,Yn), where Yi EGF(q). For high signal-to-noise ratio (SNR) values, thecorrect codeword C == (C1' ... , Ci, ... , cn ) is located, with highprobability, in a sphere of radius (dmin - 1) centered in thevector Y, where dmin is the minimum Hamming distanceof the code. Then, Y may be perturbed by a test patternT to generate a new sequence Y ' == Y EB T, in orderto find the correct codeword C within the sphere of radius(dmin - 1). Using this rule, the decoding performance mayapproximate to the complete decoding process without itscomplexity by searching the correct codeword on a sphereof radius (dmin - 1). This decoding method is executable

with Eb the average energy per information bit and No theuniform one-sided spectral density of the Gaussian receivernoise.

In order to compare the performance of systemsemploying codes with distinct rates, the system bandwidthefficiency 'TJ is used, and given by:

122

I .,

- - - - +'

A(y)ct)

yct)

Fig. 2. Block turbo decoding diagram.• (t) (t)

Fig. 3. q-ary turbo decoder for Yij -I d ij .

4. Q-ARY TURBO DECODING

4.2. q-ary turbo decoding in FFH-CDMA systems

The algorithm utilizes matrices ya and A(Y) which areobtained according to equations (5) and (8), respectively. Thestatistic Vi correspond to the nlth unquantized detector outputRnz . In the binary case, we can identify the bit value Yij by itsreliability A (Yij ) sign (positive or negative). However, in theq-ary case, the symbol value is not implicit in the reliabilitybecause the symbol can assume q possible values. Thus, theq-ary turbo decoder must be additionally fed with the receivedmatrix yeO) == Y and the q-ary auxiliary matrix ya.

In each q-ary turbo decoder iteration, the symbol value

Y~) from the tth input matrix yet) must be compared to the

symbol value dg) from the tth decoded matrix D(t). If Y~) ==dg) , the q-ary Chase decoder have not corrected/modified the

input symbol. Thus, the extrinsic information W(t) for thenext iteration must be computed as in Fig. 2, now consideringthe dashed lines. Note that the tth decoded symbol matrixvet) must be returned to the decoder input in the q-ary case.

The q-ary turbo decoding presents a particularity when Y~) -Idg). In this case, a different method to give a feedback tothe q-ary turbo decoder must be used. Thus, we propose a

procedure to q-ary turbo decoding when Y~) -I dg) which isillustrated in Fig. 3.

As we can identify the bit value by the reliability, in thebinary case we can consider that the reliability value A (y )varies in a [-1;+1] segment, as illustrated in Fig. 4a. Due tothe symbol diversity, we may not do the same considerationin the q-ary case. However, when Y~) -I dg), we can applythe proposed method, illustrated in Fig. 4b, to calculatethe reliability value for the next iteration, now denoted byA (Vij ) (t) . Observe Fig. 4b, considering the tth iteration,the point A on the abscissa axis represents the reliabilityvalue A(Yij ) (t) of the input symbol, and the point B onthe ordered axis represents the reliability value A(dij ) (t) ofthe q-ary Chase decoded symbol. We can find the point Cby plotting a straight line perpendicular to the segment ABsuch that segment AC == aCt) x AB. With some basictrigonometrical calculations, we can obtain the segment OD,whose value will represent the reliability value A(Vij )(t) ofthe next iteration. Note that, depending on A, Band a(t), thepoint D might be on abscissa A(Yij)(t) or ordered A(dij)(t)

(8)

(11)

(10)

(t) ( P(d~YIBm) ).pd·· -In('J) - ",M-l P(d~~)IB)·

L..Jr=O r#m 'tJ r

Thus, the extrinsic information is given by:

A(y)(t) == A(Y) + a(t)Z(t-l), (9)

where aCt) is a scaling factor (see [15]) varying in the range[0.0; 1.0] according to a logarithmic scale [9].

As proposed in [9], the soft output A(dij)(t) can beapproximated by

where (J(t) is a reliability factor assuming a value in the range[0.2; 1.0] according to a linear rule [9] and ~(dij )(t) is theLLR of the decision dij which is defined by:

4.1. Thrbo decoding

The rows and columns of a product code P can be decoded bya iterative method called turbo decoding [13], [14]. ConsiderA(Y) the reliability matrix of the received symbols. Thebinary turbo decoder diagram is illustrated in Fig. 2 (onlysolid lines). Firstly, the decoder performs the soft decoding(Chase decoding) of the rows (or columns) of P by usingA(Y) as the input matrix. Thus, the Chase decoded matrixD(t=l) is generated after the first iteration (t == 1). The softinput for the decoder tth iteration is given by

where y~j) is the q-ary symbol with the jth greatest aposteriori probability.

Note that, the symbol y~1) represents the ith position from

the received sequence Y and the symbol y~2) is ith positionfrom the sequence ya. Applying Bayes' rule and considering

P(y~j)) a constant value, equation (7) can be written as

123

5. SIMULATION RESULTS

Fig. 4. Turbo decoding projection: a) binary e b) q-ary.

7.06.56.05.5

---(]-- as (15,9) Chase

-.- as (15, 9) q-ary Chase

---{)-- as (31 ,25) Chase

~as(31,25) q-ary Chase

---6-- RS (31,19) Chase

-A- RS (31,19) q-ary Chase

5.04.54.03.5

10.1

10.2

10.3

P:rz1 10""al

10.5

10·e

10.7

3.0

'C

ob)

A(Yij)<t)-1----+1---+1

oa)

axis and its position will define if the next iteration symbolVij will be equal to Yij or dij , respectively.

Fig. 5. RS Chase decoding performance on AWGN channel.

Fig. 6. Decoding performance of the FFH-CDMA systemfor M == 32, L == 6, E b/ No == 25 dB. .: RS(28,20) BMdecoding .•: RS(28,20) q-ary Chase decoding. !:::.: (20,14)2binary turbo. 0: RS(28, 20)2 q-ary turbo. Ro : Cut-off rate.

0.450.400.350.300.25

~, Bandwidth Efficiency

0.20

efficiency TJ ~ 0.275 for a BER ~ 10-3 . This result isworse than that found in a RS(28, 20)2 q-ary turbo productdecoding. Park and Lee [7] also presented a study aboutiterative decoding in a FFH system. Their turbo coded systemcould accommodate U == 21 users for a BER ~ 10-4

which is inferior to the value achieved by the RS(28, 20)2q-ary turbo product decoding (U == 22 for a BER == 10-4).

However, the convolutionally coded system, also consideredin [7], increased the simultaneous users to U == 25 for thesameBER.

Finally, the results obtained in [8] are considered.Taking into account the number of simultaneous users,

10.2

10.3

~rz1

10""al

10.5

10·e

10.7

0.15

Before showing results for the FFH-CDMA system, weanalyze the performance of the proposed q-ary Chasedecoding algorithm with BPSK signalling over Gaussianchannels. The curve of Bit Error Rate (BER) versus Eb/Nofor different Reed-Solomon (RS) codes are presented in Fig.5. Note that, for all RS codes considered, the proposed q-aryChase decoding outperforms the binary Chase decoding. Forexample, a coding gain of approximately 0.5 dB is achievedby considering the RS(31,19) code for a BER == 10-6 .

Performance results for q-ary turbo decoding of productcodes with BPSK signalling over Gaussian channels will bepresented elsewhere.

The FFH-CDMA system of Fig. 1 was simulated inorder to obtain the performance of the q-ary turbo productdecoding algorithm proposed in Section 4. The systemparameters used were M == 32, L == 6 and Eb/No ==25 dB. Fig. 6 illustrates BER versus bandwidth efficiencyof the simulated codes, where plotted numbers denote U.Observe that q-ary turbo decoding of the RS(28, 20)2 productcode is more efficient than decoding the RS(28,20) codewith Berlekamp-Massey (BM) and q-ary Chase decodingalgorithms for BER ~ 5.10-3 and BER ~ 10-4,

respectively. We find that for a BER ~ 10-4 , the q-aryChase decoding accommodates 16 users, which increases to22 for the q-ary turbo product decoding. The cut-off ratecurve of the considered FFH-CDMA system is also shown inFig. 6. This curve was obtained for a coding rate Rp == 0.51,which is the rate of the RS(28, 20)2 product code.

Now we compare the above results with those of previousproposed systems. Consider the (20,14)2 product code(Rp == 0.49 ~ 0.5) and the binary turbo product decodingmethod proposed in [9]. Note that the RS(28, 20)2 code, withapproximately the same code rate, outperforms the first one inboth simultaneous users and bandwidth efficiency when theq-ary turbo product decoding is used.

Fiebig and Robertson [6] investigated various codingand decoding techniques applied to the same FFH systemsmentioned here. The best code from the set they simulatedwas a turbo code with rate Rp == 0.75 and bandwidth

124

the system with RS(28,20)2 q-ary turbo product decodingoutperforms the convolutional coded systems with softdecoding BM1 (2,1,6), BM2 (5,4,1), SM1 (5,4,1) andSM2 (5, 4,2). When the bandwidth efficiency is considered,the system with RS(28,20)2 q-ary turbo product decodingoutperforms BM1 (2, 1,6), BM2 (5, 4,1) and SM1 (5, 4,1)systems for a BER == 10-3 and SM2 (5, 4, 2) system fora BER == 10-4 • However, The SM3 (5, 3, 3) system in [8]was the most efficient code observed.

6. CONCLUSIONS

In this paper, an iterative q-ary product turbo decodingmethod was proposed. In order to cope with reliabilityinformation for a q-ary symbol, modifications of both theChase decoding algorithm and the turbo decoding wererealized. The proposed method was applied to a codedFFH-CDMA system. Performance results were obtainedand compared to results of several previous proposedcoded systems. The comparisons do not took intoaccount complexity issues. However, they indicate a goodperformance/complexity trade-off when q-ary turbo decodingis applied. It was shown in [13] that product turbo codeswith rates greater than 0.6 can work above a cutoff ratelimit on a binary input Gaussian channel. The performanceof binary turbo decoding of high-rate product codes (withsingle-error-correcting RS component codes) was obtainedin [14]. It was shown that this performance is within 1 dBof the sphere-packing lower bound of Gaussian channels. Itis expected that increasing the rate of the product code andthe cardinality of the q-ary code ~s alphabet, the proposedFFH-CDMA system can operate above its cutoff rate limit.

7. REFERENCES

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