Performance analysis of multicarrier CDMA systemswith parallel and serial concatenated coding infading channels

P.L. Kafle and A.B. Sesay

Abstract: The authors present a bit error rate performance analysis of multicarrier code divisionmultiple access (MC-CDMA) systems with turbo and serial concatenated convolutional coding(SCCC) in multipath fading channels. The performance analysis is done for maximal ratiocombining and minimum mean square error combining detection in the downlink system. Upperbounds to the average bit error probability are presented for a punctured turbo code and for anSCCC code of similar decoding complexity. These analytical bounds are derived for fullyinterleaved Rayleigh fading channels. The bit error rate performance is also verified by simulationsin the regions of low signal-to-noise ratios. The analytical and simulation results illustrate therelative merits of the turbo and SCCC codes for MC-CDMA systems and their suitability toachieve very low error rates in wireless data applications.

1 Introduction

Multicarrier code division multiple access (MC-CDMA),based on the combination of orthogonal frequency divisionmultiplexing (OFDM) and conventional CDMA, hasreceived much attention recently [1]. By using multicarriermodulation, a CDMA signal is spread over several carriers.Frequency diversity is achieved similar to path diversity inRake receivers, but with lower equalisation complexity inthe frequency domain. MC-CDMA systems are attractivebecause of their potential to support higher data rates infuture wireless systems. However, the requirement for a veryhigh quality of service for future data applications (bit errorrates of the order of 10�6 [2]) makes the use of powerfulchannel coding techniques very important for these systems.Turbo codes (or parallel concatenated convolutional codes,PCCC) and serial concatenated convolutional codes(SCCC), based on interleaved concatenated encoders anditerative decoding, are two powerful channel codingtechniques [3, 4]. These codes, in combination withMC-CDMA systems, can help achieve the high-quality,high bit-rate transmission capabilities of future wirelesssystems.

In previous work [5, 6], simulation results were providedfor coded MC-CDMA systems with turbo and SCCCcodes. Performance of these coding techniques wascompared under the constraint of similar decoding com-plexity. In this paper, we present analytical performanceresults for coded MC-CDMA systems in fading channels.The analysis is presented for fully interleaved Rayleighfading channels in the downlink system.

Analytical upper bounds, in terms of average bit errorprobability, are often used to evaluate the performance ofturbo (PCCC) and SCCC codes in the region of high signal-to-noise ratios (SNRs). This is because it is almost infeasibleto obtain bit error rates in the region of very low error ratesby simulations. Average bounds to the performance ofturbo codes using union bounds have been provided in [7]and [8]. Performance bounds on SCCC codes in additivewhite Gaussian noise (AWGN) channels are given in [4].These union bounds are very useful for the uderstanding ofthe influence of block sizes, the choice of componentencoders and the error floor behaviour of these codes in thehigh SNR regions.

The main focus of this work is to obtain the pairwiseerror probability for MC-CDMA systems in Rayleighfading channels with maximal ratio combining (MRC) andminimum mean square error combining (MMSEC) detec-tion schemes. For MRC detection, we present an accurateanalysis of the average bit error performance bounds. ForMMSEC detection, approximations to the pairwise errorprobability are obtained by using Gaussian approximations.The transfer function bounding techniques provided in [8]are for turbo codes without puncturing. In this paper, weapply the approach in [8] to the case of punctured turbocodes. In this case, computation of the weight enumeratorsfrom the transfer function of the component codes isaccomplished by taking the puncturing pattern intoaccount. We also propose a simpler method for computingthe SCCC bounds by using the transfer functions of itscomponent codes. The analytical results for the puncturedturbo code and SCCC code of equivalent decodingcomplexities are compared. The bounds discussed here arequite accurate for fully interleaved fading channels, that is,when the effect of time correlation is negligible.

2 System description

The block diagrams of the MC-CDMA transmitter andreceiver for a downlink system are shown in Fig. 1. Weassume the system is symbol and chip synchronous with K

P.L. Kafle was with the Department of Electrical and Computer Engineering,University of Calgary and is now with TR Labs, Suite 280, 3553-31 Street NW,Calgary, AB, Canada, T2L 2K7

A.B. Sesay is with the Department of Electrical and Computer Engineering,University of Calgary, Calgary, AB, Canada, T2N 1N4

r IEE, 2004

IEE Proceedings online no. 20040365

doi:10.1049/ip-com:20040365

Paper first received 7th May and in revised form 15th September 2003

IEE Proc.-Commun., Vol. 151, No. 2, April 2004 113

users. The information bit sequence of each user is firstencoded using a channel encoder. The coded bit sequence isinterleaved by a channel interleaver to reduce the effect oftime correlation and then BPSK (binary phase shift keying)modulated. For user k, each modulated symbol b(k)A{�1,+1} is spread using a code sequence of L chips,

cðkÞ¼½cðkÞ1 ; cðkÞ2 ; . . . ; cðkÞL �T, where cðkÞ1 2 f�1= Lp

;þ1 Lp

g,l¼ 1,y, L; L being the processing gain of an equivalentdirect sequence CDMA system and [.]T denotes thetransposition. Orthogonal Walsh–Hadamard codes areused for spreading.

For simplicity of notation, we consider one binary datasymbol per OFDM symbol and briefly describe the discretetime signal model. Extension to the general case incorpor-ating multiple data symbols per OFDM symbol, as in [6], isstraightforward. The spread chips of all K users are addedchip and bit synchronously to obtain the transmission signalvector s as

s ¼Xk

k¼1

bðkÞcðkÞ ð1Þ

The signal s is transmitted after MC-CDMA modulationusing L subcarriers. The MC-CDMAmodulator consists ofa serial-to-parallel conversion, frequency interleaving andIDFT (inverse discrete Fourier transform) operationfollowed by a guard-time insertion using a cyclic prefix[6]. Assuming sufficient frequency interleaving amongsubcarriers, the frequency selective fading channel ismodelled as L independent Rayleigh flat-fading subchan-nels as in [6, 9]. We also assume adequate channelinterleaving by neglecting the effect of time correlation.

After matched filtering and chip-rate sampling of thereceived signal, the samples corresponding to the cyclicprefix are first removed and an L-point discrete Fouriertransform (DFT) is performed. The equivalent received

signal vector, after demodulation, can be written as

r ¼ HXKk¼1

bðkÞcðkÞ þ n ð2Þ

where H is an L�L diagonal matrix consisting of complexchannel fading gains on the subcarriers assigned to thesignal components of one symbol b(k), i.e. H¼ diag{h1,h2,y,hL}. The noise vector n is an AWGN vector in the Lsubcarriers. After demodulation, the received signal com-ponents {rl}, l¼ 1,y, L corresponding to any one symbolof the jth user are multiplied by gains {wl} to obtain theequaliser outputs, xl¼wlrl, l¼ 1,y, L. The despreadingoperation combines the energy of the received signal over allsubcarriers. The output, after subcarrier combining for thejth user, can be written as

yðjÞ ¼XL�1

t¼0

wlcðjÞl hl

XKk¼1

bðkÞcðkÞl þ nl

!

¼ bðjÞ

L

XL�1

l¼0

wlhl|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}D

þXKk¼1k 6¼j

bðkÞXL�1

l¼0

wlhlcðjÞl cðkÞl

!|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

l

þXL�1

l¼0

wlcðjÞl nl|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}

Z

ð3Þ

where the first term, D is the desired signal, I is multiuserinterference (MUI) due to other users in the system and Z isthe effective noise contribution due to the AWGN.

spreading

c(2)b(2)d(2)

d(1) b(1)

c(1)

spreading

S/P

+interl.freq.

OFDMmodulator

(IDFT +

guard time

s1

s(t)spreading

channelinterleav.

channelencoder

channelinterleav.

channelinterleav.

sNc

d (K ) b (K )

c (K )

channelencoder

channelencoder

a

OFDMdemod.

(FFT)

channelestimator

from thechannel

r (t) deinter-leaver(freq.)

equaliser

w( j )

P

/

S

r1

rL

x1

xL

desp-reader

c(k)

d̂ ( j )LLR

estimator

channel

decoder

b

∆)

Σ

Fig. 1 Block diagram of coded-MC-CDMA downlink transmission systema Transmitterb Receiver

114 IEE Proc.-Commun., Vol. 151, No. 2, April 2004

3 Performance bounds for turbo-codedMC-CDMA systems

In this section, we obtain the bit error performance boundsof turbo coded MC-CDMA systems in a Rayleigh fadingchannel environment. We apply the transfer functionbounding technique in [8] to the case of a punctured turbocode in MC-CDMA downlink systems with MRC andMMSEC detection. Consider the union upper bound formaximum likelihood decoding of a (N/Rc, N) block code,where N is the block size of the information bits and Rc isthe code rate. An upper bound on the probability of thecodeword error is given by

Pw �Xd

tðdÞP2ðdÞ ð4Þ

where t(d) represents the number of codewords ofHamming weight d and P2(d) is the probability ofincorrectly decoding a codeword that differs from thecorrect codeword in d bit positions. Due to the presence ofan interleaver between the component encoders in turbocodes, the exact weight enumerator for a particularinterleaver is very difficult to construct. Benedetto et al.[7, 10] suggested the notion of a uniform interleaver, whichfacilitates computation of the average weight distribution.An average weight distribution is obtained by averagingover all possible interleavers, that is,

tðdÞ �XNi¼1

Ni

pðdjiÞ ð5Þ

The factor Ni

� �is the number of input codewords with

Hamming weight i and p(d7i) is the probability that an inputcodeword with Hamming weight i produces a codewordwith Hamming weight d, when averaged over all possibleinterleavers. As in [8], the average upper bound on the worderror probability is given by

Pw �Xd

tðdÞP2ðdÞ

�Xd

XNi¼1

Ni

pðdjiÞP2ðdÞ

¼XNi¼1

Ni

EdjifP2ðdÞg ð6Þ

where the expectation Ed|i {P2(d)} is with respect to thedistribution p(d7i). The bit error probability is obtained asthe average number of bit errors per codeword. Given thatthe all-zero codeword is transmitted, the decoded code-words of input weight i correspond to an error event of ibits being in error.

The bit error probability is therefore given by

Pb �XNi¼1

iN

Ni

EdjifP2ðdÞg ð7Þ

3.1 Two-codeword error probability P2(d)for MRC detectionWe first consider the two-codeword error probability P2(d)for the transmission of the MC-CDMA signal through aRayleigh fading channel with MRC combining at thereceiver. We assume that the fading coefficients, {hl},l¼ 1,y, L are known at the receiver. The MRCequalisation coefficients are wl ¼ h�l , l¼ 1,y, L [11].

For the sake of clarity, consider first a single usertransmission system. For this case, the decision variable in

(3) can be rewritten as

yðjÞ ¼ bðjÞ

L

XL�1

l¼0

jhlj2 þXL�1

l¼0

h�l cðjÞl n

¼abðjÞ þ Z ð8Þ

where a is an effective fading gain. The subcarrier fadingamplitudes, rl¼ 7hl7, l¼ 0,y, L�1 have a normalised

Rayleigh pdf, pðrÞ ¼ 2re�r2 . Denote

bl ¼ jhlj2 and a ¼ 1

L

XL�1

l¼0

bl

The random variable b is exponentially distributed. Hence,a is chi-square distributed with 2L degrees of freedom, i.e.

PAðaÞ ¼1

ðL � 1Þ!ðaÞLaL�1e�ða=aÞ ð9Þ

where �aa is the average effective fading gain.We assume that the all-zero codeword C0 is transmitted

and incorrectly decoded as Cd, which differs in d bitpositions. These d bits can be considered to be indexed from1 to d. Corresponding to a given set of effective fading gains{a1, a2,y,ad}, we denote the outputs as

yi ¼ aibi þ Zi; i ¼ 1; . . . ; d ð10Þ

where the index j referring to the jth user is omitted forconvenience.

When perfect time interleaving is assumed, {yi} areindependent for different i. For a known ai, the outputs, {yi}are Gaussian distributed with mean �a;

ffiffiffiffiffiEs

pand variance

given by

varðyijaiÞ ¼ aiNo=2 ð11Þ

The two-codeword error probability can be obtained interms of the ratio of probabilities of the output vectory¼ [y1, y2,y,yd] (for indices 1 to d) conditioned on Cd or C0

as follows:

PðC0 ! Cd ja1; . . . ; adÞ ¼ PPðyjCdÞP ðyjC0Þ

41

� �

¼ PYdi¼1

P ðyijai; bi ¼ þ1ÞP ðyijai; bi ¼ �1Þ41

( )ð12Þ

Taking the natural logarithm on both sides of the inequalityand simplifying, we obtain

P2ðdja1; . . . ; adÞ ¼ PXdi¼1

aiyi

varðyijaiÞ40

( )ð13Þ

Substituting (11) for var(yi7ai), the conditional P2(d7a1,y,ad) can be simplified to

P2ðdja1; . . . ; adÞ ¼ PXdi¼1

yi40

( )ð14Þ

Since the sum x ¼Pdi¼1

yi is Gaussian distributed with mean

�ffiffiffiffiffiEs

p Pdi¼1

ai and effective variance,

s2eff ¼ No

2

Xdi¼1

ai

IEE Proc.-Commun., Vol. 151, No. 2, April 2004 115

then

P2ðdja1; . . . ; adÞ ¼PXdi¼1

yi40

( )

¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffi2ps2eff

q Z10

e�

xþffiffiffiEi

p Pdi¼1

ai

2

2s2eff dx

¼Q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Es

No

Xdi¼1

ai

vuut0@1A ð15Þ

For a system with more than one user, the signalscorresponding to the d bit positions can be expressed as

yi ¼ aibi þ Ii þ Zi; i ¼ 1; . . . ; d ð16Þ

In this case, using (13), the conditional two-codeword errorprobability can be written as

P2ðdja1; . . . ; adÞ ¼ PXdi¼1

aiyi

s2I ;i þ s2Z;i40

( )ð17Þ

where each MUI term Ii in a K user system has beenassumed to be Gaussian distributed with zero mean and avariance s2I ;i equal to

varðIijalÞ ¼K � 1

La2i Es

and s2Z;i denotes the variance due to the noise term. For a

given ai, the summation term,

x0 ¼Xdi¼1

aiyi

s2I ;i þ s2Z;i

is a Gaussian variable with mean

Eðx0ja1; . . . ; adÞ

¼ �ffiffiffiffiffiEs

p Xdi¼1

a2iðK � 1Þ

L a2i Es þ No2ai

ð18Þ

and variance

varðx0ja1; . . . ; adÞ

¼ No

2

Xdi¼1

a2iðK � 1Þ

L a2i Es þ No2ai

ð19Þ

Hence, for the general multiuser case with MUI, theconditional two-codeword error probability is

P2ðdja1; . . . ; adÞ ¼PXdi¼1

aiyi

varðyijaiÞ40

( )

¼Q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXdi¼1

aiEs

ðK � 1ÞL aiEs þ No

2

vuuut0B@

1CA ð20Þ

To compute the actual two-codeword error probability, theconditional probability in (20) is averaged over the effective

fading gains, a1, a2,y,ad, which results in

P2ðdÞ ¼PðC0 ! CdÞ ¼Za1

� � �Zad

pAða1; . . . ; adÞ

PðC0 ! Cd ja1; . . . ; adÞda1 . . . dad

¼Za1

� � �Zad

pAða1; . . . ; adÞ

� Q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXdi¼1

aiEs

ðK � 1ÞL aiEs þ No

2

vuuut0B@

1CAda1 . . . dad ð21Þ

Assuming that the effective fading gains, {ai} over the bits1,y,d are independent, the joint probability densityfunction can be written as the product of the individualpdfs, i.e.

PAða1; . . . ; adÞ ¼Ydi¼1

pAðaiÞ ð22Þ

where pA(ai) is given by (9). To further simplify themultidimensional integral in (21), we use the following Qfunction bound [11]

Qðffiffiffix

pÞ � 1

2e�x=2; x � 0 ð23Þ

Substituting (22) and using (23), yields the followingexpression for P2(d):

P2ðdÞ �1

2

Za1

LL

ðL � 1Þ!

264

� exp � a12ðK � 1Þ

L a1 þ EsNo

# $�1

8><>:

9>=>;aL�1

1 e�La1da1

375

d

ð24Þ

where the pA(ai) has been used from (9) by assuming �aa ¼1=L for normalised fading gains. The P2(d) in (24) dependsupon the SNR (Es/No) which is related to the code rate Rc

and SNR per information bit (Eb/No) as Es/No¼Rc(Eb/No).

3.2 Two-codeword error probability P2(d)for MMSEC detectionWhen the number of users in the system is significantly highcompared to the full load capacity, only MMSEC canprovide a good performance in the downlink [9]. To findperformance bounds for turbo coded MC-CDMA withMMSEC, we first evaluate the two-codeword errorprobability using a Gaussian approximation of multiuseraccess interference.

Consider the decision variable in (3). Using the centrallimit theorem, the MUI term can be considered to be anadditive zero mean Gaussian noise component withvariance s2l . The MUI and the Z terms are independent

with their effective total variance being s2eff ¼ ðs2l þ s2ZÞ.The distribution of the random variable, {wlhl} is related tothe choice of the equalisation coefficients{wl}. ForMMSEC, the equaliser coefficients are [9]

wl ¼h�l

jhlj2 þ LNo2KEs

; l ¼ 1; . . . ; L ð25Þ

Using the expression in (3) and applying the law of largenumbers, assuming sufficiently long orthogonal spreadingsequences, the following approximations on SD, s2l and s2Z

116 IEE Proc.-Commun., Vol. 151, No. 2, April 2004

can be used for MMSEC detection [12]

SD � E½wh�2Es ¼ ½1þ cecUið�cÞ�2Es ð26Þ

s2l �K � 1

L½Efw2h2g � Efwhg2�Es

¼K � 1

Lc½1þ cecUið�cÞð1� ecUið�cÞÞ�Es ð27Þ

s2Z � Efw2gNo=2 ¼ ½�1� ð1þ cÞecUið�cÞ�No

2ð28Þ

where c is a constant term given by c ¼ LNo=2KEs andUi (x) is the exponential integral defined as [13]

UiðxÞ ¼ �Z1�x

e�z

zdz ð29Þ

We again assume that the all-zero codeword C0 isincorrectly decoded as Cd, which differs from C0 in d bitpositions. When perfect time interleaving is assumed, thesed bits can be considered to be indexed from 1 to d. ForBPSK modulation, the two-codeword error probability,similar to (14), is given by

P2ðdÞ ¼ PðC0 ! CdÞ ¼ PXd�1

i¼0

yi � 0

( )ð30Þ

The sum of the decision metric in (30) can be assumed to bea Gaussian distributed random variable with mean �d

ffiffiffiffiffiffiSD

p

and a variance ds2eff . Hence, the two-codeword error

probability can be approximated as

P2ðdÞ ¼ Q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidSD

ðs2l þ s2ZÞ

s !ð31Þ

where the terms sD, s2l and s2Z for MMSEC detection are

given by (26), (27) and (28), respectively.

3.3 Conditional probability p(d/i) for thepunctured turbo codeThe conditional probability p(d7i) of producing a codewordof weight d, given a randomly selected input sequence ofweight i, is required for computing the upper bounds in (6)and (7). The approach in [8] utilises the transfer functions ofvarious code fragments of the turbo code to obtain p(d7i).The word and bit error probability are upper boundedby a union bound that sums contributions from errorpaths of different encoded weights. The transfer functionsof the code fragments are used to enumerate all possibleinput–output weight distributions. The transfer function,T(L, I, D) is defined as [8]

T ðL; I ;DÞ ¼Xl�0

Xi�0

Xd�0

tðl; i; dÞLlI iDd ð32Þ

where t(l, i, d) denotes the number of paths of length l, inputweight i and output weight d, starting and ending at all zerostates. When puncturing is done in the component encoders,the recursive formula for enumerating the weight distribu-tions has to be obtained according to the puncturingpattern. The necessary modifications for obtaining therecursion formula are shown in the Appendix (Section 9.1),for an example of a rate 1/2 turbo code.

Assume a block of N information bits being encoded ineach code fragment. By using the recursive formula (45)(Section 9.1), we find the coefficients, t(N, i, d) containing allthe possible distributions of input and output weights afterpassing through length N in the trellis. Using thedistributions, t(N, i, d) for any component code fragment,

we find the conditional probabilities for each componentcode as

pðdjiÞ ¼ tðN ; i; dÞPdl

tðN ; i; diÞ¼ tðN ; i; dÞ

Ni

ð33Þ

where the sum of t(N, i, d) over all possible output weights,dj gives the total number of codewords of information

weight, i equal to Ni

� �.

For a turbo code with three code fragments as shown inFig. 8, we assume the output weights to be, d0, d1, and d2.Hence, the total output weight is d¼ d0+d1+d2. When thepermutations of the code fragments are selected randomlyand independently, the conditional probability for theresulting turbo code is

pðd0; d1; d2jiÞ ¼ psysðd0jiÞpcc1ðd1jiÞpcc2ðd2jiÞ ð34Þ

The conditional probability for the systematic fragment(uncoded) is psys(d07i)¼ d(i, d0). For the punctured codefragments corresponding to the identical component codes,cc1 and cc2, the coefficients t(N, i, d) are obtained by usingthe recursive formula (45). The respective pcc1(d17i) andpcc2(d27i) are obtained using (33). The conditional prob-ability, p(d7i)can be computed for 8i and 8d as

pðdjiÞ ¼X

d0;d1;d2;d0þd1þd2¼d

psysðd0jiÞpcc1ðd1jiÞpcc2ðd2jiÞ

¼Xd�i

d1¼0

pcc1ðd1jiÞpcc2ðd � i � d1jiÞ ð35Þ

4 Performance bounds for SCCC codedMC-CDMA systems

Performance evaluation of SCCC codes has been carriedout by using union upper bounds on the bit errorprobability in [4]. However, the computation of the weightenumerating function of the equivalent block code for theSCCC is quite involved, especially for a large block length.It requires finding the weight enumerators for all paths inthe trellis, including all possible re-emergences with the all-zero state. Divsalar et al. in [8] proposed a simplifiedmethod for finding the weight enumerator, which uses arecursive formula for each code fragment to obtain thetransfer function bounds for turbo codes. Here, we applythis approach to obtain the average upper bounds on the biterror probability for SCCC codes. We discuss themodifications that are required in the application of thistechnique to turbo codes.

Consider the SCCC encoder shown in Fig. 10 (see theAppendix) Assume a block size of N1 at the outer encoderof rate R1. The interleaver has a length N2¼N1/R1. Sincethe inner encoder is systematic and punctured (systematicbits are not transmitted), the block size at the output of thesecond encoder (rate 1/1) is also N2. The recursion formula,using the transfer function of the non-systematic outerconvolutional code fragment and the systematic inner codefragment, are shown in the Appendix (Section 9.2).

Using a method similar to that described in Section 3 forturbo codes, we first obtain the weight enumerator for thecode fragment of the outer encoder. This involves obtainingall possible input and output weight distributions (i1 and d1,respectively), tcc1(N1, i1, d1)at the end of trellis length N1 ofthe outer encoder using the recursive formula (46). The

IEE Proc.-Commun., Vol. 151, No. 2, April 2004 117

conditional probability, pcc1(d17i1) is then obtained as

pcc1ðd1ji1Þ ¼tcc1ðN1; i1; d1Þ

N1

i1

ð36Þ

Similarly, using the transfer function of the recursive innercode fragment to compute the tcc2(N2, i2, d2), we obtain thedistribution pcc2(d27i2) as

pcc2ðd2ji2Þ ¼tcc2ðN2; i2; d2Þ

N2

i2

ð37Þ

where the output weights d1 and d2 correspond to the outer(cc1) and inner (cc2) code fragments of the SCCC encodershown in the Fig. 10. As seen from the Figure, the inputweight, i2 for the second encoder is obtained directly fromthe first encoder after interleaving, which does not changethe Hamming distance. Also, we observe that the outputweight of the overall code is just the output from the secondcode fragment (d2). When the permutations of the codefragments are selected randomly and independently, theconditional probability p(d7i) for the resulting SCCC codecan be obtained for 8i and 8d using

pðdjiÞ ¼XN2

d 0¼0

pcc1ðd1 ¼ d 0ji1 ¼ iÞpcc2ðd2 ¼ dji2 ¼ d 0Þ

ð38ÞWhen systematic bits are also transmitted without punctur-ing in the SCCC, the final p(d7i) expression has to bemodified accordingly. In that case, the output weights alsocontain the contribution due to the systematic bits. Theabove approach for finding the conditional distributionp(d7i) simplifies the procedure for finding the union boundfor SCCC. This approach can, in general, be used for anySCCC code having an inner RSC encoder. The expressionsfor the pairwise error probability P2(d) for MRC andMMSEC obtained in Section 3 also apply in evaluating theSCCC bounds. The average upper bound on the BER forSCCC is obtained as

Pb �Xd

XN1

i¼1

iN1

N1

i

P2ðdÞpðdjiÞ ð39Þ

5 Performance results

In this Section, we discuss some numerical results on theperformance of MC-CDMA systems with turbo and SCCCcoding. The analytical upper bounds are discussed forMRC and MMSEC for the case of fully interleaved fadingchannels. Some simulation results are also given to show theperformance in the region of low signal-to-noise ratios,where the union bounds diverge.

5.1 Performance of turbo coded MC-CDMAsystemAverage bounds on the bit error rate probability forMC-CDMA systems with MRC combining in thedownlink for Rayleigh fading channels are shown inFig. 2 for various block sizes (N¼ 100, 640 and 1000).These bounds are for a rate 1/2 punctured turbo code (1, 5/7, 5/7)8. Spreading length is 32. The P2(d) expression forMRC is evaluated by numerical integration using (24).Results are shown for the single user case with perfectinterleaving. These bounds provide insight into the achiev-able performance as a function of the choice of encoder andsize of interleaver. As these BER bounds are anticipated

averages of all interleaving patterns, the choice of goodinterleavers can perform below the predicted bounds. Theerror floor seen in these curves is not flat, but rather a low-slope region where the bit error rate decreases very slowlywith increasing SNR. It is seen from the curves that theerror floor can be lowered by increasing the block size. Thebound for the rate 1/3 (1, 5/7, 5/7)8 turbo code withoutpuncturing is also shown for comparison with N¼ 640,which shows lower error floor due to its lower code rate.

Figure 3 shows the average bounds on the BERperformance with MRC for a block size of 640 bits for 1,4 and 8 users, respectively. Simulation results are alsoshown in the low SNR region for 1 and 8 users at the 8thdecoding iteration. The decoding of the turbo code is donewith log-MAP [14]. The union bounds become unsuitable,diverging from the actual BER observed through simula-tions, in the lower SNR region. Simulation results matchthe BER upper bounds at BERs below 10�4.

Figure 4 shows performance results for turbo codedMC-CDMA system with MMSEC detection for a blocksize of 640. The BER bounds are computed for MMSEC

0 2 4 6 810−10

10−8

10−6

10−4

10−2

100

Eb/No, dB

aver

age

BE

R N = 100N = 640N = 1000

N = 640 rate 1/3

rate 1/2

Fig. 2 Average BER bounds of turbo-coded MC-CDMA inRayleigh fading channel with MRC, for a single user system usinga rate 1/2 punctured turbo code with block sizes 100, 640 and 1000Bound for rate 1/3 unpunctured turbo code also shown for N¼ 640,with dotted lines

0 2 4 6 8 1010−10

10−8

10−6

10−4

10−2

100

Eb/No, dB

aver

age

BE

R

K = 1, simulation

K = 1, bound

K = 8, simulation

K = 8, bound

K = 4, bound

Fig. 3 Performance bounds of turbo-coded MC-CDMA withMRC in Rayleigh fading environment for K¼ 1, 4 and 8 users, usinga rate 1/2 punctured turbo code with a block size of 640

118 IEE Proc.-Commun., Vol. 151, No. 2, April 2004

detection using the P2(d) expression in (31). Simulationresults are shown for 8 and 32 active users in the system (atthe 8th decoding iteration) for fully interleaved Rayleighfading channels. The simulation results approach the BERbounds in the high SNR region (BER of 10�5 and less).

5.2 Performance of SCCC coded MC-CDMAsystemAn SCCC code with 4-states, rate 1/2 non-systematicconvolutional (NSC) outer code, (5, 7)8 and a 2-state,punctured rate 1/1 recursive systematic convolutional(RSC) inner code, (1, 1/3)8 is used for the performancestudies. The selection of this code is mainly to ensure thatthe SCCC and the PCCC have a similar decodingcomplexity.

Average bounds on the bit error probability of SCCCcoded MC-CDMA system with a spreading length of 32 inRayleigh fading channels are shown in Fig. 5 for MRC.The P2(d) expression for MRC is evaluated by numericalintegration using (24). The interleaver length is N¼ 1280,which is equivalent to a block size of 640 information bits.

The results are shown for 1, 4 and 8 active users,respectively. We observe that the SCCC code achieves asignificantly lower error floor than that of the PCCC codeshown in Fig. 3. Simulation results are shown in the regionof low SNRs at the 8th decoding iteration for 1 and 8 users.For the simulations, SCCC decoding is done with the Log-MAP algorithm [14].

Average bounds for the SCCC coded MC-CDMA withMMSEC are shown in Fig. 6 for 8, 16 and 32 active users,respectively. The interleaver length is 1280 bits, whichcorresponds to a block size of 640 information bits.Simulation results are also shown for 32 active users. Thesimulation results tend to approach the BER bounds in thehigher SNR regions. However, because of the difficulty inobtaining accurate low BERs, simulation results could onlybe obtained for BER ranges down to 10�6. Hence, it wasnot possible to clearly verify the error floors (seen in theBER ranges 10�8 and low) in the analytical bounds fromthe simulation results shown in Fig. 5 and 6.

5.3 Performance comparison of PCCC andSCCCFor comparison, we select a rate 1/2 PCCC with two 4-statepunctured recursive systematic convolutional codes (1, 5/7)8, which has an approximately similar decoding complex-ity as an SCCC code consisting of an outer non-systematic4-state (5, 7)8 code and an inner 2-state RSC (1, 1/3)8 code.The log-MAP SISO decoding modules in the SCCC andPCCC decoders are similar, except that computation ofcode extrinsic information are required in the outer decoderof the SCCC, which increases the decoding complexitymarginally compared to one of the component decoders ofPCCC. Since the inner encoder has two states and thelength of the trellis is twice the length of the 4-state decoderin the PCCC, decoding complexity of the inner decoder isthe same as that of one component decoder in the PCCC.Hence, the selected PCCC and SCCC can be considered tohave approximately equal decoding complexity.

Analytical bounds are used to compare the performancebehaviour of PCCC and SCCC at very low BER values(BERo10�6), where simulations to obtain the BERs areextremely difficult. Simulation results are provided in thelow SNR regions, where the bounds are not valid. Figure 7shows a comparison of the average BER bounds for thePCCC and the SCCC for a single as well as 32 active users.

0 2 4 6 8 1010−10

10−8

10−6

10−4

10−2

100

Eb/No, dB

aver

age

BE

R

K = 8, simulation

K = 8, bound

K = 16, bound

K = 32, bound

K = 32, simulation

Fig. 4 Performance bounds of turbo-coded MC-CDMA withMMSEC for downlink system in a Rayleigh fading environment forK¼ 8, 16 and 32 users, using a rate 1/2 punctured turbo code with ablock size of 640

0 2 4 6 810−12

10−10

10−8

10−6

10−4

10−2

100

Eb/No, dB

aver

age

BE

R

K =1, simulation

K =1, bound

K = 8, simulation

K = 8, bound

K = 4, bound

Fig. 5 Average BER bounds for SCCC-coded MC-CDMA inRayleigh fading environment with MRC for K¼ 1, 4 and 8 usersusing a punctured rate 1/2 SCCC code (4-state NSC outer code and2-state RSC inner code) with an interleaver length of 1280

0 2 4 6 8 1010−12

10−10

10−8

10−6

10−4

10−2

100

Eb/No, dB

aver

age

BE

R

K = 8, bound

K = 16, bound

K = 32, simulation

K = 32, bound

Fig. 6 Performance bounds of SCCC-coded MC-CDMA inRayleigh fading channel for 8, 16 and 32 users, using a rate 1/2punctured SCCC code with interleaver length of 1280Simulation result is also shown for 32 active users

IEE Proc.-Commun., Vol. 151, No. 2, April 2004 119

For the single user case, bounds are shown for the MRCwhile for the full load case, bounds are shown for theMMSEC detection. Observe that the predicted error floorof SCCC is much lower than that of the PCCC for the sameequivalent block size of 640 information bits. While thePCCC exhibits its typical error floor behaviour, SCCC BERcurves decay much faster in the range of simulations.However, in the low SNR regions where BER is higher than10�4, PCCC exhibits superior performance in all cases.Although both PCCC and SCCC can support the bit errorrate requirements of future wireless data services, thebehaviour of SCCC appears more promising in the highSNR regions because it lowers the error floor significantly.

The analytical bounds and simulation results presented inthis paper are for fully interleaved Rayleigh fading channels.Additional simulation results with respect to performancebehaviour of PCCC and SCCC codes in MC-CDMAsystem in correlated slow Rayleigh fading channels can befound in [5, 6].

6 Conclusions

We have presented performance analysis of MC-CDMAsystems with turbo codes and SCCC codes in the downlinkof a cellular mobile system. Measures of pairwise errorprobability have been derived for the MC-CDMA systemwith MRC as well as MMSEC detection. The correspond-ing analytical bounds on the BER have been discussed.Comparison of PCCC and SCCC coded MC-CDMAsystems has been done under equal decoding complexity.Although both the PCCC and SCCC codes can achievevery low error rates in wireless data applications inMC-CDMA systems, the SCCC code is capable of loweringthe error floor significantly in the region of high signal-to-noise ratios.

7 Acknowledgments

This work was supported by Alberta Informatics Circle ofResearch Excellence (ICORE), Canada and the NaturalScience and Engineering Research Council (NSERC),Canada.

8 References

1 Hara, S., and Prasad, R.: ‘Design and performance of multicarrierCDMA system in frequency-selective rayleigh fading channels’, IEEETrans. Veh. Technol., 1999, 48, (5), pp. 1584–1595

2 Dahlman, E., Gudmundson, B., Nilsson, M., and Sk.old, J.: ‘UMTS/IMT-2000 Based on wideband CDMA’, IEEE Trans. Commun., 1998,36, (9), pp. 70–80

3 Berrou, C., Glavieux, A., and Thitimajshima, P.: ‘Near Shannon limiterror-correcting coding and decoding: Turbo codes’. Proc. ICC’93,1993, pp. 1064–1070

4 Benedetto, S., Divsalar, D., Montorsi, G., and Pollara, F.: ‘Serialconcatenation of interleaved codes: Performance analysis, design anditerative decoding’, IEEE Trans. Inf. Theory, 1998, 44, (3), pp. 909–926

5 Kafle, P.L., and Sesay, A.B.: ‘Performance of multicarrier CDMAwith parallel and serial concatenated coding for wireless dataapplications’. Proc. 20th Biennial Symposium on Communications,Queen’s University, Canada, May 2000, pp. 85–89

6 Kafle, P.L., and Sesay, A.B.: ‘On the Performance of MC-CDMAwith interleaved concatenated coding and interference cancellation forhigh-rate data transmission’. Proc. IEEE ICC 2002, New York, Apr-May 2002, pp. 694–698

7 Benedetto, S., and Montorsi, G.: ‘Unveiling turbo codes: Some resultson parallel concatenated coding schemes’, IEEE Trans. Inf. Theory,1996, 42, (2), pp. 409–428

8 Divsalar, D., Dolinar, S., Pollara, F., and McEliece, R.J.: ‘Transferfunction bounds on the performance of turbo codes’. (TheTelecommunication and Data Acquisition Progress Report), JetPropulsion Laboratory, Aug. 1995, pp. 44–45

9 Kaiser, S., and Papke, L.: ‘Optimal detection when combiningOFDM-CDMA with convolutional and turbo channel coding’. Proc.IEEE ICC’96, Dallas, USA, June 1996, pp. 343–348

10 Benedetto, S., and Montorsi, G.: ‘Design of parallel concatenatedconvolutional codes’, IEEE Trans. Commun., 1996, 44, (5), pp. 591–600

11 Wozencraft, J.M., and Jacobs, I.M.: ‘Principles of communicationengineering’ (John Wiley & Sons Inc, 1965)

12 Kaiser, S.: ‘Analytical performance evaluation of OFDM-CDMAmobile radio systems’. Proc. European Personal and MobileCommunications Conf., May 1995, pp. 215–220

13 Gradshteyn, I.S., and Ryzhik, I.M.: ‘Table of integrals, series andproducts’, in Jeffrey, A. and Zwillinger, D. (Eds.) (Academic Press,New York, USA, 2000)

14 Benedetto, S., Divsalar, D., Montorsi, G., and Pollara, F.: ‘A soft-input soft-output maximum a posteriori (MAP) module to decodeparallel and serial concatenated codes’. (The Telecommunication andData Acquisition Progress Report), Jet Propulsion Laboratory, 42–127, Nov. 1996, pp. 1–20

15 Divsalar, D., and Pollara, F.: ‘Serial and hybrid concatenated codeswith applications’. Proc. Int. Symp. on Turbo Codes and RelatedTopics, Brest, France, Sept. 1997, pp. 80–87

9 Appendix

9.1 Input/output weight enumerator for apunctured codeFigure 8 shows a turbo code encoder using two recursivesystematic convolutional (RSC) codes with generatorpolynomial (1, 5/7, 5/7)8, where the three code fragmentsfor the systematic bits and parity bits are connected throughrandom interleavers, p0, p1 and p2 as a general case. Inactual encoding, interleavers p0 and p1 may not be used (nopermutation), and only the random interleaving in p2 isimportant in practice. As in [5, 6], we consider a rate 1/2punctured turbo code for performance evaluation. The stateand trellis diagrams of the (5/7)8 code fragment of the turbocode are shown in Fig. 9. Both code fragments of thecomponent code have memory, v¼ 2, with four states. Statetransitions are labelled with the input and output bits, andalso enumerate the path lengths (L), input weight (I) andoutput weight (D). Note that when the code fragment ispunctured its equivalent output bit is 0 for that instant,which does not change the output weight. We consider thespecific case of a puncturing matrix, P¼ [10] for the paritybits of the first encoder, whereas the second encoder hasP¼ [01]. This is more clearly shown by the two consecutivesections of the trellis diagram in the Fig. 9. The deletion ofalternating parity bits in the second path is shown bysquares.

From the state transitions in Fig. 9, we obtain theequivalent state transition matrix, A(L, I, D) for the P¼ [10]

0 2 4 6 8 10

10−10

10−8

10−6

10−4

10−2

100

Eb/No, dB

aver

age

BE

R

PCCC K = 1, bound (MRC)PCCC K = 1, simu. (MRC)SCCC K = 1, bound (MRC)SCCC K = 1, simu. (MRC)PCCC K = 32 bound (MMSEC)PCCC K = 32 simu. (MMSEC)SCCC K = 32 bound (MMSEC)SCCC K = 32 simu. (MMSEC)

Fig. 7 Comparison of BER performance for PCCC- and SCCC-coded MC-CDMA systems in Rayleigh fading channelsLengths of PCCC and SCCC interleaver are 640 and 1280 bitsrespectively, for the same interleaving delay of 640 information bits

120 IEE Proc.-Commun., Vol. 151, No. 2, April 2004

case as

AðL; I ;DÞ ¼L2 L2I L2I2D L2ID

L2I2 L2I L2D L2IDL2ID L2I2D L2I L2

L2ID L2D L2I L2I2

0BB@

1CCA ð40Þ

The output for the state transitions are taken after L¼ 2 inwhich the second transition does not contribute to theoutput weights. As shown in [8], the transfer function isobtained by the (0, 0) entry of the matrix

ðI þ AðL; I ; DÞ þ AðL; I ; DÞ2

þ AðL; I ; DÞ3 þ � � �ÞAð1; 1; DÞv ð41Þwhere the term A(1, 1, D)v is due to the termination of thetrellis. The term due to the trellis termination can be omitted

as in [8] for simplifying the computation. Approximating(41) by omitting the factor A(1, 1, D)v and using the factthat, (I+A+A2+A3+y)¼ (I�A)�1, we obtain

T ðL; I ;DÞ � ½I2AðL; I ;DÞ��1ð0;0Þ ð42Þ

For the case of the state transition matrix given by (40), the(0,0) entry of the inverse matrix yields the transfer functionas a ratio, T(L,I,D)EN(L,I,D)/D(L,I,D), where

NðL; I ; DÞ ¼1� 2L2I � L2I2 � L4I � L4ID2

þ L4I2 � L4I2D2 þ 2L4I3

þ L6I2 � L6I4 þ L6I2D2 � L6D2 ð43Þ

and

DðL; I ;DÞ ¼1� L2 � 2L2I � L2I2

þ L4I � L4ID2 þ 2L4I2 � 2L4I2D2

þ L4I3 � L4I3D2 þ L6I � 2L6I3

þ L6I5 � L6D2 þ L6ID2 þ L6I2D2

� 2L6I3D2 þ L6I4D2 � L6I6D2

þ L6I5D2 � L8I2 þ 2L8I4 � L8I6

þ L8D2 � 2L8I2D2 � L8I2D4

þ 2L8I4D2 þ 2L8I4D4 � 2L8I6D2

� L8I6D4 þ L8I8D2 ð44ÞUsing (43) and (44) for T(L,I,D), recursive equations forcalculating t(l, i,d) are obtained, starting with the initialvalue t(0,0,0) up to t(N,i,d). Multiplying the expression forT(L,I,D) by that of D(L,I,D) and taking the coefficients oft(l,i,d) of the resulting equation, the following recursion isobtained:

tðl; i; dÞ ¼tðl � 2; i; dÞ þ 2tðl � 2; i � 1; dÞþ tðl � 2; i� 2; dÞ � tðl � 4; i � 1; dÞþ tðl � 4; i� 1; d � 2Þ � 2tðl � 4; i � 2; dÞþ 2tðl � 4; i � 2; d � 2Þ � tðl � 4; i � 3; dÞþ tðl � 4; i� 3; d � 2Þ � tðl� 6; i � 1; dÞþ 2tðl � 6; i � 2; dÞ � tðl � 6; i � 5; dÞþ tðl � 6; i; d � 2Þ � tðl � 6; i � 1; d � 2Þ� tðl � 6; i� 2; d � 2Þ þ 2tðl � 6; i � 3; d � 2Þ� tðl � 6; i� 4; d � 2Þ þ tðl� 6; i � 6; d � 2Þ� tðl � 6; i� 5; d � 2Þ þ tðl� 8; i � 2; dÞ� 2tðl � 8; i � 4; dÞ þ tðl � 8; i � 6; dÞ� tðl � 8; i; d � 2Þ þ 2tðl � 8; i � 2; d � 2Þþ tðl � 8; i� 2; d � 4Þ � 2tðl � 8; i � 4; d � 2Þ� 2tðl � 8; i � 4; d � 4Þþ 2tðl � 8; i � 6; d � 2Þ þ tðl � 8; i � 6; d � 4Þ� tðl � 8; i� 8; d � 2Þ þ dðl; i; dÞ� 2dðl � 2; i � 1; dÞ � dðl � 2; i � 2; dÞ� dðl � 4; i � 1; dÞ � dðl� 4; i � 1; d � 2Þþ dðl � 4; i � 2; dÞ � dðl� 4; i � 2; d � 2Þþ 2dðl � 4; i � 3; dÞ þ dðl � 6; i � 2; dÞ� dðl � 6; i � 4; dÞ þ dðl� 6; i � 2; d � 2Þ� dðl � 6; i; d � 2Þ ð45Þ

where d(l,i,d)¼ 1 if l¼ i¼ d¼ 0 and d(l,i,d)¼ 0, otherwise.We have t(l, i, d)¼ 0, if any index is negative, as the initialcondition.

10

11

00

01

0/0L

1/11/1

0/0

1/0

0/1 0/1

1/0LI

LID LIDL

LI

LDLD

a

0

states

(11)

(10)

(01)

(00)1

1

10

0

1

0

1

1

0

0

0

0

1

1

input bit 0

input bit 1

b

Fig. 9 State and trellis diagrams of (5/7) code fragmenta State diagramb Trellis diagram (punching parity bits shown in boxes)

�2

�1

�0

D Du1

u0

u2D D

c2

c1

c0

+

+

+

+

+

+

Fig. 8 Turbo encoder with (1, 5/7, 5/7) code

IEE Proc.-Commun., Vol. 151, No. 2, April 2004 121

9.2 Input/output weight enumerator for SCCCcodeAn SCCC encoder structure is shown in Fig. 10. When allcode bits are transmitted, this becomes a rate 1/4 code.However, we use puncturing in the inner code by notsending the systematic bits as in [15]. Hence, the inner codeis effectively a rate 1/1 code, which results in an overall coderate of 1/2. To find the weight enumerator recursions for theoverall SCCC, we use the state transition matrices of thenon-systematic (5, 7)8 code for the outer component codeand the (1/3)8 code fragment of the systematic innercomponent code using the procedure in [8]. The statediagrams of the (5, 7)8 and (1/3)8 code fragments are shownin Fig. 11.

From the state transitions in Fig. 11a, the followingrecursive equation for the input–output weight enumeratorfor the outer (5, 7)8 code is obtained:

tðl; i; dÞ ¼tðl � 1; i; dÞ þ tðl � 1; i� 1; d � 1Þ

� tðl � 3; i � 1; d � 1Þ

þ tðl � 3; i � 1; d � 5Þ

þ dðl; i; dÞ � dðl � 1; i� 1; d � 1Þ

� dðl � 2; i � 1; d � 1Þ ð46ÞSimilarly, for the code fragment due to the RSC inner code,from the state transitions in Fig. 11b, we obtain thefollowing recursive equation for the input–output weightenumerator:

tðl; i; dÞ ¼tðl � 1; i; dÞ þ tðl � 1; i; d � 1Þ

� tðl � 2; i; d � 1Þ þ tðl � 2; i� 2; d � 1Þ

þ dðl; i; dÞ � dðl � 1; i; d � 1Þ ð47Þ

inner RSC encoder (1, 1/1+D)outer NSC encoder (1+D2, 1+D+D2)

D Du

D

c0

c1

+ +

++

�

Fig. 10 SCCC encoder with outer (5, 7)8 and inner(1, 1/3)8 code

10

11

00

01

0/00

L

0/111/11

1/00

0/01

1/10 0/10

1/01

LID

LID2 LD2

LI

LD

LIDLD

1

0

0/0

1/0

0/1

LI

L

LD

LID

1/1

1

0/1

1

0/1

a

b

Fig. 11 State diagrams for code fragments of SCCC codea (5/7)8b (1/3)8

122 IEE Proc.-Commun., Vol. 151, No. 2, April 2004