Performance of multi-code CDMA in a multipath fading channel

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Performance

of

multi-code

CDMA

in

a multipath fading

channel

D.-W-Hsiung and J.-F.Chang

Abstract: The authors provide a general investigation on the performance of multi-code code division

multiple access (MC-CDMA) over Rayleigh fading plus additive white Gaussian noise (AWGN) channels. In MC-CDMA, high speed data sources are serial-to-parallel converted to low speed streams, which are further spread by Hadamard-Walsh orthogonal sequences and random codes. Numerical results show that, in a multipath fading channel, MC-CDMA with properly assigned Hadamard-Walsh sequences outperforms conventional CDMA when signal to noise ratio

( S N R )

is low and shows comparable performance at high SNR. Furthermore, for the AWGN channel, the approximate formula developed by Lee et al. (1997) shows more variation than the authors' formula. The new approach is capable of telling the impact of code cross-correlation on system performance while the earlier approach does not.

1 Introduction

In recent few years, a new form of code division multiple access (CDMA) known as multi-code code division multiple access (MC-CDMA) has induced considerable research interests in multirate transmission [I, 21. MC-CDMA, a wireless framework, has appealing features in supporting multimedia (e.g. data, voice, image, and video). The number of codes allocated by a central control unit (e.g. base station) is linearly proportional to a user's rate. To realise this idea, high speed traffic sources are serial-to- parallel converted to low speed streams, each occupying the same bandwidth. MC-CDMA inherits the strength of CDMA in combating multipath fading and requires no need to modify the radio frequency circuitry [I, 31. Further- more, interchip interference (ICI) and intersymbol interference (ISI) are mitigated due to the usage of subcode concatenation [3] among all parallel streams. After demod- ulation, maximal ratio combining (MRC) [4] is pedormed in the correlation receivers in order to reconstruct the origi- nal source bits.

Lately, focus has concentrated on the performance including system capacity of direct sequence CDMA (DS- CDMA) scheme in a variety of channel models. [5, 61 offer the analysis of DS-CDMA in an AWGN channel and Rician fading channel, respectively. The fundamental con- cept is to regard every user except the intended one as an interferer and model it as uncorrelated Gaussian noise. A glance at the conception can also be found in the papers (e.g. [7-91). Besides, code design is a primary cause affecting the performance of variable rate CDMA. [IO, 111 present the impact of code design on the perfomiance of multirate CDMA. However, in the analysis an AWGN channel is 0 IEE, 2000

IEE Proceedings online no. 20000686 DOk lO.l049/lpcom.2~86

Paper first received 2nd April 1999 and in revised form 30th May 2000 The authors are with the Department of Electrical Engineering and the Graduate Institute of Communication Engineering, National Taiwan University, Taipei, Taiwan 10764, Republic of China

assumed that makes the results only approximate. [12] reports the capacity of a DS-CDMA system that supports various quality of services (QoSs). However, the analytical results stated in the works (e.g. [12-15]), are not satisfac- tory. These results do not reveal the impact of code assignment on system performance. That is, different primary codes and orthogonal subcodes show unnoticeable differ- ence in (e.g. BER and capacity). A more precise performance comparison between conventional CDMA and MC- CDMA in various channel models is highly helpful in dem- onstrating the strength of MC-CDMA. The objective of this paper is to offer a more comprehensive analysis of MC-CDMA in a frequency selective fading channel. The analysis developed in this paper is a generalisation of [12, 51; but, the problem treated is extended to include the multipath fading channel.

In this paper, we derive the bit error rate of MC-CDMA with higher accuracy. The result shows that MC-CDMA outperforms conventional CDMA, especially in delay non- sensitive applications. Furthermore, the approximate formula developed in [12] over an AWGN channel shows considerably more variation than ours.

s o S" m u interleaver r m z b

M interleaver x M cos(w~+"k) converter

:"

ak,m(t)

ak,M(t)

Fig. 1 Multi-code CDMA trmnitter of mer k

2 System model

2.

I

Transmitter-receiver pair and code

assignment

The transmitter and receiver pair are shown, respectively, in Figs. 1 and 3. Fig. 2 shows the main lobe spectrum of the transmitted signal. For simplicity, forward error correction

(FEC) is not considered. On the transmitting side, assume there are K users in the multi-code system, each transmit- ting at rates corresponding to code numbers M I , ..., Mk,

...,

365 IEE Pruc-Commun., Vol. 147, No. 6, December 2000

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MK, respectively. Let M denote the code number corresponding to full rate, then (1 s MI< s M). That is, for the LTh user (1 s k s K), its bit stream with duration Tik] is serial-to-parallel converted to Mk parallel streams. The new bit duration becomes T = MkTJk]. Each bit is further spread by a sequence with processing gain N . The bits in each stream are interleaved to break correlated channel errors caused by (e.g.) a multipath channel. To make the problem simple, the interleavers and the de-interleavers are not shown in Figs. 1 and 3. The chip duration T, of each concatenated subcode ak,,(t) for the mth stream can be found to be * / 1

1 :

t 9 frequency

I

other users

f-

user k

-+-

_{other users} Fig. 2 Main lobe spectrum ojiransmiited signal

cOs(wo+ek)

++I

I converter

I

ak,m(t)I

Fig.3 Multi-eo& CDMA receiver of mer k with MFjor each code Each user admitted to the system is assigned a primary code and a specific row of the Hadamard-Walsh matrix by

a central control unit. The primary codes of different users are random codes which are only approximately orthogonal. To avoid self-interference, mutually orthogonal subcodes are created by multiplying the primary code and the rows of Hadamard-Walsh matrix bitwise. This is known as subcode concatenation technique in [2]. The rows of Hadamard-Walsh matrix shared by users should be arranged so as to avoid if possible the same row being used by different users. For example, if there are ten users in the system and we use the Hadamard-Walsh matrix of dimen- sion 64, [64/10] = 6 where 1x1 denotes the largest integer less than x. If the first user starts with the 1st row, the kth user is suggested to start at the (k - 1)6

+

1st row. Define M = IN/KJ. What is meant in this example is that user 1

uses rows 1, 2,

...,

M I , while user k uses

rows

(k - 1)6

+

1, (k - 1)6

+

2, ..., (k - 1)6

+

Mk as their subcodes. This code assignment scheme can accommodate the greatest number of mutually orthogonal subcodes for a given number of users. For convenieqce, we number the mth subcode of the kth user by (k - l ) M

+

m, and the corresponding subcode may further be represented as

00

where

nrc(t)

is the unit gate function with height 1 when 0 s t 5; T,, and 0 otherwise. In eqn. 2, u&!l)k+m represents

366

the jth bit of the sequence ak,Jt). Further discussions about the primary codes are presented in Section 4.

Without loss of generality and to comply with the orthogonality condition of Hadamard sequences, the binary elements in this paper are represented by + I and -1 using the mapping 1

-

-1 and 0 -+ + I . The signal trans-

mitted from user k is

M n

S k ( t ) = d m , & ) U k , & )

cos(wot

+

el,)

( 3 )

m = l

where bk,Jt) denotes the bit series fed from the mth stream.

Also

in eqn. 3, P is the signal power, q is the carrier fre-

quency, and 0, is a random phase uniformly distributed on

[0, 2 4 . The 0,s are independent and identically distributed (i.i.d.1 for all users.

The receiver of user k sketched in Fig. 3 utilises Mk RAKE receivers for the Mk parallel streams prior to the de-interleavers and threshold devices. The RAKE receiver performs maximal ratio combining and post-detection to reconstruct the original bit stream. It is believed that no change is needed in the IFRF circuitry except for the control and adjustment of transmission power to meet different source rates and QoS objectives Fig. 1.

2.2

Channel model

A variety of channel models have been proposed and used in different transmission environments [4, 7, 161. In this paper the channel is assumed to be slowly varying Rayleigh fading with a continuum of multipaths. The impulse response of the channel carrying L paths can be described as

L

c!,

( t )

= g,,d(t

-

t k , l )

(4)

1x1

where bokk./ = crk,@@’~J is a complex Gaussian random variable

(r.v.) with zero mean and a k , / is Rayleigh distributed with variance q2. In eqn. 4, tk,[ = (1 - 1)T,

+

& /

is the delay of

the lth of the kth user. Random variables

ak,/

and q&/ are uniformly distributed on [0, T,] and [0, 274, respectively. Owing to perfect power control assumption among the MC CDMA users, ak,/, @k,f, as well as can be viewed as

i.i.d. for all k and 1.

The channel autocovariance function defined in [4] can be easily found to be

L

P k ( t ) = +(t

-

t k , l ) (5)

1=1

In this paper, we assume equal power delay profile for all paths (i.e. any path locked by the RAKE receiver is with equal power). There is no influence on the result in which one or more of the paths are locked by the matched fdters

(MFs). Notice that the spreading codes obtained by subcode concatenation have an impact on the BER that relies on the power delay profile of the channel. This impact can be ignored only for perfectly ‘random codes’.

3 Analysis

It is assumed that perfect chip, symbol, and carrier syn- chronisation are performed in the system. The received power per user stream is equal to

P,

if no fading effect is concerned. For convenience and yet no loss of generality, user 1 is assumed to be the intended receiver in the following discussions. The signal received by user 1 in a K-user system can be written as follows:

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k = l m = l 1=1

x a k , m ( t - t k , l - % ) cos(wOt+<k,l) (6) where n(t) is AWGN with zero mean and double-sided power spectral density (p.s.d.) Nd2, z k is the propagation delay of user k, and - u,)tk,/

I

u,)oozk)mod 2”.

Random variables ?& and are both umformly distrib-

uted on [O, T, and [O, 2 4 , respectively. Notice that, although

Ck,[

is correlated with tk,/ and it can be viewed

as i.i.d. for all k and 1 after modulo-2n operation.

The decision variable of user 1 on the pth stream and the nth path of the MF is = (0, + t1.n +‘T z p , n =

/

T(t)Ql,nal,,(t-tl,n) cos(wot+Cl,n)dt

.

t 1 , 7 L ( 7 ) Assuming perfect channel estimation as in [2, 51, there is no

influence on generality by setting = 0 and tl = 0; hence,

cl,/

= (6, - .y,g,,/)mod 2x. For a bit sent from the transmit- ter, the decision variable at the RAKE receiver having the first A paths locked can be represented as the sum of A MF outputs. If the pth stream is intended, we obtain

x

ZIP

= &,7L

n= 1

= NI

+

D1

+

I1

+

1 2

+

1 3 _{( 8 )}

where NI is the noise due to n(t), D l is the desired signal, I,,

12, and

Z3

are the IS1 (intersymbol interference) due to self (i.e. I , , 12) or other users’ (i.e. 13) streams. If h = L, the

RAKE receiver becomes a full-RAKE that locks the entire paths. Here, the signal to interference ratio (SIR) for the intended user is defined as [7]:

S I R E[D1]

{

Var[Nl]

+

Var[li]

1

+

V a r [ I z ]

+

V a r [ 1 3 ] } - ~ ( 9 ) As for the desired signal D , , assume b/$ as the reference

bit. Thus we have

where

r,

= Z,$, a & , is the sum of h i.i.d. chi-squared ran- dom variables. And the variance of N I can easily be seen to be Vur[Nl] = (NoT/4)Z,h,, 0,‘. In eqn. 8, the decision statis- tics

a

can be grouped into three different cases if we substitute eqns. 6 and 7 into eqn. 8 as follows:

(i) interference due to the remaining L - 1 paths, ( I # n) of

the same stream of the same user 1: I,;

(ii) interference due to the remaining L - 1 paths,

( I

#

n),

from different streams, (m z p), of the same user 1: 12; (iii) interference due to the L paths from arbitrary streams from other user (k z 1):

Z3.

3.

I

Same stream interference

I ,

In eqns. 1 and 2, owing to subcode concatenation, the intended stream of user 1 does not suffer from the self-user interference in the same path. Thus, I , and I2 arise from the remaining L - 1 paths,

I

# n. Note that the interference

experienced by the pth stream of user 1 can be expressed as

n = l l=l,l#n

IEE Pwc.-Commun., Vol. 147, No. 6, December 2000

where h ( n , 1) = Q l , l Q l , n COS(C1,l -

51,n)

t l . v + T x

l1

IL

bl,,(t

- tl,l)al,p(t - t l , l ) a l , p ( t - t1,n)dt (12) However, setting 1 = 1 and n = 2 or 1 = 2 and n = 1 in eqn. 12, we find that some terms are correlated (i.e.

COS(^^,^

- = cos(&2 - &,). Eqn. 11 can further be written as

[71:

A A A - 1 x

n=1 l=l,l#n n = l k n + l

to split h(n, I) into uncorrelated terms and utilise Gaussian approximation. From eqns. 11-13, one can write Z{,j in

terms of R,(t) as

A - 1 x

1::;

= Q1,1Ql,n cos(<1,1

-

C1,n) n = l l=n+l

+

2bFi&(li,l - ti,%)]

(14)

where b,$!] and b;”,” represent the preceding and following bit of the reference bit bf?. And RJt) and Rp(t) are contin- uous-time partial cross-correlation functions defined in [5] with subscript replaced by p. For simplicity, set = 0,

thus tl,[ - ti,,? = ( I - n)T,

+

Furthermore, we can regard

If,!,

as a stationary random process with zero mean. The

vanance of I/! can be derived as

n=l l=n+l

(15)

by performing time average over

bf;’,

b c , b1[”,”, and R,,(t,,/ - t,,,). These variables are slowly varying and inde-

pendently distributed time functions. It is adequate to make the above assumption [5, 71. A I ( / - n) and A2(I - n) are real

functions defined in [7]. Besides, q2 is defined as Var[crl,l] =

q2.

With similar mathematical manipulations, the variance of Z[$ can be expressed as:

X L

VaT

[ ]

- -

$x

a f a i A i ( l -n) (16)

n = l l = A + 1

From eqns. 11, 15 and 16, the variance of I , can be expressed as the sum of VUY[Z& and VUY[@] as follows:

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In eqn. 17, the second term of the right-hand side (RHS) is zero if we set A. = 1; namely, it can be viewed as the correlation interference between the fingers of the RAKE

receiver. However, the first term of the RHS is associated with the correlation interference among all the paths.

3.2

Other streams interference

l2

From eqn. 7, I2 can be represented as

. n # p I f n tl.n+T

x

l1,-

h , p ( t - t l , l ) a l , p ( t - tl,l)al,m(t - t1,n)dt (18) To simplify eqn. 18, the following formula is utilised [7]:

IT

b(t

- d])a,* ( t -*

7-*)um*

( t ) d t

=

{

b[*'[Cp* ,m*

(f

+

1 - N )

-

c,*

,m*

( f ) ]

+

**b[Ol[C,,m(f**

+

1) -

c,.m(f,l)

6

(19)

with m* = m, p* = p , t* = t,,/ - t l , n , f = LI.*l/i",] =

II -

y11, 6 =

I-? -JT,I

= (61,1-

SI,?)

mod T, which is uniformly distrib-

uted on [0, T,). Substitute eqn. 19 into eqn. 18 and perform derivations similar to Vur[ll], the variance of I, is

(20) where A# - nl) is defined in [7] with subscripts replaced by

m* and p*.

3.3 Other user interference

l3

I, can be regarded as the sum of the interference of the A. RAKE fingers, that is

From eqns. 6 8 , 13,p,,r takes the form:

K M I . L k = 2 m=l1=1 X J ' t1.m ti.,+T b k , m ( t

-

rk

-

t k , l ) x a k , m ( t - rk - t k , l ) a l , p ( t - tl,n)dt (22)

Utilising change of variables in the integral and regarding (zk

+

tk,r) mod T as a uniformly distributed r.v. on [0, T),

the variance of

Z,

becomes

(23)

where Y ( ~ - ~ ) & + ~ , ~ is defined in [5] with subscript replaced by

(k - 1)M

+

m and p . Notice that, in a synchronous MC-

CDMA system where ?& = 0 and L = 1, Vur[13] vanishes if

all the users are with a nonoverlapping Hadamard-Walsh row index.

Finally, combining eqns. 8, 10, 17, 20 and 23, the vari-

ance of Zlp is given by where X L n=l l=l,l#n A - 1 A n = l l=n+l X M I L

3.4 Bit error rate analysis

The performance of a MC system can be judged by SIR and BER (bit error rate). It is assumed that BPSK (binary phase shift keying) modulation is used in this discussion. Substituting eqn. 24, and Vur[Nl] into eqn. 9, the instantaneous SIR per bit becomes

where

In obtaining the average bit error rate Pe, conditional inte- gral based on the channel fading distribution should be cal- culated as

where

e(d(2Vrl))

is the instantaneous bit error probability

for BPSK modulation, and p ( W 1 ) is the density function of chi-squared distribution with 2A degrees of freedom. After some mathematical manipulations [4, 171,

(29) where

x

= U(q2)-I.

4 Numerical results and discussion

As mentioned in Section 3.4, the bit error rate of each stream depends on the cross-correlation of concatenated sequences. Borth and Pursley [6] prove that the performance of random codes lies between the best and the worst choice

of

codes in a frequency selective fading channel. However, it is not the objective of this paper to find the best code of MC-CDMA. Thus, it is adequate to adopt

IEE Proc.-Commun.. Vol. 147, No. 6, December 2000

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random codes as primary codes, which are computer gener- ated using the C library. Notice that the primary code of

each user is randomly assigned by the computer. The number of users in this system is assumed to be ten, and nine of them are the interferers with respect to the intended receiver.

Fig. 4 shows the BER against S N R when different number of concatenated subcodes are used in an AWGN

channel. In tlus example, each user has the same total power, regardless of the number of subcodes it is using.

The approximate formula in eqn. 1 of [12] is drawn here for N = 64 and 128 in lines 1 and 6, respectively, regardless of the number of subcodes that are used. Examine the case N = 64, the c w e corresponding to 4 subcodes (Mk = 4) shows the worst performance among the five curves drawn for N = 64. A similar phenomenon exists in the five curves drawn for N = 128. The deviation among the curves in

each of the two sets says that the cross-correlation property among them has an impact on system performance. That is, a stream with relatively better codes, or alternatively, lower cross-correlation, achieves lower BER. Our analysis produces results more precise than eqn. 1 of [12]. Com- pared with the approximate analysis developed in [12], our improved approach produces more refined and more accu- rate results. 10 1 10 -3 10 d -5 E 10

e

hi

g

L -9 10 -11L 10 0 5 10 15 20 25 30 SNR, d 0 Fig. 4

Curve 1: N = 64 (approximate); curve 2: N = 64, M k = 1 (improved); curve 3: N =

64, Mb = 2 (improved); curve 4: N = 64, M k = 4 (improved); curve 5: N = 64, Mk

= 8 (Improved); curve 6: N = 128 (approximate); curve 7: N = 128, Mk = I (improved); curve 8: N = 128, h f k = 2 (improved); curve 9: N = 128, Mk = 4

(improved); curve I O N = 128, Mk = 8 (improved)

MC-CDMA in A WGN cluamel with exact andupproxunute resulls

lo-'

i

-3 I

l o

b

5 10 15 20 25 30 SNR, dB

CDMA ugainsi MC-CDMA in multzpaihfkdhg c h l Curve 1 : conventional CDMA; curve 2: MC-CDMA, M k = 1; curve 3: MC- curve 6 MC-CDMA, Mk = 16; curve 7: MC-CDMA, Mk = 32; curve 8: MC- CDMA, Mk = 2; curve 4 MC-CDMA, Mk = 4; Curve 5: MC-CDMA, Mk = 8; CDMA, Mk = 64

IEE Proc.-Commun.. Vol. 147, No. 6, December 2000

In contrast to Fig. 4, Fig. 5 considers equal user rate, equal user power, equal bandwidth, and a multipath fading channel with L = 4 and it = 3, which means that the RAKE receiver can lock three out of the four paths. Unless otherwise mentioned, the spread spectrum processing gain is set to

64

for Mk = 1 and all users are with homogeneous traffic rates. After serial-to-parallel conversion, the power and rate of each stream are divided by Mk from those of the sending user. Each parallel stream is further spread to the entire' bandwidth allocated. For example, the SS

processing gain of k f k = 2 and Mk = 4 are 128 and 256,

respectively. The performance of the MC system improves as the number of streams increased. The conventional CDMA with RAKE receiver is also shown. For Mk = 1, the performance of the MC system is slightly better than the conventional RAKE due to subcode concatenation. As

demonstrated in Fig. 5, all results surpass that of the con- ventional CDMA. At hlgher SNR, however, the BER for larger Mk approaches the conventional CDMA due to an equally assigned frequency band.

Figs. 6 and 7 present the BER against

S N R

with double bandwidth and one more multipath than Fig. 5, respectively. In Fig. 6, all curves are vertically shifted downward due to trading more bandwidth for combating multipath fading. Fig. 7 shows the performance of L = 5, it = 3. The extra path that is not locked by the RAKE receiver deterio- rates the system Performance.

-3

5 10 15 20 25 30

Fi 6 Conventional CDMA ugaimt MC-CDMA in mult@uth f ding c h -

ne%ith double bandwidth

Curve 1: conventional CDMA, curve 2: MC-CDMA, Ibfk = 1; curve 3: MC- CDMA, M - 2; curve 4: MC-CDMA, M - 4; curve S: MC-CDMA, h f k = 8; curve 6: M&DMA, ~k = 16; curve 7: M&DMA, Mk = 32

SNR, dB

7

SNR, dB

CDMA against MC-CDMA in multipath f d i n g chan- Curve I: conventional CDMA, curve 2 MC-CDMA, Mk = I: curve 3: MC- CDMA h f k = 2; curve 4 MC-CDMA, Mk = 4; curve 5 MC-CDMA, M k = 8; curve 6 MC-CDMA, Mk = 16; curve 7: MC-CDMA, Mk = 32

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5 Conclusions

In this paper, a simplified MC-CDMA model with properly selected concatenation subcodes is analysed rigorously. With this arrangement, an Mk-rate user is serial-to-parallel converted to Mk mutually orthogonal streams, which further modulates the radio carrier to the same frequency band. The self-user interference on the same path vanishes and the spectral efficiency is increased due to the concatenated subcodes. Besides, path diversity is achievable with a

RAKE receiver at the receiving side. Of particular interest, the performance of the transmitterheceiver pairs in an urban environment (no direct path) is investigated. The numerical results reveal that the MC-CDMA excels the conventional arrangement when the SNR is low and shows comparable performance at high SNR. Furthermore, considerable variation is seen in eqn. 1 of [12] when compared with ours in an AWGN channel.

In the reverse link, a major problem of MC-CDMA and other popular techniques (e.g. multicarrier CDMA), is the nonconstant envelope of the transmitted signal. In t h s paper, the effect of nonconstant envelope on the performance due to (e.g. amplifier nonlinearity), is not addressed. This is a topic for further research. In addi- tion, powerful source-channel coding with burst error correction capability, especially suitable in high speed multimedia transmission is desired to further improve the system performance.

6 Acknowledgments

The authors thank the anonymous referees for helpful sug- gestions. In particular, the authors thank one referee for pointing out that the autocorrelation properties of the spreading codes have an impact on the transmission performance which depends on the power delay profile of the channel, and only for idealised ‘random’ codes this impact can be neglected.

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