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Tan F. Wong:Spread Spectrum & CDMA 6. CDMA

Chapter 6

Code Division Multiple Access

As discussed in Section 2.6.4, we can achieve code division multiple access (CDMA) based on spread

spectrum techniques. In particular, we can assign different spreading codes to different users in a DS-

SS system so that the users can share the communication channel. For a DS-SS based CDMA (DS-

CDMA) system, multiple access interference (MAI) is the major factor limiting the performance and

hence the capacity of the system. Therefore, analyses of the effect of MAI on the system performance

as well as ways to suppress MAI have been the major focus of CDMA research. Roughly speaking,

there are two different approaches to the problem. The first approach is based on the concept ofsingle-

user detection. In this approach, we identify one of the users in the system as the desired user and

treat all signals from the other users as interference. The receiver (for the desired user) detects only

the desired user signal. The second approach is calledmultiuser detection, in which all signals from

all users are detected jointly and simultaneously by the receiver. In this chapter, we will focus on the

first approach. Multiuser detection will be discussed in Chapter 7.

A common receiver based on the single-user detection approach is thematched filter receiver

matched to the desired user’s spreading signal. We note that the matched filter receiver is not optimal

(in the sense of maximizing the likelihood function) in the presence of MAI. Optimal receivers of such

a kind are discussed in [1]. However, due to their complexity, we will omit the optimal receivers here

and focus on the simple matched filter receiver. As a starting point, we will refine the crude analysis

of the symbol error performance given in Section 2.6.4. In general, it is difficult to conduct an exact

symbol error performance analysis of a DS-CDMA system based on the matched filter receiver even in

6.1


an AWGN channel. Usually we have to resort to bounds and approximations. We will discuss several

common techniques to calculate the approximate symbol error probability of a DS-CDMA system.

These approximate analyses are important because they provide us simple ways to obtain the symbol

error probability, which is crucial in determining the capacity of the system. Towards the end of the

chapter, we will also present some techniques based on the matched filter receiver to suppress MAI so

that performance of the system can be improved.

6.1 Asynchronous DS-CDMA model

In many cases, the transmissions from different users in a DS-CDMA system are not synchronized.

One of such examples is given by the uplink (reverse link) of IS-95 [2]. In order to include these

asynchronous cases, we consider a general asynchronous model of the DS-CDMA system here. We

assume that there areK actively transmitting users in the system. We associate thekth user with a

data signalbk(t) and a spreading signalak(t), where

bk(t) =q2 ~Pk

1Xi=�1

b(k)i pT (t� iT ) (6.1)

ak(t) =1X

l=�1

a(k)l (t� lT ): (6.2)

In above,~Pk is the (transmitted) power of thekth user signal,fb(k)i g is the sequence of data symbols

for the kth user, andfa(k)l g is the spreading sequence assigned to thekth user. For simplicity, we

assume that BPSK modulation is employed, i.e.,b(k)i are iid binary random variables taking values

from the setf+1;�1g with equal probabilities. We also assume that the spreading sequencefa(k)l g is

periodic with periodN , whereT = NTc, and the chip waveform (t) is time-limited to[0; Tc) and is

normalized such thatR Tc0 j (t)j2dt = Tc. Extensions to other types of modulations and long sequences

are straightforward.

The received signal at the receiver matched to the spreading signal of themth user, for1 � m � K,

is

rm(t) =KXk=1

�k;mej�k;mbk(t� �k;m)ak(t� �k;m) + n(t) (6.3)

wheren(t) is AWGN with power spectral densityN0. In (6.3),�k;m, ~�k;m, and�k;m are, respectively,

the amplitude response, the phase response, and the delay (with respect to some time reference) of

6.2


(m)

a (t)m*

^

0bz

dtT

0

mr(t) sgn(Re( ))

Figure 6.1: Matched filter receiver for themth user

the channel from the transmitter of thekth user (we will call it thekth transmitter) to the receiver of

themth user (we will call it themth receiver). Also,�k;m = ~�k;m � !c�k;m, where the term!c�k;m

is the phase difference due to the delay�k;m. We employ�k;m to model the asynchronous nature of

the system. We assume that synchronization with themth user’s signal has been achieved at themth

receiver. Hence, we can assume�m;m = 0 and�m;m = 0 without loss of generality. Fork 6= m,

we model�k;m and�k;m as independent uniform random variables on the intervals[0; T ) and[0; 2�),

respectively. Moreover, we also make the assumption that all the random variables associated with

different users are independent.

Now, let us look at themth matched filter receiver which is designed to detect themth user’s signal.

Without loss of generality, we consider the detection of the symbolb(m)0 . Let Pk;m = �k;m ~Pk be the

received power of thekth user’s signal at themth receiver. The receiver is shown in Figure 6.1. The

decision statistic for the0th symbol is given by

zm =Z T

0rm(t)a

�m(t)dt

=q2Pm;mb

(m)0 T +

KXk = 1k 6=m

ik;m + �; (6.4)

whereik;m is the interference component due to thekth signal and� is the component due to the

AWGN. It is easy to see that� is a zero-mean Gaussian random variable with varianceN0T . On the

other hand,ik;m is given by

ik;m =Z T

0�k;me

j�k;mbk(t� �k;m)ak(t� �k;m)a�m(t)dt

=q2Pk;me

j�k;mnb(k)0 [R (Tc � Æk;m)Ck;m( k;m) + R (Tc � Æk;m)Ck;m( k;m + 1)]

+ b(k)�1[R (Tc � Æk;m)Ck;m( k;m �N) + R (Tc � Æk;m)Ck;m( k;m + 1�N)]

o

6.3


=

8>>>>>>>><>>>>>>>>:

q2Pk;me

j�k;mb(k)0

h�k;m( k;m)R (Tc � Æk;m)

+ �k;m( k;m + 1)R�(Tc � Æk;m)i

if b(k)0 = b

(k)�1q

2Pk;mej�k;mb

(k)0

h�k;m( k;m)R (Tc � Æk;m)

+ �k;m( k;m + 1)R (Tc � Æk;m)] if b(k)0 = �b(k)�1;

(6.5)

where�k;m is decomposed into�k;m = k;mTc+ Æk;m with k;m 2 f0; 1; : : : ; N�1g andÆk;m 2 [0; Tc).

The development of (6.5) is similar to that in (3.14). We recall thatCk;m(�), �k;m(�), and�k;m(�) are the

aperiodic, even, and odd crosscorrelation functions between the sequencesfa(k)l g andfa(m)l g defined

in (3.15), (3.16), and (3.17), respectively.R (�) and R (�) are the partial correlations of the chip

waveform defined in (3.8) and (3.9), respectively.

6.2 Error analysis of matched filter receiver

As mentioned before, it is important to determine the error performance of a DS-CDMA system using

the matched filter receiver with the presence of MAI. Obviously, it would be most desirable if we could

obtain the exact average symbol error probability. Unfortunately, this task is exceedingly complex

for most practical scenarios in which many users are actively transmitting. Therefore, bounds and

approximations are typically employed. It is the goal of this section to give an introduction to some

common bounding and approximation techniques. We will focus on one of the receiver, namely the

mth receiver. To simplify notation, we write, fork = 1; 2; : : : ; K, ik;m asik, Pk;m asPk, �k;m as�k,

�k;m as�k, k;m as k, Æk;m asÆm, andetc.

6.2.1 Error bounds

Let us assume that the set of spreading sequences for theK users are given. For convenience, we

define a set of system parameters

Sm = fb(k)0 ; b(k)�1; �k; �k; for k = 1; 2; : : : ; m� 1; m+ 1; : : : ; Kg: (6.6)

Then the conditional symbol error probability givenSm andb(m)0 = 1 and that givenSm andb(m)

0 = �1are, respectively,

PsjSm;b

(m)0 =1

= PrnRe(zm) < 0

�� Sm; b(m)0 = 1

o

6.4


= Q

p2PmT + Imp

N0T

!(6.7)

and

PsjSm;b

(m)0 =�1

= PrnRe(zm) � 0

�� Sm; b(m)0 = �1

o

= Q

p2PmT � Imp

N0T

!; (6.8)

where

Im = Re

"KX

k = 1k 6=m

ik;m

#(6.9)

is the MAI component. Hence the average symbol error probability is

Ps = E�1

2PsjSm;b

(m)0 =1

+1

2PsjSm;b

(m)0 =�1

�

= E

(Q

p2PmT + Imp

N0T

!); (6.10)

where the expectation is taken over all the random variables in the setSm. The second equality in

(6.10) is due to the fact that the data symbolsb(k)0 andb(k)�1 for k 6= m are symmetrically distributed

about0, i.e., the distribution ofIm is symmetric about zero.

In general, the complexity of calculating of the expectation in (6.10) is exceedingly high1 even

when there are only a moderate number of users. In many cases, we have to resort to bounds which

can be calculated with a practical level of computational complexity. A simple bound based on (6.10)

is that

Ps � Q

p2PmT �maxSm jImjp

N0T

!; (6.11)

where the maximization is over all possible choices of values of the system parameters in the setSm.

For example, with the rectangular chip waveform, i.e., (t) = pTc(t), and BPSK spreading, fork 6= m

max�k;�k

jRe(ik;m)j =q2PkTc�k;m; (6.12)

where

�k;m = maxl

nj�k;m(l)j; j�k;m(l)j

o: (6.13)

1The complexity increases exponentially with the number of usersK.

6.5


Hence,

maxSm

jImj =KX

k = 1k 6=m

q2PkTc�k;m: (6.14)

Therefore,

Ps � Q

0BB@s2EmN0

26641�

KXk = 1k 6=m

sPkPm

�k;mN

37751CCA ; (6.15)

whereEm = PmT is the symbol energy of themth user. Although the bound in (6.11) or (6.15) can be

calculated easily given the set of sequences used, this bound is often not tight and hence its usefulness

is limited.

Another way to bound the average symbol error probability is to first determine and bound the

distribution function of the MAI contributionIm and then obtain bounds on the average symbol error

probability by taking the expectation in (6.10) using the bounds on the distribution function ofIm.

Using this approach, we can obtain very tight upper and lower bounds on the average symbol error

probability with a complexity which increases linearly withK [3]. In general, it is difficult to deter-

mine the distribution function ofIm. Only the distribution functions for some simple cases such as

BPSK spreading and QPSK spreading, have been worked out. Yet another way to bound the average

symbol error probability is to make use of the idea of moment-space bounds [4].

6.2.2 Gaussian approximations

Instead of obtaining bounds on the average symbol error probability, we can assume the MAI con-

tribution Im as a Gaussian random variable and obtain an approximation to the average symbol error

probability based on this assumption. There are mainly two variations to this Gaussian-approximation

approach.

Standard Gaussian approximation

The first method is to assumeIm as a zero-mean Gaussian random variable. This method is usually

known as thestandard Gaussian approximation(SGA) [5] and is applicable to situations in which

there are a large number of users with similar received powers in the system. Its validity is justified

by the central limit theorem based on the fact thatIm is a summation of independent random variables

6.6


(Re[ik;m]). When the number of usersK is large, the distribution ofIm approaches Gaussian. With

SGA, the approximate average symbol error probability is given by

PSGA = Q

0@p2PmT + E[Im]q

var(Im + �)

1A

= Q

0@p2PmT + E[Im]q

var(Im) +N0T

1A : (6.16)

Therefore, all we need is to calculate the mean and variance ofIm. It is easy to see that

E[Im] = 0 (6.17)

and

var(Im) =KX

k = 1k 6=m

var(Re[ik;m]): (6.18)

Hence, the problem reduces to the evaluation of the variance of Re[ik;m]. To do this, we use the

following simple identity

Re[ik;m] =ik;m + i�k;m

2: (6.19)

Then, fork 6= m,

var(Re[ik;m]) =1

4E[(ik;m + i�k;m)

2]: (6.20)

Let us write

A = R (Tc � Æk)Ck;m( k) + R (Tc � Æk)Ck;m( k + 1) (6.21)

B = R (Tc � Æk)Ck;m( k �N) + R (Tc � Æk)Ck;m( k + 1�N): (6.22)

From (6.5), we have

var(Re[ik;m]) =Pk2Ehej2�k(b

(k)0 A + b

(k)�1B)2 + 2jb(k)0 A+ b

(k)�1Bj2 + e�j2�k(b

(k)�0 A� + b

(k)��1 B

�)2i

= PkEhjb(k)0 A+ b

(k)�1Bj2

i= PkE[jAj2 + jBj2]: (6.23)

The second and third equalities in (6.23) are due to the independence of the random variables involved.

Substituting (6.21) and (6.22) into (6.23) and making use of the fact that k andÆk are independent,

6.7


we have

var(Re[ik;m]) = Pk

(EhjR (Tc � Æk)j2

iEhjCk;m( k)j2 + jCk;m( k �N)j2

i

+ EhjR (Tc � Æk)j2

iEhjCk;m( k + 1)j2 + jCk;m( k �N + 1)j2

i+ E

hR (Tc � Æk)R

� (Tc � Æk)

i� (6.24)

EhCk;m( k)C

�k;m( k + 1) + Ck;m( k �N)C�

k;m( k �N + 1)i

+ EhR� (Tc � Æk)R (Tc � Æk)

i�

EhC�k;m( k)Ck;m( k + 1) + C�

k;m( k �N)Ck;m( k �N + 1)i ):

Let us define

�k;m(n) =N�1Xl=1�N

Ck;m(l)C�k;m(l + n): (6.25)

With this definition, we can write (6.24) as

var(Re[ik;m]) =PkN

(�k;m(0)E

hjR (Tc � Æk)j2 + jR (Tc � Æk)j2

i

+ 2Reh�k;m(1)E[R (Tc � Æk)R

� (Tc � Æk)]

i ): (6.26)

For rectangular chip waveform, i.e., (t) = pTc(t), as in BPSK or QPSK spreading,

EhjR (Tc � Æk)j2

i= E

hjR (Tc � Æk)j2

i=T 2c

3(6.27)

and

E[R (Tc � Æk)R� (Tc � Æk)] =

T 2c

6: (6.28)

Hence, (6.26) reduces to

var(Re[ik;m]) =PkT

2

3N3

n2�k;m(0) + Re[�k;m(1)]

o: (6.29)

Combining (6.17), (6.18), and (6.29), we see that for rectangular chip waveform,

PSGA = Q

0BBBBBBBBB@

vuuuuuuuuut

2EmN0

1 +

(EmN0

KXk = 1k 6=m

PkPm

2�k;m(0) + Re[�k;m(1)]3N3

)

1CCCCCCCCCA: (6.30)

6.8


Given the set of sequences, the standard Gaussian approximationPSGA can be calculated as easily as

the simple bound in (6.15).

Sometimes, it is more convenient to have an approximation to the average symbol error probability

which does not depend on the set of sequences employed. One reasonable way to obtain such an

approximation is to assume that all the sequence elements are zero-mean iid random variables with

E[ja(k)l j2] = 1 and replace the terms�k;m(0) and�k;m(1) in (6.30) by their expectations. Forn = 0; 1,

E[�k;m(n)] =N�1Xl=1�N

E[Ck;m(l)C�k;m(l + n)]

=N�1Xl=0

E[Ck;m(l)C�k;m(l + n)] +

�1Xl=1�N

E[Ck;m(l)C�k;m(l + n)]

=N�1Xl=0

N�1�lXi=0

N�1�l�nXj=0

Eha(k)i a(m)�

i+l a(k)�j a(m)

j+l+n

i

+�1X

l=1�N

N�1+lXi=0

N�1+l+nXj=0

Eha(k)i�la

(m)�l a(k)�j�l�na

(m)j

i

=N�1Xl=0

N�1�lXi=0

N�1�l�nXj=0

Eha(k)i a(k)�j

iEha(m)�i+l a

(m)j+l+n

i

+�1X

l=1�N

N�1+lXi=0

N�1+l+nXj=0

Eha(k)i�la

(k)�j�l�n

iEha(m)�i a

(m)j

i

=

8>>><>>>:

N�1Xl=0

(N � l) +�1X

l=1�N

(N + l) if n = 0

0 if n = 1

=

8><>:N2 if n = 0

0 if n = 1:(6.31)

Thus,

E [2�k;m(0) + Re[�k;m(1)]] = 2N2: (6.32)

Replacing the term2�k;m(0)+Re[�k;m(1)] in (6.30) by the expectation in (6.32), we have the approx-

imation,

PSGA � Q

0BBBBBBBBB@

vuuuuuuuuut

2EmN0

1 +2Em3N0N

KXk = 1k 6=m

PkPm

1CCCCCCCCCA: (6.33)

6.9


0 5 10 1510

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Em

/N0

PS

GA

uniform received power and N=31

K=1

K=5

K=10

K=20

K=30

Figure 6.2: Symbol error probability based on SGA

Figure 6.2 shows the plots of the approximate symbol error probability given by the SGA in (6.33) for

the case in which all the users have equal received powers andN = 31. Note that forK = 1, the

symbol error probability is exact.

When the received powers of all users are the same and the signal-to-noise ratioEm=N0 � 1, as

indicated by the plots in Figure 6.2, we can further approximate (6.33) as

PSGA � Q

0@s

3N

K � 1

1A : (6.34)

Comparing to the approximation of the average symbol error probability given by (2.46), the stan-

dard Gaussian approximation in (6.34) gives a more optimistic estimate on the system capacity. For

example, in order to achieve an average symbol error probability of10�3, K � N=3 approximately

based on (6.34). Hence, a DS-CDMA system with a processing gainN can accommodateN=3 users

(compared toN=5 users predicted by (2.50)).

6.10


Improved Gaussian approximation

In general, the SGA is reasonably accurate if the number of users is large and the received powers of

the users are similar. However, when either the number of active users is not large or there are a few

users with received powers much higher than those of the others, the SGA does not give an accurate

approximation to the symbol error probability and another form of approximation is needed. We

would still like to approximate the MAI componentIm in the decision statistic as a Gaussian random

variable. However, we can no longer to do so by the reasoning employed before sinceIm contains of

only a few (significant) terms Re[ik;m] now. This difficulty can be circumvented by noting that each

ik;m is a summation of a large number of terms involving the sequence elements (refer to (6.5) and

(6.31)) when the processing gainN is large. We can make use of this property to obtain the desired

Gaussian approximation for the MAI by modeling the sequence elements as iid zero-mean random

variables withE[ja(k)l j2] = 1.

To aid our discussion and to conform to the notation in the literature, let us define

~Sm = f�k; Æk; for k = 1; 2; : : : ; m� 1; m+ 1; : : : ; Kg: (6.35)

to be the set of all the delays and phases of the other users2 and normalize3 the decision statisticzm in

(6.4) by the factor1=p2PmNTc. Then, the real part of the normalized decision statistic becomes

~zm =pNb

(m)0 + ~Im + ~�; (6.36)

where ~Im = Im=p2PmNTc and ~� is a zero-mean (real) Gaussian random variable with variance

NN0=2Em. It can be shown [6, 7, 8] that the normalized MAI terms Re[ik;m]=p2PmNTc are condi-

tional independent given~Sm and the desired user’s spreading sequence. Based on this, one can further

show [6, 7, 8] that conditioning on the set~Sm, the normalized MAI component~Im in (6.36) approaches

a zero-mean Gaussian random variable with varianceVm asN approaches infinity. The conditional

variance of the limiting Gaussian random variable is given by

Vm =KX

k = 1k 6=m

PkPm

~Vk; (6.37)

2The delay and phase of themth user are zero since we assume that synchronization to themth user has been achieved.3This normalization does not affect the error probability.

6.11


where ~Vk corresponds to the term due to thekth user and is a simple function (depending on the

spreading technique) ofÆk, �k, and the chip waveform [7, 8]. For example, with BPSK spreading,

~Vk =1

T 2c

hR2 (Tc � Æk) + R2

(Tc � Æk)icos2 �k

=

24 1� Æk

Tc

!2

+

ÆkTc

!235 cos2 �k; (6.38)

while with QPSK spreading,

~Vk =1

2T 2c

hR2 (Tc � Æk) + R2

(Tc � Æk)i

=1

2

24 1� Æk

Tc

!2

+

ÆkTc

!235 : (6.39)

We note that~Vk are iid random variables given our model of the delays and phases.

The discussion above implies that we can accurately approximate the MAI component~Im in the

normalized decision statistic~zm as a zero-mean Gaussian random variable with varianceVm when the

processing gainN is large. Hence, the approximate conditional symbol error probability given~Sm is

Psj ~Sm � Q

sN

Vm +NN0=2Em

!

= Q

0BBBBBBBBB@

vuuuuuuuuut

2EmN0

1 +2EmN0N

KXk = 1k 6=m

PkPm

~Vk

1CCCCCCCCCA: (6.40)

Averaging this over the delays and phases, we obtain an accurate approximation to the (unconditional)

symbol error probability

PIGA = E

26666666664Q

0BBBBBBBBB@

vuuuuuuuuut

2EmN0

1 +2EmN0N

KXk = 1k 6=m

PkPm

~Vk

1CCCCCCCCCA

37777777775: (6.41)

The approximation in (6.41) is known as the improved Gaussian approximation (IGA). We note that

the IGA is accurate, regardless of the number of active users in the system, as long as the processing

6.12


gain is large. On the other hand, the SGA is accurate, regardless of the processing gain, when there

are a large number of active users with equal received powers.

We notice that the conditional error probability depends on the delays and phases only throughVm.

Hence, an efficient way to calculatePIGA in (6.41) is to first obtain the probability density function

pVm(v) of the random variableVm and then evaluate the expectation by the integral

PIGA =ZQ

sN

Vm +NN0=2Em

!pVm(v)dv: (6.42)

The density functionpVm(v) of Vm, in turns, can be easily obtained as the(K � 1)-fold convolution

of the density functions of the independent random variablesPkPm

~Vk. Compared to the SGA in (6.31),

the computational complexity of (6.42) is still significant higher. We can reduce the computational

complexity of the IGA by further approximating the expectation in (6.41) based on a Taylor series

approximation [7, 9] of the conditional symbol error probability as below:

PIGA � 2

3Q

sN

�Vm +NN0=2Em

!+

1

6Q

0@vuut N

�Vm +p3�Vm +NN0=2Em

1A

+1

6Q

0@vuut N

�Vm �p3�Vm +NN0=2Em

1A ; (6.43)

where�Vm and�2Vm are the mean and variance of the random variableVm, respectively. From (6.37),

�Vm = E[ ~Vk]KX

k = 1k 6=m

PkPm

; (6.44)

�2Vm = var( ~Vk)KX

k = 1k 6=m

�PkPm

�2(6.45)

since ~Vk are iid. For example, with BPSK spreading,E[ ~Vk] = 1=3 and var( ~Vk) = 23=360, while

with QPSK spreading,E[ ~Vk] = 1=3 and var( ~Vk) = 1=180. Putting these into (6.43), we obtain

an approximation of the symbol error probability which is as simple computationally as the SGA in

(6.31). It is shown in [7, 9] that the approximated IGA in (6.43) is almost as accurate as the original

IGA in (6.41) in many situations.

In summary, we point out that the IGA generally gives a more accurate approximation to the

symbol error probability than the SGA does when the spreading gainN is reasonably large as in most

6.13


practical DS-CDMA systems. To illustrate this point, let us consider the symbol error probabilities of

two DS-CDMA systems with BPSK spreading and QPSK spreading, respectively. From (6.31), the

SGA predicts that the symbol error probabilities of the two systems are the same. On the other hand,

the IGA (6.43) states that the system with QPSK spreading has a smaller symbol error probability than

the system with BPSK spreading. The latter is in fact true for randomly selected sequences.

6.3 Near-far problem

Based on the SGA in (6.30), we see that the signal-to-noise ratio (SNR) of a user employing the

matched filter receiver in a DS-CDMA system withK active users is degraded by the factor

1 +

8<:EmN0

Xk 6=m

PkPm

2�k;m(0) + Re[�k;m(1)]

3N3

9=;

as compared to the case in which only the user is active. When the received powers of all users are

the same and the set of spreading sequences are properly chosen, the degradation in SNR is relatively

small if there are a moderate number of users. However, when the received powers of some of the

interferers are much larger4 than that of the desired user, the performance degradation is large. In

the context of wireless communications, this situation occurs when some of the interferers are located

close to the base station while the desired user is far away. This problem is known as thenear-far

problemin CDMA systems.

A common measure of the robustness of a receiver against the near-far problem is thenear-far

resistance, defined in [10], of the receiver. For now, we argue the intuitive idea of the near-far re-

sistance measure. The formal definition of the near-far resistance of a receiver will be given later in

Chapter 7. To understand the concept of near-far resistance, let us imagine that only one user, say

themth user, were active in the DS-CDMA system considered previously. In this case, the optimal

symbol error probability (using the matched filter receiver) would beQ(q2Em=N0), which decreases

exponentially with rate approximately equal to the SNREm=N0 when the SNR is large. We employ

this as a benchmark to compare performance of different receivers in the multiuser scenario. Going4Under the near-far situation, the SGA may not be accurate. However, a similar argument can be constructed employing

the IGA instead.

6.14


back to the realistic situation of multiple active users, the performance of any receiver will be poorer

than that of the optimal receiver of the single-user scenario just described because of the existence of

MAI. In fact, the larger the received powers of the interferers, the poorer is the performance. We look

at the exponential rate5 of decrease of the symbol error probability given by a certain receiver as the

SNR increases to some very large values. The ratio of exponential decrease rate toEm=N0 tells us how

efficient the receiver is compared to the optimal receiver of the single-user scenario. If the received

MAI power increases, this ratio will get smaller. The ratio of the exponential decrease rate toEm=N0

in the limiting case of extremely large MAI power is the near-far resistance of the receiver. A receiver

with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver

of the single-user scenario (the best that could be done). A near-far resistance of 0 indicates that the

receiver will break down in a near-far situation. For the matched filter receiver, we can employ (6.31)

or (6.43) to conclude that the symbol error probability levels off when the SNR increase (for example,

see Figure 6.2). Hence the exponential decrease rate is 0 and the near-far resistance is 0.

When the matched filter receiver does not give an adequate level of performance, there are basically

two ways to tackle the near-far problem. One of the ways is to control the transmitted powers of all

the users so that the received powers are the same. This method is known aspower control, and is

employed in all current practical CDMA systems, such as IS-95. Another way to tackle the near-far

problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to

develop better receivers that are near-far resistant.

6.4 Multiple access interference suppression

In this section, we discuss receivers based on the single-user detection approach. Receivers based on

the multiuser detection approach will be introduced in Chapter 7.

The optimal signal-user receiver described in [1] is near-far resistant, but it is too complex for

practical implementation. Its development is rather involved and interested readers are referred to [1].

Here, we focus on suboptimal receivers that are less complex than the optimal signal-user receiver. The

basic working principle of these receivers is to exploit some structures of the MAI that are different5If the symbol error probability does not decreases exponentially, the “exponential” decrease rate is 0.

6.15


from the desired signal and to utilize the difference to remove or suppress the MAI component from the

received signal. The structures of MAI we can utilize depend the design of the spreading sequences and

the availability of resources such as multiple receive antennas. A main dichotomy on MAI suppression

receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based

DS-CDMA systems. In a short-sequence-based system, a simple structural differentiation between the

desired signal and the MAI is the difference in the sequences employed. Since the sequences repeat

every symbol period, this structural differentiation is invariant from symbol to symbol. As a result, we

can easily extract and utilize this structural difference by using , for example, some standard adaptive

signal processing techniques, making the receiver implementation simple and desirable. It turns out

that most of these short-sequence-based single-user MAI suppression techniques can be interpreted

as special cases of their multiuser counterparts. To avoid repetition, we leave their development to

Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user

receivers.

For DS-CDMA systems employing long sequences, although the sequences are still different, the

structural differentiation mentioned above is not very useful since the structural difference varies from

symbol to symbol, making it hard to extract and utilize. Therefore, we have to employ some other

forms of structural differentiation between the MAI and the desired signal that are invariant from sym-

bol to symbol. To illustrate this idea, we will present a simplified development of the MAI suppression

receiver suggested in [11].

The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-

CDMA systems. It is based on the matched filter receiver. Its working principle is to exploit the

structural difference between the desired signal and the MAI caused by the fact that the delays of the

signals are different. The problem of symbol-by-symbol varying nature of the spreading sequences

is solved by obtaining the averaged (over all possible choices of sequences) structural difference in-

stead of the instantaneous one. This is possible since segments (over a symbol duration) of the long

sequences look like random. Hence the statistical averaged structural difference can be approximated

by the “easy-to-obtain” time-averaged structural difference.

The abstract statement above can made precise by looking at the output signal from the matched

filter. First, the signal model we consider here is exactly the same as the one described in Section 6.1

6.16


except that the period of the spreading sequences is now much large than the spreading gainN . To

simplify our discussion, we model the sequence elementsa(k)l as zero-mean iid random variables with

E[ja(k)l j2] = 1. In the context of long-sequence-based systems, this model basically means that the

sequences are aperiodic and are picked randomly from the set of all possible sequences. Of course,

this is an (good) approximation to the actual set of sequences used in practice. Let us focus on the

detection ofb(m)0 , the0th symbol of themth user. In this case, the impulse response of the matched

filter is given by

h(m)0 (t) =

N�1Xj=0

a(m)�j �(�t� jTc): (6.46)

Note that we have chosen to employ a non-causal filter here to simplify our notation later. Of course,

we have to use a causal filter in practice and the amount of delay in the causal filter, for exampleT ,

has to be added to all the results we are going to present.

Let us denote the output signal of the matched filter as

~rm(t) = ~y(t) + ~n(t); (6.47)

where ~y(t) is the component due to the desired user and~n(t) = ~nI(t) + ~nW (t) is the MAI-plus-

thermal-noise component with the subscriptsI andW standing for MAI and thermal noise, respec-

tively. Conditioning on the set~Sm of delays and phases defined in (6.35), it is easy to show that the

autocorrelation function of the matched filter output is given by

R~r(t; s) = R~y(t; s) +R~n(t; s); (6.48)

whereR~n(t; s) = R~nI (t; s) +R~nW (t; s),

R~y(t; s) = ~ (t) ~ �(s); 6 (6.49)

R~nI (t; s) =KX

k = 1k 6=m

PkNPm

1Xj=�1

~ (t� jTc � Æk) ~ �(s� jTc � Æk); (6.50)

and

R~nW (t; s) =N0pPm

~ (t� s): (6.51)

6The expression in (6.49) is actually an approximation, which is accurate whenN is large (see [11] for details).

6.17


Tc

Desired signal component

Interference components

Figure 6.3: Structural difference between the desired signal and MAI

The function (�) is the autocorrelation of the chip waveform defined by

(t) =1

Tc

Z 1

�1 (s) �(s� t) ds: (6.52)

Also, ~ (t) =pP0T (t) is just a scaled version of (t). We note that since the chip waveform (t)

is time-limited to[0; Tc), both (t) and ~ (t) are time-limited to(�Tc; Tc) and hence the summation

in (6.50) contains only a finite numbers of terms for any fixed values oft ands. For example, when

the chip waveform is rectangular (i.e., (t) = pTc(t)), then (t) is the triangular waveform stretching

from�Tc to Tc.

Now, the structural difference between the desired signal and the MAI can be readily observed by

considering the difference between the autocorrelation functions,R~y(t; s) andR~nI (t; s), of the desired

signal component~y(t) and the MAI componentnI(t), respectively. The autocorrelation functions are

made up of products of different delayed versions of the functions~ (t) and ~ (s). This difference

can be easily visualized by considering the intuition provided in Figure 6.3, which shows the delayed

versions of the function~ (t) that make up the desired signal and MAI autoccorrelation functions for

the two-user case (K = 2) with rectangular chip waveform. The blue triangle corresponds to the

~ (t) due to the desired signal. The four red triangles correspond to the delayed versions of~ (t) due

6.18


[r ]j i~

[g]~i*

r (t)h (t)j(m)m bj

(m)

Σi

Figure 6.4: MAI suppression receiver for long-sequence-based DS-CDMA systems

to the interferer. The reason that there are four triangles corresponding to the interferer is that for

0 < Æk < Tc, (k 6= m) as shown in the figure each interferer contributes exactly four non-zero terms in

the second summation in (6.50) for the observation interval of�Tc � t; s � Tc, in which the desired

signal autocorrelation function is non-zero. ForÆk = 0, there is only one triangle corresponds to the

interferer. This triangle coincides with the blue triangle corresponding to the desired user. In this case,

there is no structure difference between the desired signal and the MAI, and hence the MAI cannot be

suppressed.

The discussion above reveals the structural difference between the desired signal and the MAI

induced by the different delays of the signals. We can utilize this structure to suppress the MAI

component in the received signal by observing the matched filter output. Since the structure does not

depend on the spreading sequences, it is invariant from symbol to symbol and hence simple adaptive

algorithms can be employed. A simple way [11] to suppress MAI based on this structure is to sample

the matched filter output about the peak at each symbol (see Figure 6.3) and then weigh the samples

to form a decision statistic for that symbol as shown in Figure 6.4. We note that the matched filter

receiver is a special case of this method with only one sample taken. The weights are chosen so that

the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized. The

suppression of the MAI is performed implicitly in the process of minimizing the MSE.

More precisely, suppose that we sample the matched filter output2M + 1 times at

t = �MTs;�(M � 1)Ts; : : : ;�Ts; 0; Ts; : : : ; (M � 1)Ts;MTs;

whereTs < Tc, for the detection of the0th symbol of themth user. For example, Figure 6.3 shows that

case in whichTs = Tc=2 andM = 2. For convenience, we arrange the samples into the vector~rm and

6.19


work using vector and matrix notation7. An estimateb(m)0 of the transmitted symbolb(m)

0 is obtained

as the weighted sum of the samples taken at the output of the matched filter, i.e.,

b(m)0 =

Xj

[~g]�j [~rm]j = ~gHj ~rm; (6.53)

where[~rm]j = ~rm(jTs) is the sample at timejTs and[~g]j is the weight for that sample8. The weight

vector~g is chosen to minimize the MSE defined by

E��b(m)

0 � b(m)0

��2� = E��b(m)

0 � ~gHj ~rm

��2� ; (6.54)

where the expectation is conditioned over the set~Sm. It can be easily shown [11] that the optimal

choice of weight vector is the solution of the following set of equations:

[~ ]i =Xj

[R~r]ij[~g]j or ~ = R~r ~g; (6.55)

where [R~r]ij = R~r(iTs; jTs) and [~ ]j = ~ (jTs) are samples of the autorrelation functions of the

matched filter output~r(t) and the signal~ (t), respectively. As the existence of the thermal noise

component in the signal~r(t) guarantees the invertibility of the correlation matrixR~r, the optimal

weight vector is given by

~g = R�1~r~ : (6.56)

Writing the samples of the MAI-plus-thermal-noise autocorrelation functionR~n(t; s) as the matrixR~n

(i.e., [R~n]ij = R~n(iTs; jTs)), we can employ the the matrix inversion lemma9 to show that

~g =1

1 + R�1

~n~ ; (6.57)

where

= ~ HR�1~n~ : (6.58)

It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the

decision statistic~gHj ~rm. The maximum SNR value achieved is exactly given by (6.58). Suppose that

we can employ the improved Gaussian approximation here to approximate the MAI component in the7For further convenience, we index vectors and matrices with negative indices.8The notation[A]ij represents the(i; j)th element of the matrixA.9Assuming all the required invertibilities,(A�BDC)

�1= A�1 +A�1B

�D�1�CA

�1B��1

CA�1.

6.20


0 2 4 6 8 10 12 14 1610

−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Em

/N0 (dB)

Ps

matched filter receiver

MAI suppression receiver

single−user performance

Figure 6.5: Symbol error probability achieved by the MAI suppression receiver in a two-user system

with near-far problem

decision statistic as a Gaussian random variable. Then the conditional (conditioned on~Sm) symbol

error probability is given byQ(p2 ). For example, Figure 6.5 gives the plot of the symbol error

probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme

as shown in Figure 6.3 in the case where there are two active users in the system, the fractional delay

of the interferer isTc=2, and the received power of the interferer is 20dB above that of the desired

user. Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter

receiver and the symbol error probability of the single-user scenario. We see that the presence of

the strong MAI causes the error rate of the matched filter levels off as the SNREm=N0 increases.

With the MAI suppression receiver, the symbol error probability decreases exponentially as the SNR

increases, showing that the MAI is suppressed by the receiver. In fact, with the particular set of system

parameters, the performance of the MAI suppression receiver is only 3dB worse than the single-user

scenario.

6.21


We evaluate the robustness of the receiver against the near-far problem based on a measure similar

to the near-far resistance suggested in Section 6.3. First, we note that the SNR obtained by the receiver

in any situation cannot be larger than the SNR obtained when themth user is the only active user, i.e.,

� m4

= Em=N0. Following the idea of the near-far resistance in Section 6.3, we define the “near-far

efficiency” (NFE)10 as

NFE = inff = mjPk � 0; k = 1; 2; : : : ; m� 1; m+ 1; : : : ; Kg: (6.59)

We note that0 � NFE � 1 and it is a function of the delays and phases in the set~Sm. For example, in

the two-user case (K = 2), it can be shown [11] that the NFE of the receiver discussed here is upper

bounded by(k 6= m)

NFE1(Æk) = 1�1X

l=�1

j (lTc + Æk)j2; (6.60)

where this bound can be achieved in the limit by sampling finer and finer. For example, theNFE1

for the rectangular chip waveform is shown in Figure 6.6 as a function ofÆk (k 6= m). We see that

since the NFE is positive when the system is not chip-synchronous (i.e.,Æk 6= 0), the receiver is robust

against the near-far problem when the two users are asynchronous. In general, the robustness of this

receiver degrades when more and more strong interferers are in the system.

As a closing note, we emphasize that as the structural difference between the MAI and the desired

signal is invariant from symbol to symbol, we can employ standard adaptive algorithms to obtain the

optimal weight vector described in (6.56). For example, when a training data sequence is available,

we can employ the LMS algorithm to obtain the weight vector. In fact, it is shown in [11] that a blind

adaptive algorithm, which does not require the availability of a training sequence, can be developed to

obtain the weight vector.

6.5 References

[1] H. V. Poor and S. Verdu, “Single-user detectors for multiuser channels,”IEEE Trans. Commun.,

vol. 36, no. 1, pp. 50–60, Jan. 1988.10There is a subtle difference between NFE and the near-far resistance defined in [10]. This difference, which will

become apparent in Chapter 7, is caused by the fact that the SNR here is obtained over all the possible choices of sequences.

Hence, NFE measures the “averaged robustness” of the receiver against the near-far problem.

6.22


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

NF

E∞

δk/T

c

Figure 6.6:NFE1 of the MAI suppression receiver in a two-user channel

[2] TIA/EIA/IS-95 Interim Standard,Mobile Station-Base Station Compatibility Standard for Dual

Mode Wideband Spread Spectrum Cellular System, Telecommunications Industry Association,

Washington, D.C., Jul. 1993.

[3] J. S. Lehnert, “An efficient technique for evaluating direct-sequence spread-spectrum multiple-

access communications,”IEEE Trans. Commun., vol. 37, no. 8, pp. 851–858, Aug. 1989.

[4] K. Yao, “Error probability of asynchronous spread spectrum multiple access communication

systems,”IEEE Trans. Commun., vol. 25, no. 8, pp. 803–809, Aug. 1977.

[5] M. B. Pursley, “Performance evaluation for phase-coded spread-spectrum multiple-access com-

munication — Part I: System analysis,”IEEE Trans. Commun., vol. 25, no. 8, pp. 795–799,

Aug. 1977.

[6] R. K. Morrow and J. S. Lehnert, “Bit-to-bit error dependence in slotted DS/SSMA packet sys-

tems with random signature sequences,”IEEE Trans. Commun., vol. 37, no. 10, pp. 1052–1061,

Oct. 1989.

6.23


[7] T. M. Lok and J. S. Lehnert, “Error probabilities for generalized quadriphase DS/SSMA com-

munication systems with random signature sequences,”IEEE Trans. Commun., vol. 44, no. 7,

pp. 876–885, Jul. 1996.

[8] T. M. Lok and J. S. Lehnert, “An asymptotic analysis of DS/SSMA communication systems with

random signature sequences,”IEEE Trans. Inform. Theory, vol. 42, no. 1, pp. 129–136, Jan.

1996.

[9] J. M. Holtzman, “A simple, accuratemethod to calculate spread-spectrum multiple-access error

probabilities,”IEEE Trans. Commun., vol. 40, no. 3, pp. 461–464, Mar. 1992.

[10] S. Verdu,Multiuser Detection, Cambridge University Press, 1998.

[11] T. F. Wong, T. M. Lok, and J. S. Lehnert, “Asynchronous Multiple Access Interference Suppres-

sion and Chip Waveform Selection with Aperiodic Random Sequences,”IEEE Trans. Commun.,

vol. 47, no. 1, pp. 103–114, Jan. 1999.

6.24

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