Abstract

An ecient series that is used to calculate the probability of error for a BPSK

modulated DS CDMA system with chip timing and carrier phase errors in a slowly

fading, multipath channel is derived. The receiver is assumed to be a coherent RAKE

receiver. Three types of diversity schemes are considered: selection diversity, equal

gain diversity combining and maximal ratio diversity combining. The error probability

derivation does not resort to the widely used Gaussian approximation for the inter-

symbol interference and multiple access interference and is very accurate. The derived

series for probability of error calculations is used to assess the reduction in the sys-

tem capacity due to dierent levels of synchronization errors. For all three diversity

combining schemes considered, the degradation in the system performance is expressed

as an eective reduction in the system processing gain. Systems of 1.25MHz, 5MHz

and 10MHz are considered for dierent number of diversity branches and it is shown

that the percentage reduction in the system capacity due to synchronization errors is

approximately the same for all these systems.

yThis research was supported by a grant from the Canadian Institute for Telecommunications Research

under the NCE program of the Government of Canada.

zM. Oguz Sunay was with the Department of Electrical and Computer Engineering, Queen's University,

Kingston, Ontario, K7L 3N6, Canada. He is now a Member of Technical Sta in Bell Laboratories in

Whippany, New Jersey, 07981, USA. Peter J. McLane is with the Department of Electrical and Computer

Engineering, Queen's University, Kingston, Ontario, K7L 3N6, Canada.

I Introduction

Direct-sequence code division multiple access (DS CDMA), which was primarily used in

military communications until the late 80's, has for some time been the center of attention

in cellular radio communications [1-3]. In fact, Europe, Japan and Korea as well as North

America have all decided to base at least one of their third generation wireless standards on

the DS CDMA technology [4].

In DS CDMA systems, bandwidth spreading is accomplished by direct modulation of a

data modulated carrier by a wideband spreading code. Here, the signals all occupy the full

allocated bandwidth at all times. Interferers are therefore assumed to come from all directions.

The correlation properties of the spreading codes of dierent users provide multiple

access interference immunity in an ideal channel.

In mobile radio environments, the received signals are subjected to multipath fading

which severely degrades the system performance. If the system bandwidth is larger than

the coherence bandwidth of the channel, fading is frequency selective [5]. In this case, the

multipath components in the received signal are resolvable with a resolution in the time delay

of Tc, the chip duration. With CDMA techniques, the resolvable paths can be demodulated

individually by a RAKE receiver which exploits the excess redundancy due to the presence

of independent channel outputs from the multipaths. In a RAKE receiver, information

obtained from each branch is combined in a certain way to minimize the interference and

further mitigate the fading [6].

The performance of a DS CDMA system is usually measured in terms of the bit error

rate as a function of the number of active users for a given signal to noise ratio (SNR). There

have been a number of papers on the calculation of error probabilities for DS CDMA systems

in the recent literature for both additive white Gaussian noise channels and multipath

fading channels [7-14]. Most of these papers use Gaussian approximations for intersymbol

interference and multiple access interference and assume that the systems in question enjoy

perfect synchronization of the chip timing and the carrier phase. We have previously developed

an accurate, innite series expression for the calculation of error probabilities for a

DS CDMA system that uses coherent reception in an AWGN channel without making use

of a Gaussian approximation for the interference [16-19]. Furthermore, we did not neglect

the presence of synchronization errors in our derivation. In this paper, extending on the

procedure of [19], we present a performance analysis technique for a DS CDMA system in

1

a slowly fading multipath environment. We assume that the system uses coherent reception

with a RAKE receiver. We consider three dierent diversity combining schemes, namely,

selection diversity, equal gain combining and maximal ratio combining. As in [17-19], we do

not neglect the presence of synchronization errors in the system but rather investigate the

degradation in the system capacity due to such errors. We show that a synchronization error

for such systems can be represented as an eective loss in the processing gain.

An outline of the paper is as follows. The system-fading and multipath model is described

in Section II. Section III presents the coherent RAKE receiver and Section IV outlines the

use of the Fourier series to develop the innite series for the bit error rate calculations. The

system sensitivity to synchronization errors is discussed in Section V. Finally, conclusions

are drawn in Section VI.

II System Model

This paper is concerned with the calculation of the probability of symbol error for a DS

CDMA system in a frequency-selective multipath fading environment where each resolvable

path is independently Rayleigh faded. The DS CDMA system model examined in this paper

is similar to that used in [7] and [8] and is illustrated in Figure 1. Referring to this gure, let

us assume that there are K users transmitting signals in the system. Then, each transmitter

will transmit a signal in the form,

sk(t) = q2Pkbk(t Tk)ak(t Tk) cos(!ct + k) (1)

where !c is the common signaling frequency, Tk is the initial message starting times, k is

the initial phase osets and ak(t) and bk(t) are the user specic data and spreading signals,

respectively. The power of the transmitted signal, sk(t) can be calculated as

}k = lim

T!1

1

2T Z T

T jsk(t)j2dt = lim

T!1

1

T

Pk T +

1

2wc

sin(2wcT + 2k) = Pk: (2)

The data modulation ak(t) is a sequence of nonoverlapping rectangular pulses of duration

T, each of which has an amplitude of 1 or -1 with equal probability. Mathematically ak(t)

can be represented as,

ak(t) =

1 X

i=1

ak;i pT (t iT ): (3)

2

Here, ak;i is one symbol of the data modulation and takes on values f1g randomly and

pT (t) is the rectangular pulse of duration T. Similarly, the long PN sequence bk(t) has the

form,

bk(t) =

1 X

i=1

bk;i (t iTc) (4)

where bk;i is one chip of the PN sequence and takes on values f1g randomly. The chip

waveform (t) has duration Tc = T=G where G is the system processing gain. Note that

since the PN sequences are assumed to be long, the period of bk(t) is much larger than T.

Mobile radio channels are eectively modeled as a continuum of multipath components,

and thus the lowpass equivalent impulse response of the channel hk( ; t) can be written as,

hk( ; t) =

1 X

l=1

hk;l(t)( tkl(t)) (5)

where the tap gains hk;l(t) are complex Gaussian random variables and the time delays tkl(t)

are uniformly distributed over the interval [0; T] [23, 24]. When a wide-sense stationary

channel with uncorrelated scattering is considered, hk;l(t) are independent, identically distributed

random variables since they are modeled to have Gaussian distributions [27]. For a

multipath delay spread of Tm, (5) can be truncated at L = jTm

Tc k + 1 [5]. Here bxc denotes

the largest integer that is less than or equal to x. For a slowly varying channel, one can

assume that hk;l(t) = hk;l and tkl(t) = tkl during at least an entire duration of one symbol.

Since hk;l are complex Gaussian random variables, the channel model can equivalently be

written as,

hk(t) =

L

Xl=1

kl expfj#klg(t tkl) (6)

where kl is the path gain, #kl and tkl are the phase and time osets introduced by the

multipath channel on the l'th path of the k'th user's signal. In this case, k;l is Rayleigh

distributed with Ef2

klg = 20 and has a probability density function

fkl(x) =

x

0

exp

x2

20!u(x) (7)

where u(x) is the unit step function. Note that the frequency-nonselective channel is a

special case of the channel described by (6) with L = 1.

The total received signal can then be written as,

r(t) =

K

Xk=1 Z 1

1

hk(')sk(t ')d' + n(t)

3

=

K

Xk=1

L

Xl=1q2Pkklbk(t Tk tkl)ak(t Tk tkl) cos(!ct + k + #kl !ckl)

+n(t) (8)

where n(t) is the additive white Gaussian noise (AWGN) introduced by the channel. The

net time delay, kl, and the net phase oset, kl, are obtained by summing their respective

transmitter and channel parts such that,

kl = Tk + tkl; (9)

kl = k + #kl !ckl: (10)

The distributions of the random variables kl and kl are discussed in the next section for

all values of k and l.

III Receiver Model

We want to capture the signal from user 1, namely, a1(t). The received signal goes through

a RAKE receiver as shown in Figure 2. Here, the received signal is despread independently

for each multipath component by multiplying the spreading code of the rst user delayed by

an amount equal to the delay of the multipath component. The signal is then stripped o

its carrier and passed through a bank of correlators.

For the analysis, we consider a data symbol interval as [0; T] for convenience. In this

case, from Figure 2, the input to the decision device from the j'th path is,

Z1j = 2 Z T+^1j

^1j

b1(t ^1j)r(t) cos(2fct + ^1j)dt

= D1j + I1j +M1j + N1j (11)

where ^1j and ^1j are the estimates of 1j and 1j , respectively. In (11), D1j; I1j;M1j and

N1j correspond to the terms that consist of the desired signal plus self interference caused

by synchronization errors, intersymbol interference caused by multipath, multiple access

interference and additive white Gaussian noise, respectively. These terms can be expressed

as,

D1j = q2P11j cos(1j ^1j) Z T+^1j

^1j

b1(t ^1j)b1(t 1j)a1(t 1j)dt; (12)

4

I1j =

L

Xl=1

l6=j q2P11l cos(1l ^1j) Z T+^1j

^1j

b1(t ^1j )b1(t 1l)a1(t 1l)dt; (13)

M1j =

K

Xk=2

L

Xl=1q2P1kl cos(kl ^1j) Z T+^1j

^1j

b1(t ^1j)bk(t kl)ak(t kl)dt; (14)

(15)

and

N1j = 2 Z T+^1j

^1j

n(t)b1(t ^1j) cos(2fct + ^1j)dt: (16)

A system with synchronization errors will not be able to estimate the time delays and

phase delays corresponding to the individual paths in the RAKE receiver correctly. The chip

timing and carrier phase errors for the j'th path will be, 1j ^1j and 1j ^1j, respectively.

Without any loss of generality, we assume that ^1j = ^1j = 0. Then, for our purposes, 1j

and 1j are the chip timing and carrier phase errors, respectively.

We now simplify D1j . Using (3), (12) can be rewritten as,

D1j = q2P11j cos(1j)a1;1 Z 1j

0

b1(t)b1(t 1j)dt

+q2P11j cos(1j)a1;0 Z T

1j

b1(t)b1(t 1j)dt: (17)

Now, using the following partial correlation functions dened in the literature by Pursley

[26],

Rij ( ) = Z

0

bi(t)bj(t )dt; (18)

^R

ij ( ) = Z T

bi(t)bj(t )dt (19)

we can rewrite (17) as,

D1j = q2P11j cos(1j)a1;1b1;0b1;1R11(1j)

+q2P11j cos(1j)a1;0^R

11(1j) (20)

For random PN sequences with rectangular shaped chips it is possible to further simplify

(20). By making use of (4) and the fact that j1j j < Tc in order for the spread spectrum

system to successfully operate, we get,

D1j = q2P11j cos(1j)a1;1b1;0b1;11j

5

+q2P11j cos(1j)a1;0G(Tc 1j)

+q2P11j cos(1j)a1;0

G1

Xm=1

b1;mb1;(m1)1j : (21)

We dene the following random variables,

0 4=

a1;1b1;0b1;1; (22)

i 4=

a1;0b1;ib1;(i1); i = 1; 2; : : : ; G 1: (23)

Then, i; i = 0; : : : ; (G 1) are iid random variables taking on values f1g with equal

probability. Thus (21) can be rewritten as,

D1j = D1ja + D1jb (24)

where

D1ja = q2P11j cos(1j)G(Tc 1j)a1;0 (25)

and

D1jb = q2P11j cos(1j)1j

G1

Xm=0

m: (26)

Here, D1ja is the desired signal term and D1jb is the self interference term caused by the

non-zero chip timing error. Note that if the system is free of synchronization errors,

D1ja = q2P11jTa1;0; (27)

D1jb = 0: (28)

Now, we simplify the intersymbol interference term, I1j, dened in (13). Once again,

using (2) and (3), (13) can be rewritten as,

I1j =

L

Xl=1

l6=j q2P11l cos(1l) ha1;1R11(1l) + a1;0^R

11(1l)i: (29)

where 1l; l 6= j are modeled to be iid random variables, uniformly distributed over [0; Tm]

where Tm is the multipath delay spread of the channel. Similarly, we model 1l to be

iid random variables, uniform in [0; 2]. Then, the autocorrelation functions in (29) can

easily be evaluated. We dene t1l = 1l mod Tc and R = (1l t1l)=Tc. In this case,

t1l is a random variable that has a probability density function (pdf) dependent on the

6

specic value of the maximum multipath delay spread, Tm. If Tm is an integer multiple

of the chip duration, Tc, t1l is uniform in [0; Tc]. Otherwise, the pdf of t1l will see a

drop in its value in the interval [Tm mod Tc; Tc] and correspondingly, the pdf value in the

interval [0; Tm mod Tc] will increase as shown in Figure 3. This deviation from the uniform

distribution will be negligibly small since Tm >> Tc, and hence, for our purposes, we assume

that t1l is uniformly distributed in [0; Tc]. We also dene two random variables li = b1;ib1;x

and

li = b1;ib1;(x1) where x is an integer number that is dependent on the value of 1l.

Regardless of the value of x, however, li and

li are iid random variables taking on the

values f1g randomly. Then,

R11(1l) =

R1

Xi=0

[

lit1l + li(Tc t1l)] +

lRt1l (30)

^R

11(1l) = lR(Tc t1l) +

G1

X i=R+1

[

lit1l + li(Tc t1l)] (31)

We now dene the following random variables,

~li = 8<

:

a1;1li i = 0; : : : ; R

a1;0li i = R + 1; : : : ; G 1

li = 8<

:

a1;1

li i = 0; : : : ; R 1

a1;0

li i = R; : : : ; G 1

~li and li are iid random variables that take on values f1g randomly. Then,

I1j =

L

Xl=1

l6=j

G1

Xi=0 q2P11l cos(1l) [t1l~li + (Tc t1l)li] (32)

Now, we simplify the multiple access interference term, M1j , dened in (14). For all

practical reasons, all of the information signals that cause interference may be considered to

be random and thus be imbedded into the long PN sequences. Thus we dene,

qk(t) = ak(t)bk(t) =

1 X

i=1

qk;i (t iTc) (33)

where, for our purposes, qk;i is a set of random variables that randomly take on the values

f1g. Then, without any loss of generality, the random variables kl; k = 2; : : : ; K can be

assumed to be iid and uniformly distributed over the interval [0; Tc]. The random variables,

7

kl are assumed to be iid and uniform over [0; 2]. Therefore we get,

M1j =

K

Xk=2

L

Xl=1q2Pkkl cos(kl)

G1

Xj=0 Z (j+1)Tc

jTc

b1;j

1 X

i=1

qk;i (t jTc) (t iTc kl)dt

=

K

Xk=2

L

Xl=1

G1

Xj=0 q2Pkkl cos(kl) hb1;jqk;(j1)kl + b1;jqk;j(Tc kl)i: (34)

We now dene the following random variables,

kj = b1;jqk;(j1)

kj = b1;jqk;j

Then, kj and kj are iid random variables that take on values f1g randomly. Thus,

M1j =

K

Xk=2

L

Xl=1

G1

Xj=0 q2Pkkl cos(kl) [kjkl + kj(Tc kl)] (35)

Finally, the AWGN term, N1j is a Gaussian random variable with zero mean and N0T

variance [27].

IV Bit Error Rate Analysis

Suppose that a1;0 = 1 represents the binary symbol 1 and a1;0 = 1 represents the binary

symbol 0. The decision device in Figure 2 produces the symbol 1 if the decision variable

Z > 0 and the symbol 0 if Z < 0. An error occurs if Z < 0 when a1;0 = 1 or if Z > 0

when a1;0 = 1. Since a1;0 is assumed to take on values f1g with equal probability, the

probability of error is simply equal to the probability of having Z > 0 when a1;0 = 1,

P(E) = P(Z > 0ja1;0 = 1): (36)

Recall from (11) and (24) that the input signal to the decision device from the j'th path of

the receiver is Z1j = D1ja + D1jb + I1j + M1j + N1j where D1ja is the desired signal, D1jb

is the self interference due to chip timing errors, I1j is the intersymbol interference due to

the multipath, M1j is the multiple access interference and N1j is the AWGN term with zero

mean and N0T variance.

Once all Z1j's are obtained, diversity combining is performed in the receiver and the

decision variable Z is formed. We consider three diversity combining schemes: selection

diversity, maximal ratio diversity combining and equal gain diversity combining.

8

IV.1 Selection Diversity

When selection diversity is employed, the receiver simply selects the receiver path with the

highest path gain, 1j, and uses the information from this path to estimate the transmitted

signal a1(t). The other paths are not used in the decision making process. In other words,

the decision variable Z is equal to,

Z = Max1jfZ1jg: (37)

Since only the j'th path is used in the decision making, the chip timing and carrier phase

errors in the receiver are, 1j and 1j , respectively. When the j'th path gain is the maximum

of the L gains where the individual gains are Rayleigh distributed, the probability density

function of the j'th path gain will be in the form,

f1j (x) = L

L1

Xk=0 L 1

k (1)kx

0

exp

x2(k + 1)

20 !u(x) (38)

as shown in equation (5.2-7) of [23].

The error probability conditioned on the random variables I1j;M1j; 1j and m is,

P(EjI1j;M1j; 1j; m) = Q2

4

p2P11j cos(1j) (Tc 1j)G 1j PG1

m=0 mI1jM1j

pN0T 3

5

(39)

where Q(x) is the Q-function dened as,

Q(x) = Z 1

x

et2=2

p2

dt: (40)

The random variables I1j and M1j all arise from dierent phyisical sources. Hence they are

independent. Thus, using the total probability theorem [27], the error probability conditioned

only on 1j and m is written as,

P(Ej1j; m) = Z 1

1 Z 1

1

P(EjI1j;M1j; 1j; m)fI1j (i)fM1j (m) di dm (41)

The probability density functions fI1j (i) and fM1j (m) are dicult to determine. Instead,

we proceed to rewrite the conditional error probability given in (39) using a Fourier series

expansion of Q(x) [22]. We dene the error function Q(x) to be,

Q(x) '

1 X

m=1

cmejm!x (42)

9

where ! is the Fourier series frequency and cm are the Fourier series coecients and are

given by,

cm =8>

><

>

>:

1

j2mem2!2=2 m> 0 and m odd

0 m> 0 and m even

1

2 m = 0

(43)

with cm = cm;m > 0 [22]. (42) becomes exact in the limit as w goes to zero.

If we substitute (42) into (41), we obtain,

P(Ej1j; m) =

1 X

m=1

cme

jm!p2P11j cos(1j )((Tc1j )G1jPG1

m=0

m)

pN0T

Z 1

1 Z 1

1

ejm! i+m pN0T fI1j (i)fM1j (m) di dm: (44)

But, the characteristic function of a random variable, P is dened as,

P (!) = Efexp(j!P)g = Z 1

1

ej!pfP (p) dp: (45)

Then,

P(Ej1j; m) =

1 X

m=1

cme

jm!p2P11j cos(1j )((Tc1j )G1jPG1

m=0

m)

pN0T

I1j

m!

pN0T ! M1j

m!

pN0T ! (46)

Thus, we need only to nd the characteristic functions of I1j and M1j. To this end, we let,

il = q2P11l cos(1l)

G1

Xi=0

[t1l~i + (Tc t1l)i] (47)

to get

I1j =

L

Xl=1

l6=j

il: (48)

Since the il are independent random variables,

I1j (!) = [i

l(!)]L1 (49)

If (47) is studied, it is seen that the random variables 1l; 1l and t1l remain constant

throughout the duration of G chips whereas the random variables ~i and i vary independently

from chip to chip. In this case, the characteristic function of il is dened as,

il (!) = Eej!p2P11l cos(1l)PG1

i=0 [t1l ~i+(Tct1l)i]

= E(cos(!q2P11l cos(1l)Tc) + cos(!q2P11l cos(1l)(2t1l Tc))G)(50)

10

Using the binomial expansion, it is possible to rewrite (50) as,

il(!) =

E8<

:

G

Xp=0 G

p !cosp(!q2P11l cos(1l)(2t1l Tc)) cosGp(!q2P11l cos(1l)Tc)9=

;

(51)

It is possible to perform the expectation on the random variable t1l analytically. Using

(2.513.3) and (2.513.4) on page 132 of [28], we get,

il(!) = E8<

:

G=2

Xp=0 G

2p! 2p

p ! cosG2p(!p2P11l cos(1l)Tc)

22p

+

G

Xp=1 G

p ! cosGp(!p2P11l cos(1l)Tc)

2p1!p2P11l cos(1l)Tc

bp1

2 c

Xq=0 p

q! sin((p 2q)!p2P11l cos(1l)Tc)

p 2q 9>=

>

;

: (52)

The expectation relative to the random variables 1l and 1l can be performed using numerical

integration. A simple trapezoidal rule provides accurate results in a reasonably fast

manner. Once the numerical integration is performed, the characteristic function of I1j can

simply be found using (49).

We now nd the characteristic function of M1j. To this end, we let,

mkl = q2Pkkl cos(kl)

G1

Xj=0

[kjkl + kj (Tc kl)] : (53)

to get

M1j =

K

Xk=2

L

Xl=1

mkl: (54)

Similar to il in (48), the mkl are iid random variables as well. Therefore,

M1j (!) = [mkl (!)](K1)L (55)

where the characteristic function of mkl is found as,

mkl(!) = E ej!p2Pkkl cos(kl)PG1

j=0 [kjkl+kj (Tckl)]

= E (cos(!q2Pkkl cos(kl)Tc) + cos(!q2Pkkl cos(kl)(2kl Tc))G)

11

= E8<

:

G

Xp=0 G

p !cosp(!q2Pkkl cos(kl)(2kl Tc))

cosGp(!q2Pkkl cos(kl)Tc) (56)

As before, the expectation on the random variable, kl can be performed analytically. Using

[28] we get,

mkl (!) = E8<

:

G=2

Xp=0 G

2p! 2p

p ! cosG2p(!p2Pkkl cos(kl)Tc)

22p

+

G

Xp=1 G

p ! cosGp(!p2Pkkl cos(kl)Tc)

2p1!p2Pkkl cos(kl)Tc

bp1

2 c

Xq=0 p

q! sin((p 2q)!p2Pkkl cos(kl)Tc)

p 2q 9>=

>

;

: (57)

The expectation relative to the random variables kl and kl in equation (46) can be performed

using numerical integration. Note that the characteristic functions of il and mkl

become identical if P1 = Pk. Once again, trapezoidal rule can be used to perform these

integrations.

An alternative way to nd the characteristic functions of I1j and M1j would be to make

some independence assumptions as was done earlier in [19, 18] at the expense of some degradation

in the accuracy of the technique. If one assumes that the products 1l cos(1l)[t1l~i

+(Tc t1l)i] and kl cos(kl) [kjjklj + kj(Tc jklj)] vary independently from chip to

chip, it is possible to nd closed form expressions for the characteristic functions of I1j and

M1j and these expressions would be signicantly faster to compute. Under these assumptions

we rst dene,

ili = q2P11l cos(1l) [t1l~i + (Tc t1l)i] (58)

to get

I1j =

L

Xl=1

l6=j

G1

Xi=0

ili: (59)

and since ili are assumed to be iid, this would result in,

I1j (!) = [i

li(!)]G(L1) : (60)

12

Thus, we nd the characteristic function of ili. By using the equations (9.1.18) on page 360

of [29] and (6.629) on page 716 of [28] we get,

ili (!) = Ef

1

2

J0(!q2P11lTc) +

1

2

J0(!q2P11l(2t1l Tc))g

=

1

2

e!2P10T2

c

+

1

2

1 X

n=0

n!(!pP10Tc)2n

(2n + 1)!

M(n + 1; 2(n + 1);!2P10T2

c ) (61)

where J0(x) is the zeroth order Bessel function and M(a; b; z) is the con

uent hypergeometric

function and is dened as,

M(a; b; z) =

1 X

k=0

(a)kzk

(b)kk!

(62)

where (a)0 = 1 and (a)n = a(a+1) : : : (a+n1) [29]. Note that (61) is in its closed form and

does not require numerical integration. Once (61) is computed, the characteristic function

of I1j can easily be found using (60).

We continue with the derivation of the characteristic function for the multiple access

interference term, M1j. For this purpose we dene,

mklj = q2Pkkl cos(kl) [kjkl + kj(Tc kl)] : (63)

From (29),

M1j =

K

Xk=2

L

Xl=1

G1

Xj=0

mklj (64)

Once again, using the equations (9.1.18) on page 360 of [29] and (6.629) on page 716 of [28]

we get,

mklj (!) = E1

2

J0(!q2PkklTc) +

1

2

J0(!q2Pkkl(2kl Tc))

=

1

2

e!2Pk0T2

c

+

1

2

1 X

n=0

n!(!pPk0Tc)2n

(2n + 1)!

M(n + 1; 2(n + 1);!2Pk0T2

c ) (65)

and since mklj are iid random variables, from (64),

M1j (!) =

K

Yk=2

L

Yl=1

G1

Yj=0

mklj (!) (66)

13

Having found the expressions for the characteristic functions of I1j and M1j, we can now

nd the innite series for the error probability. Using (43) and (46),

P(Ej1j; m) =

1

2

2

1 X

m=1

modd

1

m

em2!2=2

sin0

@

m!p2P11j cos(1j) (Tc j1j j)G j1j jPG1

m=0 m pN 0T 1

A I (

m!

pN0T

) M(

m!

pN0T

) (67)

where the characteristic functions of I1j and M1j can be found using either (49) and (55)

or (60) and (66). We need to integrate P(E) over the distributions of 1j and m;m =

0; : : : ; G1 to get the unconditional error probability expression. Recall that when selection

diversity is employed 1j has a probability density function of the form given in (38). m,

on the other hand, are iid random variables that take on values f1g randomly. We dene,

=

G1

Xm=0

m: (68)

Then, is a random variable with binomial distribution. Since 1j and are independent

random variables,

P(E) = Z 1

1 Z 1

1

P(Ej1j; )f1j (y)f(x) dy dx (69)

By using equations (3.952.1) on page 495, (3.323.2) on page 307 and (3.462.6) on page 338

of [28] successively and performing some substitutions we get,

P(E) =

1

2

2

1 X

m=1

m odd

1

m

em2!2=2L

L1

Xk=0 L 1

k (1)kCmk

k + 1 s

D2

mk + 1=(2G)

1

D2

mk

D2

mk + 1=(2G)!exp( C2

mkD2

mk

D2

mk + 1=(2G) C2

mk) 1

p2G

I1j (

m!

pN0T

) M1j (

m!

pN0T

) (70)

where

Cmk =

m!p2P1 cos(1j)(Tc j1j j)G

2pN0T k+1

20 1=2 ; (71)

Dmk =

m!p2P1 cos(1j)j1j j

2pN0T k+1

20 1=2 : (72)

14

We use both methods of nding the characteristic functions for the interference terms

to compute the system capacities at dierent chip timing and carrier phase errors. System

performance as a function of the number of active users is graphed for both methods in Figure

4 for systems at various synchronization error levels. Here, systems with 2 multipaths and

20dB SNR are considered. As can be seen from Figure 4, the probability of error calculated

from the two methods is slightly dierent; the independence assumption results in slightly

optimistic values. The dierence, however, is never large enough to grant a discrepency in

the system capacity calculated using these methods. We nd that when an error probability

of 103 is desired, the two methods give exactly same value for the system capacity at all

synchronization error levels.

IV.2 Maximal Ratio Combining

When maximal ratio combining is employed, the receiver, having perfect knowledge of the

individual path gains, weighs each path with its corresponding path gain and then sums

these weighted terms. It is this sum that is used in the decision making. Then,

Z =

L

Xj=1

1jZ1j

=

L

Xj=1

1jfD1j + I1j +M1j + N1jg

= D + I +M + N (73)

where D is the sum of the desired signal terms from all branches, I is the intersymbol

interference, M is the multiple access interference and N is the AWGN term with variance

equal to,

2N

= 2N0T0Lf1 + (L 1)

20

4 g (74)

since Ef2

1jg = 20 and Ef1jg = 0p20=2. Since all of the branches are used in the

decision making, each branch in the receiver structure of Figure 2 has the potential to have

synchronization errors. Thus the chip timing errors are dened as, 11; 12; : : : ; 1L, and

correspondingly, the carrier phase errors are dened as, 11; 12; : : :; 1L.

Following the same procedure outlined for the selection diversity receiver, one can write

the error probability for maximal ratio combining conditioned on the L path gains and the

15

Gaussian random variable as,

P(Ej11; : : :; 1L; ) =

1

2

2

1 X

m=1

m odd

1

m

em2!2=2

sin "m!p2P1PLj

=1 2

1j cos(1j)[G(Tc 1j ) 1j ]

N #

I

m!

N M

m!

N : (75)

Once again, we only need to nd the characteristic functions of I and M to nd the error

probability expression. These functions can be found either semi-analytically or using the

independence assumption discussed in the previous section. Using the equations (9.1.18) on

page 360 of [29] and (6.629) on page 716 of [28], we obtain,

I (!) = "1

2

e!2P10T2

c (PL

j=1

1j)2

+

1

2

1 X

n=0

n!(!pP10Tc(PLj

=1 1j))2n

(2n + 1)!

M(n + 1; 2(n + 1);!2P10T2

c (

L

Xj=1

1j)2)3

5

G(L1)

; (76)

and

M(!) = "1

2

e!2Pk0T2

c (PL

j=1 1j)2

+

1

2

1 X

n=0

n!(!pPk0Tc(PLj

=1 1j))2n

(2n + 1)!

M(n + 1; 2(n + 1);!2Pk0T2

c (

L

Xj=1

1j)2)3

5

GL(K1)

: (77)

It is possible to uncondition (75) from the random varible analytically. The error probability

conditioned only on the individual path gains is given by,

P(Ej11; : : :; 1L)

=

1

2

2

1 X

m=1

m odd

1

m

em2!2=2 exp(m2!2P1G(

L

Xj=1

2

1j cos(1j)1j)2=2N

)

sin "m!p2P1PLj

=1 2

1j cos(1j)G(Tc 1j)

N #I

m!

N M

m!

N (78)

Unconditioning (78) from the individual path fains requires numerical integration. We use

the simple trapezoidal rule to perform this integration.

16

IV.3 Equal Gain Combining

When equal gain combining is employed, the receiver simply sums each path term and uses

this sum in the decision making. Then,

Z =

L

Xj=1

Z1j

=

L

Xj=1

D1j + I1j +M1j + N1j

= D + I +M + N (79)

where D is the sum of the desired signal terms from all branches, I is the intersymbol

interference, M is the multiple access interference and N is the AWGN term with variance

equal to,

2N

= N0TL: (80)

Similar to the maximal ratio combining case, when equal gain combining is employed, it is

clear from (79) that all of the L paths are used in the decision making process. Therefore,

the chip timing errors are dened as, 11; 12; : : : ; 1L and similarly, the carrier phase errors

are dened as, 11; 12; : : :; 1L.

Following the same procedure outlined for the selection diversity receiver one can write

the error probability for equal gain combining conditioned on the L path gains and as,

P(Ej11; : : :; 1L; ) =

1

2

2

1 X

m=1

m odd

1

m

em2!2=2

sin "m!p2P1PLj

=1 1j cos(1j)[G(Tc 1j ) 1j ]

N #

I

m!

N M

m!

N : (81)

The characteristic functions are found the same way as in the two previous cases as,

I (!) = [

1

2

e!2P10T2

c +

1

2

1 X

n=0

n!(!pP10Tc)2n

(2n + 1)!

M(n + 1; 2(n + 1);!2P10T2

c )]G(L1); (82)

and

M(!) = [

1

2

e!2Pk0T2

c +

1

2

1 X

n=0

n!(!pPk0Tc)2n

(2n + 1)!

M(n + 1; 2(n + 1);!2Pk0T2

c )]GL(K1): (83)

17

Similar to the maximal ratio case, it is possible to uncondition the error probability expression

from the variable analytically,

P(Ej1j) =

1

2

2

1 X

m=1

m odd

1

m

em2!2=2 exp(m2!2P1G(

L

Xj=1

1j cos(1j)1j)2=2N

)

sin "m!p2P1PLj

=1 1j cos(1j)G(Tc 1j )

N #I

m!

N M

m!

N (84)

However, as before, unconditioning (75) from the individual path gain variables requires

numerical integration. Once again, we use the trapezoidal rule for this purpose.

IV.4 Computational Aspects

The error probability expressions for the three diversity combining schemes, (70), (78) and

(84) can be proved to be absolutely convergent [30]. Note once again that ! is the Fourier

series frequency. Thus the above expressions for the probability of error are strictly true in

the limit as ! goes to zero. In our case, we nd that it is sucient to assume ! = =25

and correctly calculate error probabilities greater than or equal to 107 with negligible error.

The accuracy of the technique is clearly bounded by the truncation of the innite series. In

our case, taking the rst 21 terms in the series into consideration was sucient to achieve

the desired level of accuracy. Bounds on the truncation error have been discussed in detail

in [22].

V Degradation of the System Capacity due to Syn-

chronization Errors

For the three dierent receiver structures considered in this paper the probability of error,

as can be observed in 70), (78) and (84), is a function of the received signal to noise ratio,

SNR=20P1T=N0, the number of users present in the system, K, the number of diversity

paths, L, the synchronization errors, 1j and 1j and the processing gain, G. Thus, it is

possible to nd the system capacity for dierent values of SNR, number of diversity paths,

synchronization errors and processing gain. When G, SNR, L, 1j and 1j are xed, we can

evaluate the error probability of the system for increasing number of users. The capacity of

18

the system is simply the maximum number of users that will still yield an error probability

below a certain threshold.

We consider a voice transmission at 9600 bits/sec that requires an error probability

103 and system bandwidths of 1.25 MHz, 5MHz and 10MHz. The 1.25MHz system has a

processing gain of 128, the 5MHz system has a processing gain of 512 and the 10MHz system

has a processing gain of 1024. We assume that the maximum multipath delay spread is in

the range of 25 to 200 nanoseconds [31]. Thus, we conclude that the 1.25MHz channel has

only one, the 5MHz channel has two and the 10MHz system has three resolvable paths. We

assume a slowly fading channel.

For all three diversity combining schemes, multipath fading aects the system performance

dramatically. The 1.25MHz system at 20dB SNR has a capacity of zero if no coding

and diversity is employed and no voice activity factor is taken into account. The same system

was shown to have a capacity of 39 users in an AWGN environment [9, 19]. As stated

previously, the 1.25MHz system has no inherent diversity through multipath; other means

such as the use of multiple antennas at the receiver are to be employed to achieve a nonzero

capacity. When an articial diversity of 2 is achieved, the system has a capacity of 4 if

selection diversity is employed, 9 if equal gain combining is employed and 13 if maximal

ratio combining is employed.

The wideband systems of 5MHz and 10MHz, on the other hand, both have resolvable

multipath branches of 2 and 3, respectively. If all of these branches are utilized by means

of selection diversity at the receiver, the 5MHz system has a capacity of 17 and the 10MHz

system has a capacity of 58, respectively.

Synchronization errors eect the system capacity signicantly. For example, for the

1.25MHz system with 2 articial multipaths, a chip timing error of 10% will reduce capacity

of the selection diversity system to 3, the capacity of the equal gain combining system to

7 and the capacity of the maximal ratio combining system to 10, a 20% reduction in the

system capacity in all cases. This percentage loss in the system capacity is approximately

the same as the percentage loss of the same system in the AWGN environment [19]. Here,

we assume that for maximal ratio and equal combining systems, all of the receiver branches

suer from the same level of synchronization errors, i.e., 11 = 12 = : : : = 1L and 11 =

12 = : : : = 1L. From 70), (78) and (84), it can be seen that the synchronization errors eect

the system performance by potentially reducing the energy of the desired signal component

19

of the received signal and by introducing self interference. We have numerically found that

the self interference, in comparison to the signal energy reduction is negligibly small. The

reduction in the desired signal energy level can alternatively be interpreted as an eective

processing gain loss. Then, the eective processing gain is approximately a linear function

of the system capacity for a given level of maximum allowable error probability.

System error probabilities as a function of SNR are plotted for the three diversity combining

schemes when perfect synchronization is achieved in Figure 5. For the selection diversity

receiver, the capacity losses for various levels of synchronization errors are listed in tabular

form in Table 1 for the 1.25MHz system, in Table 2 for the 5MHz system and in Table 3

for the 10MHz system, respectively, all at 20dB SNR. From these tables, it can be seen that

the percentage capacity loss due to a certain level of synchronization error is approximately

the same in all three systems independent of the number of diversity branches considered.

The occasional discrepencies between the values are due to the quantization inherent in the

process of nding the system capacity from the bit error rate. This is because, the system

capacity can only take on integer values.

In Tables 4 and 5, the capacity losses of the three diversity combining schemes are

compared for various chip timing and carrier phase error values when the system bandwitdh

is 1.25MHz and 5MHz, respectively. It is seen that the percentage capacity losses due to

a certain level of synchronization error are approximately the same for the three diversity

schemes. Thus, it can be concluded that all three diversity combining schemes are equally

sensitive to synchronization errors.

In a practical system, both chip timing and carrier phase errors will be present. If we

dene the capacity loss of a system as the dierence between the capacity when there are

no synchronization errors and the capacity when synchronization errors are present, it can

be seen from Tables 1-5 that the capacity loss from the presence of both chip timing and

carrier phase errors is approximately the sum of individual losses for all values of SNR for

all systems considered.

Error probability as a function of the number of active users present in the system is

plotted in Figures 6, 7 and 8 for the 1.25MHz, 5MHz and 10MHz systems, respectively

when selection diversity is employed. The graphs show that as the number of users increase,

the performance of systems with dierent number of diversity paths converge. However,

when low error probabilities are required, as is the case for reliable transmission of voice,

20

data and video signals, having multiple diversity branches increase the system performance

that was reduced by fading. However, diversity combining on its own, is not sucient to

gain back all of the capacity that is lost due to fading. Additional means such as coding,

interleaving and/or antenna diversity are necessary to further increase the capacity. It is seen

from Figures 7 and 8 that when the multipath branches inherent in the 5MHz and 10MHz

systems are not utilized (i.e. when L=1 for the 5MHz system and when L=1,2 for the 10MHz

system), the performance of these systems become almost as poor as the 1.25MHz system

that does not employ any diversity.