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 di erent levels of synchronization errors. For all three diversity
combining schemes considered, the degradation in the system performance is expressed
as an e ective reduction in the system processing gain. Systems of 1.25MHz, 5MHz
and 10MHz are considered for di erent 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 di erent 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, in nite 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 di erent 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 e ective 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 in nite 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 o sets and ak(t) and bk(t) are the user speci c 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 e ectively 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 o sets 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 Ef 2
klg = 20 and has a probability density function
f kl(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=1q2Pk klbk(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 o set, 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 = q2P1 1j 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 q2P1 1l cos(1l 􀀀 ^1j) Z T+^1j
^1j
b1(t 􀀀 ^1j )b1(t 􀀀 1l)a1(t 􀀀 1l)dt; (13)
M1j =
K
Xk=2
L
Xl=1q2P1 kl 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 = q2P1 1j cos(1j)a1;􀀀1 Z 1j
0
b1(t)b1(t 􀀀 1j)dt
+q2P1 1j cos(1j)a1;0 Z T
1j
b1(t)b1(t 􀀀 1j)dt: (17)
Now, using the following partial correlation functions de ned 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 = q2P1 1j cos(1j)a1;􀀀1b1;0b1;􀀀1R11(1j)
+q2P1 1j 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 = q2P1 1j cos(1j)a1;􀀀1b1;0b1;􀀀11j
5
+q2P1 1j cos(1j)a1;0G(Tc 􀀀 1j)
+q2P1 1j cos(1j)a1;0
G􀀀1
Xm=1
b1;mb1;(m􀀀1)1j : (21)
We de ne the following random variables,
0 4=
a1;􀀀1b1;0b1;􀀀1; (22)
i 4=
a1;0b1;ib1;(i􀀀1); 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 = q2P1 1j cos(1j)G(Tc 􀀀 1j)a1;0 (25)
and
D1jb = q2P1 1j cos(1j)1j
G􀀀1
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 = q2P1 1jTa1;0; (27)
D1jb = 0: (28)
Now, we simplify the intersymbol interference term, I1j, de ned in (13). Once again,
using (2) and (3), (13) can be rewritten as,
I1j =
L
Xl=1
l6=j q2P1 1l 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 de ne 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
speci c 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 de ne two random variables li = b1;ib1;x
and
li = b1;ib1;(x􀀀1) 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) =
R􀀀1
Xi=0
[
lit1l + li(Tc 􀀀 t1l)] +
lRt1l (30)
^R
11(1l) = lR(Tc 􀀀 t1l) +
G􀀀1
X i=R+1
[
lit1l + li(Tc 􀀀 t1l)] (31)
We now de ne 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
G􀀀1
Xi=0 q2P1 1l cos(1l) [t1l~ li + (Tc 􀀀 t1l) li] (32)
Now, we simplify the multiple access interference term, M1j , de ned 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 de ne,
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=1q2Pk kl cos(kl)
G􀀀1
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
G􀀀1
Xj=0 q2Pk kl cos(kl) hb1;jqk;(j􀀀1)kl + b1;jqk;j(Tc 􀀀 kl)i: (34)
We now de ne the following random variables,
kj = b1;jqk;(j􀀀1)
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
G􀀀1
Xj=0 q2Pk kl 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 = Max 1jfZ1jg: (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,
f 1j (x) = L
L􀀀1
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
p2P1 1j cos(1j) (Tc 􀀀 1j)G 􀀀 1j PG􀀀1
m=0 m􀀀I1j􀀀M1j
pN0T 3
5
(39)
where Q(x) is the Q-function de ned as,
Q(x) = Z 1
x
e􀀀t2=2
p2
dt: (40)
The random variables I1j and M1j all arise from di erent 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(Ej 1j; 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 de ne 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
j2me􀀀m2!2=2 m> 0 and m odd
0 m> 0 and m even
1
2 m = 0
(43)
with c􀀀m = 􀀀cm;m > 0 [22]. (42) becomes exact in the limit as w goes to zero.
If we substitute (42) into (41), we obtain,
P(Ej 1j; m) =
1 X
m=􀀀1
cme
jm!p2P1 1j cos(1j )((Tc􀀀1j )G􀀀1jPG􀀀1
m=0
m)
pN0T
 Z 1
􀀀1 Z 1
􀀀1
e􀀀jm! i+m pN0T fI1j (i)fM1j (m) di dm: (44)
But, the characteristic function of a random variable, P is de ned as,
P (!) = Efexp(j!P)g = Z 1
􀀀1
ej!pfP (p) dp: (45)
Then,
P(Ej 1j; m) =
1 X
m=􀀀1
cme
jm!p2P1 1j cos(1j )((Tc􀀀1j )G􀀀1jPG􀀀1
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 = q2P1 1l cos(1l)
G􀀀1
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(!)]L􀀀1 (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 de ned as,
il (!) = Eej!p2P1 1l cos(1l)PG􀀀1
i=0 [t1l ~ i+(Tc􀀀t1l) i]
= E(cos(!q2P1 1l cos(1l)Tc) + cos(!q2P1 1l 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(!q2P1 1l cos(1l)(2t1l 􀀀 Tc)) cosG􀀀p(!q2P1 1l 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 ! cosG􀀀2p(!p2P1 1l cos(1l)Tc)
22p
+
G
Xp=1 G
p ! cosG􀀀p(!p2P1 1l cos(1l)Tc)
2p􀀀1!p2P1 1l cos(1l)Tc

bp􀀀1
2 c
Xq=0 p
q! sin((p 􀀀 2q)!p2P1 1l 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 = q2Pk kl cos(kl)
G􀀀1
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 (!)](K􀀀1)L (55)
where the characteristic function of mkl is found as,
mkl(!) = E ej!p2Pk kl cos(kl)PG􀀀1
j=0 [kjkl+kj (Tc􀀀kl)]
= E (cos(!q2Pk kl cos(kl)Tc) + cos(!q2Pk kl cos(kl)(2kl 􀀀 Tc))G)
11
= E8<
:
G
Xp=0 G
p !cosp(!q2Pk kl cos(kl)(2kl 􀀀 Tc))
 cosG􀀀p(!q2Pk kl 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 ! cosG􀀀2p(!p2Pk kl cos(kl)Tc)
22p
+
G
Xp=1 G
p ! cosG􀀀p(!p2Pk kl cos(kl)Tc)
2p􀀀1!p2Pk kl cos(kl)Tc

bp􀀀1
2 c
Xq=0 p
q! sin((p 􀀀 2q)!p2Pk kl 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 signi cantly faster to compute. Under these assumptions
we rst de ne,
ili = q2P1 1l cos(1l) [t1l~ i + (Tc 􀀀 t1l) i] (58)
to get
I1j =
L
Xl=1
l6=j
G􀀀1
Xi=0
ili: (59)
and since ili are assumed to be iid, this would result in,
I1j (!) = [i
li(!)]G(L􀀀1) : (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(!q2P1 1lTc) +
1
2
J0(!q2P1 1l(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 de ned 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+n􀀀1) [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 de ne,
mklj = q2Pk kl cos(kl) [kjkl + kj(Tc 􀀀 kl)] : (63)
From (29),
M1j =
K
Xk=2
L
Xl=1
G􀀀1
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(!q2Pk klTc) +
1
2
J0(!q2Pk kl(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
G􀀀1
Yj=0
mklj (!) (66)
13
Having found the expressions for the characteristic functions of I1j and M1j, we can now
nd the in nite series for the error probability. Using (43) and (46),
P(Ej 1j; m) =
1
2 􀀀
2

1 X
m=1
modd
1
m
e􀀀m2!2=2
 sin0
@
m!p2P1 1j cos(1j) (Tc 􀀀 j1j j)G 􀀀 j1j jPG􀀀1
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; : : : ; G􀀀1 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 de ne,
=
G􀀀1
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(Ej 1j; )f 1j (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
e􀀀m2!2=2L
L􀀀1
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 di erent 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 di erent; the independence assumption results in slightly
optimistic values. The di erence, 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 10􀀀3 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 Ef 2
1jg = 20 and Ef 1jg = 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 de ned as, 11; 12; : : : ; 1L, and
correspondingly, the carrier phase errors are de ned 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(Ej 11; : : :; 1L; ) =
1
2 􀀀
2

1 X
m=1
m odd
1
m
e􀀀m2!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(L􀀀1)
; (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(K􀀀1)
: (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(Ej 11; : : :; 1L)
=
1
2 􀀀
2

1 X
m=1
m odd
1
m
e􀀀m2!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 de ned as, 11; 12; : : : ; 1L and similarly, the carrier phase errors
are de ned 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(Ej 11; : : :; 1L; ) =
1
2 􀀀
2

1 X
m=1
m odd
1
m
e􀀀m2!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(L􀀀1); (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(K􀀀1): (83)
17
Similar to the maximal ratio case, it is possible to uncondition the error probability expression
from the variable analytically,
P(Ej 1j) =
1
2 􀀀
2

1 X
m=1
m odd
1
m
e􀀀m2!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 10􀀀7 with negligible error.
The accuracy of the technique is clearly bounded by the truncation of the in nite 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 di erent 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 di erent 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
 10􀀀3 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 a ects 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 arti cial 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 e ect the system capacity signi cantly. For example, for the
1.25MHz system with 2 arti cial 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
su er 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 e ect
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 e ective
processing gain loss. Then, the e ective 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
de ne the capacity loss of a system as the di erence 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 di erent 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.