B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

632 SPREAD SPECTRUM COMMUNICATIONS
After application of the matched filter (despreading), the ith receiver output at the sampling
instant t = k T , is
JkTs
r{ i) = c;(t)y(t) dt
(k-l)Ts
M
= L g;/P;st J s
kT
c;(t)c1(t) dt + J s
j=I (k- l)T s (k-l)T,
kT
c;(t)n(t) dt
" (i)
= gJ/P;R;j(k)s k
+ n;(k)
j=I
(11.28b)
For notational convenience, we have defined the (time-varying) cross-correlation coefficient
between two spreading codes as
JkTs
R;j(k) = c;(t)c1(t) dt
(11.28c)
(k-l)T s
and the ith receiver noise sample as
kT
n;(k) = J '
(k-l)T s
c;(t)n(t) dt
(1 1 .28d)
It is important to note that the noise samples of Eq. (11.28d) are Gaussian with mean
J k Ts
n;(k) = c;(t)n(t) dt = 0
(k-l)T,
The cross-correlation between two noise samples can be found as
(11 .29a)
(11.29b)
Equation ( 11.29) shows that the noise samples at the DSSS CDMA receiver are temporally
white. This means that the Gaussian noise samples at different sampling time instants are
independent of one another. Therefore, the optimum detection of {st ) } can be based on the
samples {r{ i) } at time t = kT.
For short-code CDMA, {c;(t)} are periodic and the period equals T s . In other words, the
PN spreading signals { c;(t)} are identical over each period [ (k - 1 )T s , kT s J. Therefore, in