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

626 SPREAD SPECTRUM COMMUNICATIONS
shift register with suitable feedback connection, has a length L = 2 m - 1 bits, the maximum
period for such a finite state machine. Figure 11.9b shows a shift register encoder for
m = 6 and L = 63. For such "short" PN sequences, the autocorrelation function is nearly an
impulse and is periodic
{ T s
Rc(r) = J
o c(t)c(t + r) dr = - T : r = 0, ±LT c , · ·.
{ LT
r =fa 0, ±LT c , . ..
(11.13)
As a matter of terminology, a DSSS spreading code is a short code if the PN sequence
period equals the data symbol period T s . A DSSS spreading code is a long code if the PN
sequence period is a (typically large) multiple of the data symbol period.
Single-User DSSS Analysis
The simplest analysis of DSSS system can be based on Fig. 11.8. To achieve spread spectrum,
the chip signal c(t) typically varies much faster than the QAM symbols. As shown in Fig. 11.8,
there are multiple chips of ±1 within each symbol duration of Ts . Denote the spreading
factor
T c = chip period
Then the spread signal spectrum is essentially L times broader than the original modulation
spectrum
Be = (L + I )B, L · Bs
Note that the spreading signal c(t) = ± 1 at any given instant. Given the polar nature of the
binary chip signal, the receiver, under an A WGN channel, can easily "despread" the received
signal
y(t) = sos(t) + n(t) = SQAM (t)c(t) + n(t)
(11.14)
by multiplying the chip signal with the received signal
r(t) = c(t)y(t)
= SQAM(t)c 2 (t) + n(t)c(t)
= SQAM (t) + n(t)c(t)
'-v-'
x(t)
(11.15)
(11.16)
Thus, this multiplication allows the receiver to successfully "despread" the spread spectrum
signal. The analysis of the DSSS receiver depends on the characteristics of the noise x(t).
Because c(t) is deterministic, and n(t) is Gaussian with zero mean, x(t) remains Gaussian
with zero mean. As a result, the receiver performance analysis requires finding only the PSD
of x(t).