$1\mathrm{.}33$ Spread spectrum communication systems reduce the effects of narrow bandwidth

interference by spreading the bandwidth of the transmitted signal out over a range

of frequencies much greater than the bandwidth occupied by an AMDSB$-$SC signal

alone. Consider the AMDSB$-$SC signal in Eq$.(1\mathrm{.}2\mathrm{.}1),$ where $m(t)$ is the message

with $|M()|$ as in Fig. $1\mathrm{.}1$a$.$

Let $p(t)$ be a sequence of $+-1$ pulses with $F\{p(t)\}=P()$ as in Fig. $1\mathrm{.}31,$ where

${}_p{}_c{}_m.$ The spreading of the bandwidth is accomplished by multiplying

by $p(t)$ to obtain $s_{DSB}(t)p(t).$

$($a$)$ Find the band of frequencies occupied by $s_{DSB}(t)p(t).$

$($b$)$ If there is no distortion, the transmitted signal $s_{DSB}(t)p(t)$ appears at the

receiver input. Show that multiplying by $p(t)$ again produces $S_{DSB}(t),$ from

which we can obtain the message $m(t).$

$($c$)$ Assume that there is additive distortion $n(t)$ with $F\{n(t)\}=N()$ such that

Show that if the received waveform is $r(t)=s_{DSB}(t)p(t)+n(t),$ then multiply$-$

ing by $p(t)$ at the receiver reduces the noise power in the bandwidth of interest

by a factor proportional to $\frac{1}{{}_p}.$

$s_{DSB}(t)=A_{c}m(t)cos{}_{c}t.$

Fig. $1\mathrm{.}1$ Frequency content of the message and double sideband signals: a Message signal; $b$ double

sideband signal

Fig. $1\mathrm{.}31$ Pulse Fourier

Transform for Problem $1\mathrm{.}33$