Page 74 - Fundamentals of Communications Systems

2.26 Chapter Two

Problem 2.2. Plot, find the period, and find the Fourier series representation of
the following periodic signals

(a) x(t) = 2 cos(200πt) + 5 sin(400πt)
(b) x(t) = 2 cos(200πt) + 5 sin(300πt)
(c) x(t) = 2 cos(150πt) + 5 sin(250πt)
Problem 2.3. Consider the two signals

x 1 (t) = m(t) cos(2π f c t) x 2 (t) = m(t) sin(2π f c t)

where the bandwidth of m(t) is much less than f c . Compute the simplest form
for the following four signals
(a) y 1 (t) = x 1 (t) cos(2π f c t)
(b) y 2 (t) = x 1 (t) sin(2π f c t)
(c) y 3 (t) = x 2 (t) cos(2π f c t)
(d) y 4 (t) = x 2 (t) sin(2π f c t)

Postulate how a communications engineer might use these results to recover
a signal, m(t), from x 1 (t)or x 2 (t).
Problem 2.4. (Design Problem) This problem gives you a little feel for microwave
signal processing and the importance of the Fourier series. You have at your
disposal

(1) a signal generator that produces ± 1 V amplitude square wave in a 1 sys-
tem where the fundamental frequency, f 1 , is tunable from 1 kHz to 50 MHz
(2) an ideal bandpass filter with a center frequency of 175 MHz and a bandwidth
of 30 MHz (±15 MHz).
The design problem is

(a) Select an f 1 such that when the signal generator is cascaded with the filter
that the output will be a single tone at 180 MHz. There might be more than
one correct answer (that often happens in real life engineering).
(b) Calculate the amplitude of the resulting sinusoid.

Problem 2.5. This problems exercises the signal and system tools. Compute the
Fourier transform of

(a)

A 0 ≤ t ≤ T p
x(t) = (2.86)
0 elsewhere