A pure tone x(t) = 4 cos(1000t) is transmitted using an amplitude mod-ulation communication system with a carrier cos(10000t). The output of the AM system is

y(t) = x(t) cos(10000t)

At the receiver, the sent signal y(t) needs first to be separated from the thousands of other signals that are received. This is done with a band-pass filter with a center frequency equal to the carrier frequency. The output of this filter then needs to be demodulated.

(a) Consider an ideal band-pass filter H(jΩ). Let its phase be zero, determine its bandwidth, center frequency, and amplitude so we get as its output 10y(t). Plot the spectrum of x(t), 10y(t), and the magnitude frequency response of H(jΩ).

(b) To demodulate 10y(t), we multiply it by cos(10000t). You need then to pass the resulting signal through an ideal low-pass filter to recover the original signal x(t). Plot the spectrum of z(t) = 10y(t) cos(10000t), and from it determine the frequency response of the low-pass filter G(jΩ) needed to recover x(t). Plot the magnitude response of G(jΩ).