ENGN 1570 Fall 2014 Homework 8

ENGN 1570 Fall 2014 Homework 8
Problem 1
In this problem we will consider what happens when we demodulate an AM signal using
a sinusoidal signal with frequency that is similar but doesn’t exactly match the carrier
frequency. Let y(t) be the AM modulation of x(t) with carrier frequency wc ,
y(t) = x(t) cos(wc t).
Suppose we demodulate y(t) by multiplying it with a sinusoidal of frequency wd followed by
lowpass filtering,
w(t) = y(t) cos(wd t),
z(t) = w(t) ∗ h(t).
Here h(t) is an ideal lowpass filter with cutoff frequency W . Denote the difference between
the frequencies in the modulator and demodulator by ∆w = wd − wc . Suppose x(t) is band
limited with X(w) = 0 for |w| ≥ wM and assume the cutoff frequency W is such that
wM + ∆w < W < 2wc + ∆w − wM
(a) Show that the demodulated signal z(t) is proportional to x(t) cos(∆wt).
(b) Suppose x(t) is an audio signal. What does the demodulated signal sound like?
Problem 2
AM modulation and demodulation requires multiplication of signals. Multiplication can be
hard to implement in practice and some systems use alternative nonlinear operations instead
of multiplications.
One example for AM modulation involves transforming the input signal x(t) by squaring
the sum of x(t) and the carrier signal, and then bandpass filtering the result to obtain the
amplitude modulated signal z(t). That is, we take y(t) = (x(t) + cos(wc t))2 and then use a
bandpass filter on y(t) to obtain z(t).
Assume that x(t) is band limited with X(w) = 0 for |w| ≥ wM . Determine what the
Fourier transform of the bandpass filter should look like so that z(t) = x(t) cos(wc t). Specify
any necessary constraints (if any) on wc and wM .