Solve the following problem:
The frequency-response characteristic of a low pass channel can be approximated by
H(ƒ)= {1 + α cos 2πƒt0 |a|< 1, |ƒ|≤ W
{0 otherwise
where W is the channel bandwidth. An input signal s(t) whose spectrum is band-limited to W Hz is passed through the channel.
a. Show that
y(t) = s(t) + 1/2 α[s(t-t0) + s(t + t0)]
Thus, the channel produces a pair of echoes.
b. Suppose that the received signal y(t) is passed through a filter matched to s(t). Determine the output of the matched filter at t = kT , k = 0, ±1, ±2, ... , where T is the symbol duration.
c. What is the ISI pattern resulting from the channel if t0 = T ?