In many practical situations a signal is recorded in the presence of an echo, which we w...

In many practical situations a signal is recorded in the presence of an echo, which we would like to remove by appropriate processing. For example, in Figure P7.41 (a), we illustrate a system in which a receiver simultaneously receives a signal x(t) and an echo represented by an attenuated delayed replication of x(t). Thus, the receiver output is

Assume that x(t) is band limited [i.e., X(jω) = 0 for and that | α | < 1.

(a) If , and the sampling period is taken to be equal to determine the difference equation for the digital filter hire] such that yc(t) is proportional to x(t).

(b) With the assumptions of part (a), specify the gain A of the ideal lowpass filter such that

(c) Now suppose that . Determine a choice for the sampling period T, the lowpass filter gain A, and the frequency response for the digital filter h[n] such that is proportional to x(t).