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

Chapter 7, Problem 7.41

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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 s(t) = x(t) + ax(t- T0 ), where Ia I < 1. This output is to be processed to recover x(t) by first converting to a sequence and then using an appropriate digital filter h[n], as indicated in Figure P7.4l(b). Assume that x(t) is band limited [i.e., X(jw) = 0 for lwl > WM] and that lal < 1. (a) IfT0 < 7T/wM,andthesamplingperiodistakentobeequaltoT0 (i.e.,T =To), determine the difference equation for the digital filter h[n] such that Yc(t) is proportional to x(t). (b) With the assumptions of part (a), specify the gain A of the ideallowpass filter such that Yc(t) = x(t). (c) Now suppose that 7TIWM

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