In long-distance telephone communication, an echo is sometimes encountered due to the tr...

In long-distance telephone communication, an echo is sometimes encountered due to the transmitted signal being reflected at the receiver, sent back down the line, re-flected again at the transmitter, and returned to the receiver. The impulse response for a system that models this effect is shown in Figure P9.60, where we have as-sumed that only one echo is received. The parameter T corresponds to the one-way travel time along the communication channel, and the parameter a represents the attenuation in amplitude between transmitter and receiver.

(a) Determine the system function H(s) and associated region of convergence for the system.

(b) From your result in part (a), you should observe that H(s) does not consist of a ratio of polynomials. Nevertheless, it is useful to represent it in terms of poles and zeros, where, as usual, the zeros are the values of s for which H(s) = 0 and the poles are the values of s for which For the system function found in part (a), determine the zeros and demonstrate that there are no poles.

(c) From your result in part (b), sketch the pole-zero plot for H(s).

(d) By considering the appropriate vectors in the s-plane, sketch the magnitude of the frequency response of the system.