A signal x[n] is processed by a linear time-invariant system H(z) and then downsampled by a factor of 2 to yield y[n], as shown in Figure. The pole-zero plot for H(z) is shown in Figure.

(a) Determine and sketch h[n], the impulse response of the system (z).

(b) A second system is shown in Figure, in which the signal x[n] is first time compressed by a factor of 2 and then passed through an LTI system G(z) to obtain r[n].

Determine whether G(z) can be chosen so that y[n] = r[n] for any input x[n]. If your answer is no, clearly explain. If your answer is yes, specify G(z). If your answer depends on the value of M, clearly explain how. (M is constrained to be an integer greater than or equal to 2.)

24 y[n] = w[2n] H(z) w [n] x[n] Im Unit Mth-order pole at z = 0, M2 2 and M is an integer circle z-plane Re 2 24 G(2) x[n] r[n]