A signal x[n] is processed through an LTI system H(z) and then down sampled by a factor of 2 to yield y[n] as indicated in Figure. Also, as shown in the same figure, x[n] is first down sampled and then processed through an LTI system G(z) to obtain r[n].

(a) Specify a choice for H(z) (other than a constant) and G(z) so that r[n] = y[n] for an arbitrary x[n].

(b) Specify a choice for H(z) so that there is no choice for G(z) that will result in r[n] = y[n] for an arbitrary x[n].

(c) Determine as general a set of conditions as you can on H(z) such that G(z) can be chosen so that r[n] = y[n] for an arbitrary x[n]. The conditions should not depend on x[n]. If you first develop the conditions in terms of h[n], restate them in terms of H(z).

(d) For the conditions determine in part (c), what is g[n] in terms of h[n] so that r[n] = y[n].

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H(2) 12 y[n] w[2n] win] x[n] G(z) +2 x [n] s[n] = w[2n] %3! r{n]