A sequence x[n] is the output of a linear time-invariant system whose input is s[n]. This system is described by the difference equation?

x[n] = s[n] ? e^{? 8?} s[n ? 8],?

where 0

(a) Find the system function, and Plot is poles and zeros in the z-plane. Indicate the region of convergence.

(b) We wish to recover s[n] from x[n] with a linear time-invariant system. Find the system function, such that y[n] = s[n]. Find all possible regions of convergence for H₂(z), and for each, tell whether or not the system is causal and/or stable.

(c) Find all possible choices for the impulse response h₂[n] such that

y[n] = h₂[n] * x[n] = s[n].

(d) For all choices determined in part (c), demonstrate, by explicitly evaluating the convolution in, that when s[n] = ?[n], y[n] = ?[n].?

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Part A X(z) H(2) = S(2) Part B Y(2) H2(z) = X(2) %3!