If the input and output of a causal LTI system satisfy the difference equation
y[n] = ay[n − 1] + x[n],
then the impulse response of the system must be h[n] = anu[n].
(a) For what values of a is this system stable?
(b) Consider a causal LTI system for which the input and output are related by the difference equation
y[n] = ay[n − 1] + x[n] − aNx[n − N],
where N is a positive integer. Determine and sketch the impulse response of this system. Hint: Use linearity and time-invariance to simplify the solution.
(c) Is the system in part (b) an FIR or an IIR system? Explain.
(d) For what values of a is the system in part (b) stable? Explain.
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