# Signals: Fourier series and frequency response

## Homework Statement

A recursive DT system with input x[n] and output y[n] is given by

y[n] = -0.25y[n-2] + x[n]

a) Determine and plot the impulse response h[n] such that y[n] = x[n]*h[n]

b) how would you classify the system and why?

c) What modification, if any, should be made so that the system has unit DC gain?

d) Another recursive DT LTI system has an impulse response given by

g[n] = δ[n-1] - (1/4)δ[n-2] + (1/16)δ[n-3] - (1/64)δ[n-4] + (1/128)δ[n-5]....

By comparing with part (a), or otherwise, what is a recursive system which has g[n] as an impulse response?

## Homework Equations

1) Assume that if x[n] is 0 while n < 0, y[n] is also 0 while n < 0

2) H(z) = $\sum$h[k]z-k from k = -∞ to ∞

## The Attempt at a Solution

I've got part a) down: substituting x[n] = δ[n] and starting from n = 0, we see that h = 1, h = -1/4, h = 1/16, etc, with h[n] = 0 for odd values of n. We can represent this mathematically as:

h[n] = (-0.25)0.5n

Parts b) and c) are the ones that are confusing me right now, because despite consulting the textbook many times I am conceptually having enormous difficulty understanding the conversion between an impulse response and the corresponding frequency response, particularly for a DT system like what's being shown here. For b) and c), I expect I'd need an equation that would express the frequency response directly in terms of frequency, rather than evaluate a given number of frequencies and analyse them.

I think the difficulty comes from the complex number z. Even if I was to evaluate the expressions manually, what would I change z to each time? The same question holds true for evaluating it for any given z. I think fundamentally I'm not appreciating the role that the complex number plays in the analysis.