Question

An LTI system is governed by

$$ y[n]=\sum_{k=0}^{4}(-1)^{k} x[n-k] $$

(a) Find the system's transfer function \(H(z)\) and express it as the ratio of degree- 4 polynomials in \(z\).

(b) Use the partial geometric series to re-write the answer to (a) as the ratio of degree-5 polynomials in \(z\).

(c) From your answer to (b) the values of \(z\) for which \(H(z)=0\). Express each of them in the form \(R e^{j \theta}\). Caution: \(z=-1\) isn't one of them. (Suggestion: \(\left(-e^{j k 2 \pi / 5}\right)^{5}=-1\) for all \(\left.k .\right)\)

Answer 1

a)

Apply z-transform on both sides

b)

Using Geometric series summation with no. of terms N = 5, common ratio r = -1/z

, z ≠ -1

c)

H(z) = 0

for z^5 = -1 = e^(±j(2k+1)?) {Euler's form}, k = 0,1

z = e^(±j(2k+1)?/5)

z1 = e^(j?/5)

z2 = e^(-j?/5)

z3 = e^(j3?/5)

z4 = e^(-j3?/5)