Number 5

An LTI system has unit impulse response $ h[n] = u[n] - u[n - 2]\, $.

a)Compute the system's function H(z).

b)Use the answer from a) to compute the system's response to the input $ x[n] = cos(\pi n)\, $


Answer

a)

$ H(z) = \sum_{k=-\infty}^{\infty}h(n)z^{-n}\, $

After graphing out the two spikes:

$ = 1 * z^{-0} + 1 * z^{-1}\, $

$ = 1 + \frac{1}{z}\, $


b)

$ x[n] = cos(\pi n) = (-1)^{n}\, $

Given this, z = -1, so:

System response $ = H(z)z^{n} = (1 + \frac{1}{-1}) = 0\, $

Alumni Liaison

Basic linear algebra uncovers and clarifies very important geometry and algebra.

Dr. Paul Garrett