Exam 1 Problem 5

I found this problem to be the most difficult for me because when I saw that H(z), my mind completely died like a hard drive read/write head crashing into a metal platter spinning at 10000RPM. A horrible screeching noise ensued, causing data loss in epic proportions.

The question as stated: An LTI system has unit impulse response $ h[n]=u[n]-u[n-2] $. Compute (a) the system's function $ H(z) $ and (b) the system's response to the input $ x[n]=\cos(\pi n) $.

Solution to Problem 5

Apparently, the solution wasn't as dreadful as I made it out to be but at that moment, my mind had crashed...therefore, here goes attempt number two...

Part A

$ H(z)=\sum_{n=-\infty}^{\infty}h[n]z^{-n} $ $ =1 z^0 + 1 z^{-1} \ $ $ =1+ \frac{1}{z} \ $

Part B

$ cos(\frac{\pi}{n}) = (-1)^n \ $

$ (-1)^n -> H(-1)(-1)^n = 0 \ $