Contents
A DT-LTI System
$ y[n] = x[n] + x[n-3]\, $
Unit Impulse Response
$ y[n] = x[n] + x[n-3]\, $
$ x[n] = \delta[n]\, $
$ h[n] = \delta[n] + \delta[n-3]\, $
Frequency Response
$ y[n] = \sum^{\infty}_{\infty} h[n] * x[n] dn\, $ where $ x[n] = e^{jwn} \, $
$ y(t) = \sum^{\infty}_{-\infty} (\delta[n] + \delta[n-3]) * e^{jwn} dn\, $
$ y(t) = \sum^{\infty}_{-\infty} (\delta[m] + \delta[m-3]) e^{jw(n-m} dm\, $
$ y(t) = e^{jwn} \sum^{\infty}_{-\infty} (\delta[m] + \delta[m-3]) e^{-jwm} dm\, $
$ H(s) = \sum^{\infty}_{-\infty} (\delta[m] + \delta[m-3]) e^{-jwm} dm\, $
$ H(s) = e^{-jw0} + e^{-jw3}\, $
$ H(s) = 1 + e^{-jw3}\, $
Response of the DT system defined in Q1
CT Periodic Signal : $ x[n] = 5\cos(6\pi n + \pi) + 7\cos(3\pi n)\, $
$ x(t) = \sum^{\infty}_{k = -\infty} a_k e^{jk\pi n}\, $ where $ a_k = -5 , k = 2,4,6,8,10....\, $ and $ a_k = 7 , k = 1,3,5,7,9,11....\, $
$ y(t) = \sum^{\infty}_{k = -\infty} a_k H(s) e^{jk\pi n}\, $
$ y(t) = \sum^{\infty}_{k = -\infty} a_k (1 + e^{-jw3}) e^{jk\pi n}\, $
