Revision as of 20:02, 23 September 2008 by Jpfister (Talk)

DT LTI System

$ y[n] = \sum_{n=-\infty}^{\infty}\frac{1}{2}x[n] \; \; $     (scaled DT integral)

h[n]

$ h[n] = \sum_{n=-\infty}^{\infty}\frac{1}{2}\delta [n] = \frac{1}{2}u[n] $

H(z)

$ H(z) = \sum_{m=-\infty}^{\infty}h[m] e^{-j \omega m} = \sum_{m=-\infty}^{\infty} \frac{1}{2}u[m] e^{-j \omega m} = \sum_{m=0}^{\infty} \frac{1}{2}e^{-j \omega m} = \sum_{m=0}^{\infty} (\frac{1}{2 e^{j \omega}})^m = \frac{1}{1-\frac{1}{2 e^{j \omega}}} $     (geometric series $ r^n $ where $ |r| < 1 $)

Response to x[n]

Input $ x[n] $ is the following signal:

SawDTJP ECE301Fall2008mboutin.jpg

The Fourier series coefficients for $ x[n] $ are:

$ a_{0} = 1 $

$ a_{1} = -\frac{1}{2} $

$ a_{2} = 0 $

$ a_{3} = -\frac{1}{2} $

$ y[n] = \sum_{k=0}^{3} a_{k} F(e^{jk \omega_{0}} $

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