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==DT LTI System== | ==DT LTI System== | ||
| − | <math>y[n] = \sum_{n=-\infty}^{\infty}x[n] \; \;</math> (DT integral) | + | <math>y[n] = \sum_{n=-\infty}^{\infty}\frac{1}{2}x[n] \; \;</math> (DT integral) |
==h[n]== | ==h[n]== | ||
| − | <math>h[n] = \sum_{n=-\infty}^{\infty}\delta [n] = u[n]</math> | + | <math>h[n] = \sum_{n=-\infty}^{\infty}\frac{1}{2}\delta [n] = \frac{1}{2}u[n]</math> |
==H(z)== | ==H(z)== | ||
| − | <math>H(z) = \sum_{m=-\infty}^{\infty}h[m] e^{-j \omega m} = \sum_{m=-\infty}^{\infty} u[m] e^{-j \omega m} = \sum_{m=0}^{\infty} e^{-j \omega m} = \sum_{m=0}^{\infty} (\frac{1}{e^{j \omega}})^m</math> | + | <math>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</math> |
Revision as of 19:50, 23 September 2008
DT LTI System
$ y[n] = \sum_{n=-\infty}^{\infty}\frac{1}{2}x[n] \; \; $ (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 $
