(→h[n] and H(z)) |
(→h[n] and H(z)) |
||
| Line 4: | Line 4: | ||
=== h[n] and H(z) === | === h[n] and H(z) === | ||
| − | + | <br> | |
We obtain <math> h[n] </math> by finding the response of <math> x[n] </math> to the unit impulse response (<math> \delta[n] </math>). | We obtain <math> h[n] </math> by finding the response of <math> x[n] </math> to the unit impulse response (<math> \delta[n] </math>). | ||
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<math> \,\ H[z] = \sum_{m=-\infty}^\infty h[m] * Z</math><sup>(<math>-m</math>)</sup><br><br> | <math> \,\ H[z] = \sum_{m=-\infty}^\infty h[m] * Z</math><sup>(<math>-m</math>)</sup><br><br> | ||
| − | <math> \,\ H[z] = \sum_{m=- | + | <math> \,\ H[z] = \sum_{m=-\infty}^{\infty} (5*\delta[n-5] + 6*\delta[n+6]) * Z</math><sup>(<math>-m</math>)</sup> |
| + | |||
| + | By the sifting property, this sum equals:<br> | ||
| + | <math> \,\ H[z] = 5*Z</math><sup>-5</sup><math> \,\ + 6*Z</math><sup>6</sup> | ||
Revision as of 17:57, 25 September 2008
Define a DT LTI System
$ \,\ x[n] = 5*u[n-5] + 6*u[n+6] $
h[n] and H(z)
We obtain $ h[n] $ by finding the response of $ x[n] $ to the unit impulse response ($ \delta[n] $).
$ \,\ h[n] = 5*\delta[n-5] + 6*\delta[n+6] $
$ \,\ H[z] = \sum_{m=-\infty}^\infty h[m] * Z $($ -m $)
$ \,\ H[z] = \sum_{m=-\infty}^{\infty} (5*\delta[n-5] + 6*\delta[n+6]) * Z $($ -m $)
By the sifting property, this sum equals:
$ \,\ H[z] = 5*Z $-5$ \,\ + 6*Z $6
