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Given the following LTI DT system | Given the following LTI DT system | ||
| − | <math>\,s[ | + | <math>\,s[n]=x[n]+x[n-1]\,</math> |
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<math>\,H(z)=\sum_{m=-\infty}^{\infty}h[m]z^{-m}\,</math> | <math>\,H(z)=\sum_{m=-\infty}^{\infty}h[m]z^{-m}\,</math> | ||
| + | |||
| + | <math>\,H(z)=\sum_{m=-\infty}^{\infty}(\delta[m]+\delta[m-1])z^{-m}\,</math> | ||
| + | |||
| + | using the sifting property | ||
| + | |||
| + | <math>\,H(z)=z^{0}+z^{-1}\,</math> | ||
| + | |||
| + | <math>\,H(z)=1+z^{-1}\,</math> | ||
== Part B == | == Part B == | ||
Revision as of 18:20, 25 September 2008
Given the following LTI DT system
$ \,s[n]=x[n]+x[n-1]\, $
Part A
Find the system's unit impulse response $ \,h[n]\, $ and system function $ \,H(z)\, $.
The unit impulse response is simply (plug a $ \,\delta[n]\, $ into the system)
$ \,h[n]=\delta[n]+\delta[n-1]\, $
The system function can be found using the following formula (for LTI systems)
$ \,H(z)=\sum_{m=-\infty}^{\infty}h[m]z^{-m}\, $
$ \,H(z)=\sum_{m=-\infty}^{\infty}(\delta[m]+\delta[m-1])z^{-m}\, $
using the sifting property
$ \,H(z)=z^{0}+z^{-1}\, $
$ \,H(z)=1+z^{-1}\, $
