Revision as of 18:24, 25 September 2008 by Jkubasci (Talk)

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}\, $


Part B

Compute the system's response to (from problem 2 HW4.2 Jeff Kubascik_ECE301Fall2008mboutin)

$ \,x[n]=\sum_{k=-\infty}^{\infty}\delta[n-4k] + \pi\delta[n-1-4k] - 3\delta[n-2-4k] + \sqrt[e]{\frac{\pi^3}{\ln(5)}}\delta[n-3-4k]\, $


Using the Fourier coefficients calculated in problem 2, we can express the system's response to $ \,x[n]\, $ as

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