# Find the Output of an LTI System Given Input and Impulse Response

## Homework Statement

Consider a LTI system for which the filter coefficients are ##{\mathrm{h}_k}=\{1,2,1\}##. Find the output when the input is ##\mathrm{x}[n]=3\mathrm{cos}(\frac{\pi}{3}n-\frac{\pi}{2})-3\mathrm{cos}(\frac{7\pi}{8}n)##. Identify two frequencies in this composite signal. Show the frequency response with respect to these two frequencies.

## The Attempt at a Solution

Input:
##\mathrm{x}=3\mathrm{cos}(\frac{\pi}{2})-3\mathrm{cos}(0)=-3##
##\mathrm{x}=3\mathrm{cos}(\frac{-\pi}{6})-3\mathrm{cos}(\frac{7\pi}{8})=-0.1736##
##\mathrm{x}=3\mathrm{cos}(\frac{\pi}{6})-3\mathrm{cos}(\frac{7\pi}{4})=0.4768##

Is the output then ##\mathrm{y}[n]=\mathrm{h}_k*\mathrm{x}[n]=\{-3, -0.3472, 0.4768\}##?

Are two frequencies in this composite signal ##\frac{\pi}{3}## & ##\frac{\pi}{8}##?

I'm not sure what the last part is asking for. Are frequency response and output the same thing?

Also, is the given ##\mathrm{h}_k## called the impulse response? I'm trying to get my vocabulary down.

-Captain1024

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