# Find Output of LTI System Given Input and Impulse Response

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1. Jan 30, 2016

### Captain1024

1. The problem statement, all variables and given/known data
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.

2. Relevant equations

3. The attempt at a solution
Input:
$\mathrm{x}[0]=3\mathrm{cos}(\frac{\pi}{2})-3\mathrm{cos}(0)=-3$
$\mathrm{x}[1]=3\mathrm{cos}(\frac{-\pi}{6})-3\mathrm{cos}(\frac{7\pi}{8})=-0.1736$
$\mathrm{x}[2]=3\mathrm{cos}(\frac{\pi}{6})-3\mathrm{cos}(\frac{7\pi}{4})=0.4768$

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

2. Feb 1, 2016

### Captain1024

I figured it out.