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I would like to find the impulse response, h[n], of an LTI system given the input

x[n]=[1,−3,2]

and the output

y[n]=[1,−1,−4,4]    

I know that y[t]=x[t]∗h[t], but I am having hard time to figure out the right way to calculate the impulse response. I know very little about signal processing, so if you don't mind giving an easy explanation, then I appreciate it. Or, if it's possible to do an example, that's better.

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If $x[n]$ is non-zero for $n=0\ldots2$, and $y[n]$ is non-zero for $n=0\ldots3$, then $h[n]$ must have non-zero components for $n=0$ and $n=1$. This is a property of the convolution.

So now you have to solve $y[n] = h[0]x[n] + h[1]x[n-1]$ for the relevant values of $n$, and there you go.

Start with the edges ($n=0$ and $n=3$).

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