# How to deduce a linear system's impulse response from a set of input/output signals?

I want to know how to solve those types of problems.. is it by inspection ?

Consider the linear system below. When the inputs to the system $x_1[n]$, $x_2[n]$ and $x_3[n]$, the responses of the systems are $y_1[n]$, $y_2[n]$ and $y_3[n]$ as shown.

a. Determine whether the system is time invariant or not. Just your answer.

b. What is the impulse response?

Edit: Assuming a general case where the given inputs don't contain a scaled impulse like $x_2[n]$

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Hint: Use $x_2[n]$ and $y_2[n]$ to determine what the impulse response of $T$ must be (since $x_2[n]$ is just a scaled impulse). That gives you the answer to part (b). Then, check the other two cases to see if the inputs/outputs are consistent with that impulse response (using the superposition property of a linear system) to get an answer for part (a). –  Jason R Jan 3 '13 at 14:42
That's a more difficult problem in the general case. If they are all short like this, you know an upper bound on the duration of the impulse response, and you have enough input/output pairs, then you could set up a system of linear equations that you could solve to arrive at the unknown impulse response values. –  Jason R Jan 3 '13 at 14:50
In the general case it's also quite possible that there is no FIR solution or no solution at all. Hint: check the DC values of x1[n] and y1[n]. –  Hilmar Jan 3 '13 at 15:52
Hint: What does the signal $x_2[n]-x_2[n-2]$ look like? For an LTI system, the response should be $y_2[n]-y_2[n-2]$, no? Is it? Also, note that for a discrete-time linear time-variant system, there is not one unit-pulse response but an infinitude of unit-pulse responses, one for each time instant when the unit pulse occurs. –  Dilip Sarwate Jan 3 '13 at 16:29
@DilipSarwate: I agree that this is a dreadful homework problem. However, the system does look causal. While $y_3[n]$ is nonzero for $n=-2$, so is $x_3[n]$, so the system output isn't leading the input in time. –  Jason R Jan 5 '13 at 18:15

There seems to be an image which is gone now and hence I might be missing something.

1. In order to declare if the system is invariant you should see if a delay of the input yields only a delay in the output.
In you case, does the input x1[n-m] yields y1[n-m] and etc...

2. If the input signals are band limited and their bandwidth is less than your system you won't be able to restore the impulse response.
You will be able only to get the response in the frequencies the input has energy in.
This could be done by frequency analysis of the input and the output.
If your system is indeed LTI the connection between input and output is given by convolution with the impulse response.
Convolution is multiplication in the frequency domain, hence you could easily get the impulse response (Again, only at frequencies the input has energy in).

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The image is back now. It seems you have a very specific question. Hence my answer which was much more general isn't focused enough. –  Drazick Apr 14 '14 at 22:13