# 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 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 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 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 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 at 18:15