# Finding the impulse response given response to another signal

I was trying to solve this question :

I respresented $$x(t) = u(t+1)-u(t-1)$$

writing the convolution as $$[u(t+1)-u(t-1)]*h(t) = y(t)$$

I then used the property of differentiation to convert from the step to the impulse function :

$$[\delta(t+1) -\delta(t-1)]*h(t) = y'(t)$$

$$\implies h(t+1) - h(t-1) = y'(t)$$

However , I am now stuck here. Moreover , the fact that the width of $$y(t)$$ is the same as that of $$x(t)$$ , bugs me a little.

How can I find the impulse response $$h(t)$$ using the given info ?

• Could you also show the inputs $x_2(t)$ and $x_3(t)$? Feb 28, 2021 at 17:29
• @MattL. made the necessary edits Feb 28, 2021 at 17:31
• Are you familiar with Fourier transforms? Mar 1, 2021 at 6:46
• @cjferes: I'm afraid that this won't help much in this case. Mar 1, 2021 at 10:11

This exercise is meant to help the student appreciate the fact that if the response $$y_1(t)$$ of an LTI system to an input $$x_1(t)$$ is known, then the response to an input

$$x_2(t)=\sum_{k=1}^Ka_kx_1(t-t_k)\tag{1}$$

is given by

$$y_2(t)=\sum_{k=1}^Ka_ky_1(t-t_k)\tag{2}$$

which is a direct consequence of linearity and time-invariance.

Consequently, if an input signal can be expressed in the form given by $$(1)$$, it is not necessary to compute the system's impulse response in order to determine the output signal.

Also note that this exercise is almost a copy of problem P3.8 in this MIT Signals and Systems problem set. Only the input signals have been changed.

It now appears that the given signal $$x_2(t)$$ can't be expressed in the form $$(1)$$, whereas signal $$x_3(t)$$ can. I believe that this a mistake in the exercise, because as far as I can see, any other method to determine the solution is beyond what can be expected from a student at this level.

• I am just not able to represent $x_2(t)$ as an linear combination through just time shifting and amplitude scaling . Seems as if time scaling would be required but that would change the response to $x_1(t)$ Feb 28, 2021 at 17:56
• @Starboy: If that is the case then it's a mistake in the exercise. Check problem P3.8 of these MIT signals and systems problems. It's too close to be a coincidence. Feb 28, 2021 at 19:03