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I was trying to solve this question : enter image description here

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 ?

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

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  • $\begingroup$ 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 $x_1(t)$ $\endgroup$ – Starboy Feb 28 at 17:56
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    $\begingroup$ @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. $\endgroup$ – Matt L. Feb 28 at 19:03

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