# Convolution - Fourier Transform

## Homework Statement

An LTI system has an impulse response h(t) = e-|t|
and input of x(t) = ejΩt

## Homework Equations

Find y(t) the system output using convolution
Find the dominant frequency and maximum value of y(t)

## The Attempt at a Solution

I have tried using the Fourier transform to get y(t) but when you try to find X(Ω), I get infinity
as X(Ω) = ∫ x(t) * e-jΩtdt = ∫ e jΩt*e-jΩtdt = ∫ 1 dt = t between inf and -inf
h(t) I could find as 2/(1+Ω2)

Any ideas on how to solve this?

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You need to write the transform as ## X(\omega)=\int x(t) e^{-i \omega t} dt ##. The result is ## X(\omega)=2 \pi \, \delta(\omega-\Omega) ##. You need to read about delta functions. If you google it, you should find some useful formulas like the one I just gave you.

You need to write the transform as ## X(\omega)=\int x(t) e^{-i \omega t} dt ##. The result is ## X(\omega)=2 \pi \, \delta(\omega-\Omega) ##. You need to read about delta functions. If you google it, you should find some useful formulas like the one I just gave you.
So then using that you would get that Y(w) = 4π/(1+w2) * δ(w-2), but then how would you get that back into the time domain?

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## H(\omega)=\int\limits_{0}^{+\infty} h(t) e^{-i \omega t} dt ##, since ## h(t)=0 ## for ## t<0 ##. (Perhaps they didn't tell you this (## h(t)=0 ## for ## t<0 ##) in the problem statement, but it is clear that that's what they want). Try recomputing ## H(\omega) ##.(You have it incorrect). ## \\ ## Now ## Y(\omega)=H(\omega)X(\omega) ##. (That part you have correct.) ## \\ ## Use an inverse transform to get ## y(t)=\frac{1}{2 \pi} \int\limits_{-\infty}^{+\infty} Y(\omega) e^{i \omega t} \, d \omega ##. ## \\ ## Wait until you process everything to put in the value for ## \Omega ##.

## H(\omega)=\int\limits_{0}^{+\infty} h(t) e^{-i \omega t} dt ##, since ## h(t)=0 ## for ## t<0 ##. Try recomputing ## H(\omega) ##.(You have it incorrect). ## \\ ## Now ## Y(\omega)=H(\omega)X(\omega) ##. (That part you have correct.) ## \\ ## Use an inverse transform to get ## y(t)=\frac{1}{2 \pi} \int\limits_{-\infty}^{+\infty} Y(\omega) e^{i \omega t} \, d \omega ##. ## \\ ## Wait until you process everything to put in the value for ## \Omega ##.
Sorry the reason why I got what i did for H(w), was because I forget to put in an absolute around the t. I've changed in now

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Sorry the reason why I got what i did for H(w), was because I forget to put in an absolute around the t. I've changed in now
An ## h(t) ## with an absolute value would be unphysical. That is saying it responds before the impulse. These functions always begin at ## t=0 ##. If they gave you the problem in such a fashion, it is unphysical.