# Signals - Fourier Analysis

## Homework Statement

The signal y(t) is generated by convolving a band limited signal x1(t) with another band limited signal x2(t) that is y(t)=x1(t)*x2(t) where:

--> X1(jω)=0 for|ω| > 1000Π
--> X2(jω)=0 for|ω| >2000Π

Impulse train sampling is performed on y(t) to obtain:
--> yp(t)= [summation from n = (−∞,∞)] y(nT)δ(t− nT)

Specify the range of values for sampling period T which ensures that y(t) is recoverable from yp(t).

## Homework Equations

All of the equations that I would are most likely showing.

## The Attempt at a Solution

My thoughts were to plug in (nT) for every t in both x1(t) and x2(t) and then take the Fourier transform of that, cut of the edges where the transforms are equal to zero and then that is where I go blank...

I imagine that is the right implementation to start the problem with, but please correct me if I am wrong. Thank you in advance to all who may be able to help - it is much appreciated!!!

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vela
Staff Emeritus
Homework Helper
Hint: convolution theorem.

I can see where convolution comes into play, but how can I implement the CTF transforms that are given?

vela
Staff Emeritus
Homework Helper
What sets the lower limit on the sampling rate if you want to be able to recover the original signal?

The Nyquist rate, to sample at the perfect rate (without aliasing/oversampling to occur) - it would be = 2fmax

I am pretty confused at what you're trying to hint at here. I appreciate the help, but my mind is still blank.

vela
Staff Emeritus