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Fourier transform problems

1. A continuous time signal x(t) has the Fourier transform X(ω) = 1/(jω+b) where b is a constant. Determine the Fourier transform for v(t) = t^2 x(t).

2. A continuous time signal x(t) has the Fourier transform X(ω) = 1/(jω+b) where b is a constant. Determine the Fourier transform for v(t) = x(t) * x(t).

3. Compute the Fourier transform for x(t) = te^-t u(t)

4. Compute the inverse Fourier transform for X(ω) = cos 4ω.

5. A signal with the highest frequency component at 10 kHz is to be sampled. To reconstruct the signal, the sampling must be done at a minimum frequency of?

Solution Preview

Please see the attached file for detailed solutions.

1. Find the Fourier Transform FT of if the FT of the function is given by

We have the FT

We need to find the time domain function representation of this which can be found from FT look up tables as

A simple proof of this result can be found in the appendix

Thus the FT of becomes the FT given by the integral of

Again splitting the integral into two domains, below and above zero, we get to evaluate

As represents the unit step function which is given by

, for

, for

The first integral disappears (as it equals zero) then we get to evaluate

This is the product of two functions of so we need to evaluate this expression using the method of integration by parts (IBP)

As the IBP formula states that

We make so that

We also make so that

Making these substitutions into our IBP formula we need to solve

Applying the IBP formula again to the integral
we let so that

We also make so that

Then we obtain

Combining all these results we obtain

Applying the limits

Everything in these limits ...

Solution Summary

The continuous time signal for Fourier transforms are found. The expert computes the inverse Fourier transform for a function.

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