Fourier transform problems
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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?
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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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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 ...
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