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    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 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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