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Fourier Transform - Solving for Impulse Response!

  1. Nov 10, 2012 #1
    1. The problem statement, all variables and given/known data
    I'm trying to Solve for an impulse response h(t) Given the excitation signal x(t) and the output signal y(t)

    x(t) = 4rect(t/2)
    y(t) = 10[(1-e-(t+1))u(t+1) - (1-e-(t-1))u(t-1)]
    h(t) = ???

    y(t) = h(t)*x(t) --> '*' meaning convolution!

    I am unsure how to take the Fourier Transform of the elements in the output signal. I have posted my attempts below and I would like to know if I am going this correctly or not, Thanks!
    2. Relevant equations
    Using the multiplication - convolution duality I know that we need to take the Fourier transform of each element giving us the following...

    Y(f) = H(f)X(f)

    Which then allows us to solve for H(f) by Y(f)/X(f)


    3. The attempt at a solution

    First I distributed the unit step functions in y(t) giving...

    y(t) = 10[u(t+1)-e-(t+1)u(t+1) - u(t-1) + e-(t-1)u(t-1)

    Now I take the Fourier transform of each element in y(t)

    F(u(t+1)) = 1/(jω+(02))(ej2∏f)

    F(e-(t+1)u(t+1)) = 1/(jω+(12))(e-j2∏f)

    I got this by using the following definition of the Fourier Transform
    e-Atu(t) <---> 1/(jω+A2) for A > 0


    I was curious as to if anyone could give me some insight on whether I am performing these operations correctly or not. I apologize if I left out any information!
     
    Last edited: Nov 10, 2012
  2. jcsd
  3. Nov 10, 2012 #2
    Find the Fourier transforms of x(t) and y(t),then use the convolution theorem and inverse transform.
     
  4. Nov 10, 2012 #3
    Ok, but my question was more about whether I am performing the transform correctly , thank you for your response
     
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