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I found this question in a previous test paper:

enter image description here

I'm having a few confusions:

1) Low pass filters are supposed to block signals below a certain frequency. Here that certain frequency is \$w_c=a\$. I'm not sure how to approach this problem. What would be the frequency of \$v(t) = e^{-at}\$? Wolfram Alpha says that the Fourier transform of functions of the form \$e^{-at}\$ do not have any closed form. Also, Fourier series of the signal wouldn't exist since it's not periodic. So, how should I find the energy dissipated in the resistor?

2) In the second part of the question, what exactly is meant by \$h(t)\$ and \$y(t)\$, any idea?

Please note that I've read the answers to How Fourier transform be able to deal with transients?, but it doesn't really answer my question.

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  • \$\begingroup\$ 1.) The Fourier transform of e^(-a|t|) is easy to find, do the integral. 2.) h(t) is the system impulse response, either do time domain covolution of x(t) and h(t) or take the Laplace transform, multiply, and X(w) and H(w) and take the inverse transform. \$\endgroup\$ Commented Apr 21, 2018 at 13:43
  • \$\begingroup\$ @Captainj2001 Thanks. That makes sense. Also, what is \$y(t)\$ ? \$\endgroup\$
    – user186505
    Commented Apr 21, 2018 at 13:49
  • \$\begingroup\$ @user554252 y(t) is the output. Just treat the filter like a brick wall filter. Change the limits on the energy integral to +/- wc \$\endgroup\$ Commented Apr 21, 2018 at 14:26

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