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Hello I borrowed the title for another post.

I cannot figure out how to find the inverse fourier transform from this spectrum. spectrum form my homework problem

I know what the transform is formula

I'm sorry for the plot being hard to read, it is how we were given it and I understand the frustration. The magnitude plot goes from -w0 to w0 with a amplitude 1. The phase plot has the same interval but alternates from pi/2 to -pi/2. Apologies for formatting I am new and on mobile. Can someone help me I am totally lost.

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    $\begingroup$ sorry, we can't even read your plots. $\endgroup$ – Marcus Müller Mar 13 '19 at 7:43
  • $\begingroup$ Are you familiar with the angle ($\angle$) and magnitude ($|.|$) signs and how these work with the Discrete Fourier Transform? $\endgroup$ – A_A Mar 13 '19 at 7:53
  • $\begingroup$ I'm sorry for the plot being hard to read, it is how we were given it and I understand the frustration. The magnitude plot goes from -w0 to w0 with a amplitude 1. The phase plot has the same interval but alternates from pi/2 to -pi/2. Apologies for formatting I am new and on mobile. $\endgroup$ – J1smi48 Mar 13 '19 at 12:10
  • $\begingroup$ Additionally I only know how to construct the fourier transform from the plot and work from there. This is what is done in all the examples, if there is some other method then no I am not aware of it. $\endgroup$ – J1smi48 Mar 13 '19 at 12:12
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That's actually a fairly a tricky one. Two ways you can go about it

Method 1

  1. Magnitude is that of a ideal lowpass filter. Inverse of that is sinc function
  2. Phase is that of an hilbert transformer (times -1). Impulse response is (roughly) $1/\pi t$
  3. convolve the two

Method 2

  1. Write out the equation for inverse Fourier transform
  2. Pop in values from the graph and simplify a bit
  3. You get either sum or integral for unity amplitude sine waves up to $\omega_{0}$

Neither one is pretty.

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