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Recently I stumbled upon Fourier transforms which means I am very new to it. I've been given a question from my professor to find the inverse Fourier transform of a frequency spectrum which is given by:

enter image description here

I can go through each step to explain what I've done so far.

At first, I wrote the basic formula for inverse Fourier transform which is shown below enter image description here

I know that omega is always positive, I rewrote the inverse fourier transform like this

enter image description here

After a little algebra and integration, I got to this equation which is shown below enter image description here

Now the biggest equation is this - I can surely see that the signal seems to be in the complex form. I tried rewriting the exponential part in terms of sine and cosine function and then split the imaginary part and the real part. Is this the correct way to graph it? The equations which contain real part of the function and imaginary part are shown below

enter image description here

Also another thing I thought about is to find the magnitude of the complex function once I have real and imaginary part. Question is if this approach is correct. When I found the module of this function, I got this result (assuming that omega zero is equal to 1)

enter image description here

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  • \$\begingroup\$ What is "the module" of the function? \$\endgroup\$
    – JRE
    Commented Mar 7, 2023 at 16:49
  • \$\begingroup\$ Magnitude of the complex number (we call it ''modulo") \$\endgroup\$
    – tom_ger
    Commented Mar 7, 2023 at 16:51
  • \$\begingroup\$ Fourier transform works in complex signals, but we often use real signals anyway (imaginary part is 0). When you Fourier-transform a real signal, the output is a frequency spectrum and the magnitude/modulo tells you the amplitude and the argument/angle tells you the phase. It's harder to say what the imaginary part means for inverse Fourier transform, but consider that forward and inverse are almost identical, it could still be interpreted the same way. \$\endgroup\$ Commented Mar 7, 2023 at 16:52
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    \$\begingroup\$ @JRE Slovak language - Modul môže byť: absolútna hodnota (reálneho či komplexného čísla alebo prvku v čiastočne usporiadanom vektorovom priestore alebo vektora). I know, it's quite weird. Should have told you that the magnitude is the concern here. \$\endgroup\$
    – tom_ger
    Commented Mar 7, 2023 at 16:58
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    \$\begingroup\$ "I know that omega is always positive" No, it's not. Redo your integration from \$-\omega_0\$ to \$+\omega_0\$ \$\endgroup\$
    – Ben Voigt
    Commented Mar 7, 2023 at 17:26

1 Answer 1

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Complex formulation of Fourier transforms need also negative frequencies! They are there a must. "I know omega is positive" should be written "I do not know why there should be negative frequencies, so I leave them out".

Do again the inverse transform integration, but include the full range i.e. enter image description here

Qualitatively one should see in 2 seconds that the spectrum belongs to an ideally low pass filtered impulse. That's constant amplitude vs. frequency, only a certain bandwidth taken into the account and linear phase which means constant frequency independent delay.

Do not worry. Like the rest of us, also you will see it soon if you continue with Fourier transforms. And you will if you are going to understand the signal theory.

Not asked, but if we look backwards we see that negative frequency has puzzled people several times. Presenting a plain common sinusoidal voltage as a sum of positive and negative frequency component gives to us a possibility to use complex exponentials and that makes numerous common formulas and equations incredibly simple and symmetric and that's utilized well in communication signal theory books.

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