I am trying to understand intuitively why convolution is multiplication in frequency domain. I started at the mathematical derivation of this, but didn't understand what is happening intuitively. This question may seem very easy for someone or very vague for someone else, so i would also like to put my efforts here in understanding the same.

I would be glad if someone is able to understand what i understand as of now and give answer in similar way, but any other thought-process is also fine. Also, i would prefer more wordy explanation rather than seeing math derivation. If anyone explain derivation of this theorem, by showing the steps in derivation and explaining each step, that is also fine.

Current Understanding:

I know fourier transform is a process of wrapping a signal x(t) around a circle which is nothing but obtained by $$x(t)\cdot e^{-j\omega t}$$ where $\omega$ is frequency with which we are wrapping x(t) around circle. Now for a given frequency $\omega$, we will get some 2D shape in complex plane. If i integrate this over dt, i will get $$\int{x(t)\cdot e^{-j\omega t}dt}$$, which can be though as if i am calculating location of "center of mass" of that shape in 2D complex plane. So i will get some complex number. All this we did is for a single value of $\omega$ (winding frequency). If i repeat same procedure for all $\omega$, i will get frequency plot.

Now, let's say if i want to convolve 2 same signals, meaning of this in frequency domain turns out to be X($\omega$) times X($\omega$) as stated by convolution theorem. Now, in time domain its equivalent will be y-axis showing the value of convolution integral and x-axis showing the value of shift between 2 signal, which in this case are same signals.

Now in frequency domain, for a given value of $\omega$, if i multiply X($\omega$) with itself, it means multiplication of 2 complex numbers. Although 2 complex numbers here are same, generally thinking, multiplying 2 complex numbers results in addition of phases and multiplication of amplitudes. Also, each complex number here represents the "center of mass" of the shape we got when we wrapped up the whole signal around a circle.

After this, i am not able to think!!!

Final Goal: My final goal is to understand wiener khinchin theorem. Difference between the asked question and the wiener khinchin theorem is, X($\omega$)$\cdot$ X($\omega$) is convolution and X($\omega$)$\cdot$ $X^{*}$($\omega$) is wiener khinchin theorem.


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