2

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 w is frequency with which we are wrapping x(t) around circle. Now for a given frequency w, 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 w (winding frequency). If i repeat same procedure for all w, 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(w) times X(w) 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 w, if i multiply X(w) 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!!!

Omibuddyy
  • 106
  • 6
  • @KingDuken I mean dot product. Maybe i will just remove the word bit-wise for simplicity. Basically plain multiplication. – Omibuddyy Dec 02 '19 at 21:12
  • 5
    I'm voting to close this question as off-topic because this is a question about mathematics not specific to the application in engineering. – The Photon Dec 02 '19 at 21:16
  • @ThePhoton Sure. Is there any way to transfer the question from here to mathematics exchange site? – Omibuddyy Dec 02 '19 at 21:18
  • Okay "dot product" has to with vectors in space. But I agree with Photon. I think you're trying to ask for a proof of something, which would belong to our Math SE. You can delete this question and ask it on there. –  Dec 02 '19 at 21:18
  • @Omibuddyy, you can click "flag" below the question text and ask the mods to move it. (But first search the math site to make sure the question hasn't been asked before) – The Photon Dec 02 '19 at 21:27
  • "x-axis showing the value of shift between 2 signal, which in this case are same signals. ": That would be correlation, in convolution you have to mirror one of the signals by the y-axis first. – yar Dec 02 '19 at 21:56
  • @yar (Correct me if i am wrong) You are correct. But eventually in both convolution and correlation case, x-axis indicates amount of shift, so shouldn't matter as far as question is concerned. – Omibuddyy Dec 02 '19 at 22:00

1 Answers1

0

Not remotely a mathematical proof, but I gather from the post that you understood the math and wanted a more intuitive understanding of the relationship...

When you convolve, you create a time series, but the time axis isn't real time, it's time difference between the two time-based functions. You then integrate the product across the time axis for each time difference to get your result in the time domain.

A time difference is a period, and the inverse of period is frequency. Each point in X(f) represents a frequency integrated across the time axis. In the frequency domain, therefore, X(f) and Y(f) have already been integrated with a reference function across the time axis.

Now that that integration is complete, the multiplication is the remaining operation. Since you've already integrated across the time axis, multiplication gives you your result...but it's still in the frequency domain. An inverse Fourier transform is still needed to get the time result.

Interestingly, it's symmetric...multiplication in time is convolution in frequency. But we invented a different word for it...heterodyne.

Cristobol Polychronopolis
  • 9,622
  • 9
  • 17