Circular vs. Linear Convolution: What's the Difference? [DSP #08]

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In this video, we are presenting the circular convolution and how it differs from the linear convolution.
In case of any doubt in understanding, please, refer to the article above 🙂
00:00 Introduction
00:34 Convolution property of the discrete Fourier transform
00:50 Circular convolution example
01:17 Where does circular convolution come from?
03:03 Circular convolution formula
03:41 Samples of circular convolution corresponding to linear convolution
05:45 Circular convolution as the basis of fast convolution
05:59 Summary
#dsp #convolution

[@WolfSoundAudio]

Have I helped you with this video? If yes, please, consider buying me a ☕ coffee at www.buymeacoffee.com/janwilczek
Thanks! 🙂

[@amber1862]

Love your explanations and pacing!

[@WolfSoundAudio]

Thank you, I'm happy to hear that!

[@StefanRinger]

For me the nicest way to think about this is to think of the conv as matrix muliplication and then do eigenanalysis on it. Explains really well the connection between shifting, padding and Fourier/DFT transform and why conv becomes multiplication

[@WolfSoundAudio]

Nice, thanks!

[@MrSocialish]

Could you elaborate a little more on the eigenanalysis portion?

[@StefanRinger]

@@MrSocialish Convolution = Sum(Signal*impulse response) where you shift the impulse response over time. You can write this as a matrix multiplication in which you have the signal as vector times a matrix that row by row shifts the impulse response by one entry.
In the case of linear convolution you pad the shifted impulse response by zeros, resulting in a toeplitz matrix. In the case of circular convolution you get a circulant matrix.
Now you can do eigendecomposition of those matrices. The toeplitz one gives you eigenvectors of the fourier transform, the circulant has the eigenvectors of the DFT.
If you rearrange the equations a bit you can see the convolution theorem: Convolution in time domain is equal to multiplication in the frequency domain. Or without rearranging: to filter a signal decompose it into sinusoidals (eigenvec matrix^-1 in decomposition), change their amplitudes according to the filter (diagonal eigenvalue matrix in decomposition) and then put sinusoidals back into time domain (eigenvec matrix in decomposition)
The deeper reason being the connection of derivatives and polynomials when looking at ODEs which are linear systems hence describable by linear algebra (y'' -> x^2 -> solve for x -> complex numbers -> fourier)

[@lounes9777]

underrated video !!!!!

[@MrSocialish]

Dude you are awesome

[@krishc.1808]

At 2:08, we do we squash the 5 points into 4 points?

[@vabhi8190]

Finally found the difference ..!!

[@WolfSoundAudio]

I'm happy to hear that!

[@Vitonolable]

I feel bad discarding poor samples 😢
Bit on the serious note, what are the applications of circular convolution? Does it mean that output signal is 'shorter' than would have been with liniar ?

[@oguzcan815]

sup