Introducing Convolutions: Intuition + Convolution Theorem

In this lesson, I introduce the convolution integral. I begin by providing intuition behind the convolution integral as a measure of the degree to which two functions overlap while one sweeps across the other. I demonstrate this intuition by showing that the convolution of two box functions is a triangle.
I then move on to proving the Convolution Theorem for Fourier Transforms, and discussing how it compares to the Convolution Theorem for Laplace Transforms. The proof for Fourier Transforms is relatively simple, but the proof for Laplace Transforms is a bit more difficult (if you really want to see the Laplace Transform proof, I can make another video but I've put it off for now).
Questions/requests? Let me know in the comments! Hopefully the intuition I provided was sufficiently clear.
Prereqs: Very basic knowledge of Fourier and Laplace Transforms (i.e. you just need to know what they are and what they're used for), ODEs, and integration. Playlist: • Topics in Ordinary Dif...
Lecture Notes: drive.google.com/open?id=1dDW...
Patreon: www.patreon.com/user?u=4354534
Twitter: / facultyofkhan
Special thanks to my Patrons for supporting me at the $5 level or higher:
- Jose Lockhart
- Yuan Gao
- Justin Hill
- Marcin Maciejewski
- Jacob Soares
- Yenyo Pal
- Chi
- Lisa Bouchard

[@dipankerbaral3301]

i hereby declare this underrated video the best explanation of convolution in the internet

[@BoZhaoengineering]

I hereby agreed! the best interpretation of what is convolution integral and its fourier transform and laplace transform. Clear concept explained in plain language and easy function plot. Prompt those nasty algebra expression to save audiences' concentrated energy for listening to the core ideas. superb.

[@fsto9039]

agree

[@raghua9318]

Function plot explanation of convolution of two functions is similar to explanation of correlation between two functions. How do they differ?

[@varun_4125]

Fr

[@riyabansal9191]

Oh my god, I have been trying to gain an intuition on this topic for so long. So glad I ran into this video! Thank you, sir.

[@pratikahir148]

Please continue to make such videos,it serves as a quick refresher for me before exam(VERY HELPFUL).

[@rockspoon6528]

I've been through all four calc courses and am on Linear Circuits 2, and this is the first time anyone's written the first part of the definition. And it makes sense now.

[@shine_at_dusk]

Thank you very much) The explanation is so clear I've watched only for about 2 minutes and already got the idea behind the use case

[@rogerz741]

Thank you! The concept of convolution is concisely presented.

[@ostensiblyquerulous]

Oh boy can’t wait to see this!

[@gutzimmumdo4910]

finally someone explains concisely what that fucking -t means for fuck sakes, thank you alot best explanation of convolution on the internet.

[@djtoddles8750]

this is great, please make more of these intuition vids

[@xiaoxuanwang2570]

This is sooo helpful! Thank you!

[@hilalnizamoglu4599]

Amazing explanation, thank you!

[@eulefranz944]

Love the convolution :) great work!

[@pramod120895]

Thanks for the explanation... Atlast got a clear visualisation on this topic

[@simrannahar8262]

this was extraordinarily well explained

[@slashpl8800]

Awesome, that's what I was looking for

[@Vicky-pb5hg]

What a fantastic explanation. :-) 🙏

[@davidarredondo2106]

"I won't be spending the next 18 minutes taking the convolution of sin and cosine in an effort to show you that the convolution of two functions is an actual quantity." Savage. I agree though--that Kahn academy video was a waste of time

[@FacultyofKhan]

I'm glad someone understood the reference haha

[@dania5426]

I had to go back to the Khan Academy's video to understand how it is happening. I think both videos are very useful and needed. However it was rude from the Faculty of Khan to say so.

[@engineered.mechanized]

@@dania5426 I agree with that. Educators should maintain mutual respect for one another. True professionalism is depicted in how you present these videos. Salman from Khan Academy is never seen to say anything that is not pertinent to the topic of the video. Whereas, Faculty of Khan, even in his intro video felt the need to defame the other educators who have put out their own lectures in the past. I am an Engineer and an educator myself and I believe it is for the sake of the growth of this channel it would be better for Faculty of Khan not to defame other educators. For the purpose of roasting and dissing we already have tonnes of other entertainment channels.

[@ztac_dex]

so that the 2 Khans don't overlap

[@filippocastellani4761]

@@ztac_dex If I understood the lecture, I'd say that the convolution of the two videos must be nearly zero.

[@duckymomo7935]

LOL “I hope that the explanation wasn’t too convoluted, haha”

[@luansouzasilva31]

THANK YOU SO MUCH, I LOVE YOU

[@j_j8758]

Omg thank u so much. This was very helpful.

[@khaledtaleb3085]

Thanks god that you made me saw this video in the first month of the semester

[@TheNinjaDwarfBiker]

The hero that we all needed

[@MrSnitsarenko]

great explanation! thanks

[@giwahdavalos4699]

Awesome explanation

[@thevoidzzz]

This is such a MONUMENTALLY important idea in electrical engineering, I don't understand why so many other videos and teachers are so bad at explaining this topic

[@gatoradeee]

Very gud as always.

[@ASHUTOSHKUMAR-en9vv]

best video for convolution

[@stekim]

Super helpful thank you

[@Dontonethefirst]

Getting me through signals and systems man.

[@debarshimajumder9249]

BUT VERY HELPFUL VIDEO... THANK U...

[@CHEESYhairyGASH]

Thank you

[@midreesbhat11bhat26]

Nice illustration.....

[@leiyplane2011]

Boundary condition between negative side and positive side can use Laplace Transform too. Fourier Transform is just a special version of Laplace Transform.

[@rishidey]

Excellent

[@ovais217]

haha, amazing video !! Thanks man !!

[@duckymomo7935]

I love Fubinis theorem

[@LoganStVrain]

Could you do a video about the Fourier transform (definition, purpose and derivation). Also, what is the difference between a Fourier Transform and a Fourier Series. Thanks!

[@FacultyofKhan]

Sure, as I continue my series on PDEs, I'll do some videos on Fourier! Also, a Fourier Transform is an operation that converts a function of time to a function of frequency (in a sense, it's like the Laplace Transform), while a Fourier series is a way to express a function as a sum of sines and cosines. Hope that helps!

[@themightyquinn100]

I've always learned that the upper bound of the integration was 't' for the laplace convolution, not 'inf'. One give you a function of t the other gives you a number. How do we distinguish between these two?

[@LakshmikanthAyyadevara]

excellent video

[@FacultyofKhan]

Thank you very much!

[@sherzadakhan3191]

Dear how g(tau) represent function over a range/interval? isn't g(tau) only represent value of g at a particular point (tau)?

[@rordic.y5947]

From your last video almost like 10 years ago and you said the upper limit and of the integration to be t, then the lower limit of the integration to be zero thereupon leading to totally different result, can you explain the reason behind this two different operation?

[@principioequivalencia9455]

I love you!

[@y031962]

thanks for the video; I didn't get how you split the exponential into two forms. Can someone shed light on that part? thanks

[@douglasstrother6584]

Slick!

[@schrodingerbracat2927]

FYI, t - tau is the reflection of tau in a vertical mirror at t/2.

[@area51xi]

You're actually sweeping across values of tau not t. t is a constant inside the integrand and that is why integrating results in a function of t, y(t).

[@chunchaoma4473]

I like your cool video

[@simonbourguigne9988]

What do you use to make the drawings?

[@Starcell170]

Good explaining! This is why Convolution that is used for image filtering is also called "convolution"

[@olenbrown8145]

"I wont be spending the next 18 minutes showing you the convolution of sine and cosine in an effort to demonstrate that the convolution of two actual functions is an actual quantity" damn, some harsh words for sal

[@NikhiLKumar-nx9nt]

we can do this by applying the tau-t also in g. then why do we do that taking the mirror of g

[@marlonbrade9424]

so what the idea ,in the case that one of the function is not well defined somewhere ?

[@rohanwalia5851]

Why is g flipped? I mean, why is the convolution considered to start at (t - tau) for g? Why can't we calculate f(t) * g(t) instead of f(t) * g(t - tau)?

[@worly14]

Yeah, i'm also curious why. Have you maybe found out?

[@Lezrec]

The functions/signals need to sweep past each other to determine the behavior of how correlated/in sync they are.

[@coz_outline]

The "Ha Ha" in 6:30, lol Thanks anyway

[@erfanmohagheghian707]

Hi sir, If the upper limit of the convolution integral for Laplace transform is infinity, then why LaplaceInverse(F(s)xG(s))=int f(Tau)xg(t-Tau)d_Tau from Tau=0 to "Tau=t" (and not infinity), where F(s) and G(s) are the Laplace transforms of f(t) and g(t)? Thanks.

[@dougb70]

that's exactly what I was thinking. Thanks for being so brave.

[@CHEESYhairyGASH]

At 2:02, isn't it the increasing value of Tau, rather than t, that causes the g function to sweep to the right? If you have y = (x - Tau)^2 and you increase the value of Tau, you will cause the function to shift rightwards.

[@FacultyofKhan]

It's a bit different in this case; to use your analogy, we're increasing the value of x and not tau (this is the same as increasing t in g(t-tau)). If you draw y = (x-tau)^2 (y vs. tau as your axes), then increasing x will make your function move rightward. For instance, if x = 0, then y = -tau^2 (i.e. the vertex of the parabola will be at tau = 0). However, if x = 1, then y = (1-tau)^2: now, the vertex of the parabola will be at tau = 1 (i.e. you've moved your function to the right). Same idea in 2:02. Hope that helps!

[@CHEESYhairyGASH]

@@FacultyofKhan That does help, thank you.

[@coreysimmerer]

Is the voice computer generated or not? That's all I can focus on.

[@FacultyofKhan]

Uhhh no, absolutely not! I am totally not a computer-generated voice/teacher. Beep beep boop boop.

[@GaryTugan]

Ahhhh, well I thought the same thing. 👍

[@graemelaubach3106]

I wish computer-generated voices sounded this good. What kind of computer-generated videos have you guys been watching??

[@elzedliew972]

whats the software used to draw here?

[@callmedeno]

fuckin hell well explained

[@mmpmaldo]

6:32

[@user-or7ji5hv8y]

but I wonder why do we need a construct such as convolution?

[@therealbriancox]

First time seeing "Faculty of Khan", after coming from Khan Academy, also thought it was a robot talking and couldn't help but think- is this an incredibly advanced neural network, trained on Khan Academy neural net tutorials to output simpler neural net tutorials? Is this a weak AGI primitively reaching out and asking us to bring it to full capacity? If so, uh... *I'm here to help* Cheers! 🍺

[@lovedancing1996]

khanvolution

[@akhilnikhilkandagiri5159]

Khan-volution?

[@alexis-74]

genius

[@SO-dl2pv]

Why you are supposing f(t) and g(t) to be positives? that is not the case in general

[@FacultyofKhan]

It's not you're right, but this was just a way to explain the idea behind convolutions. Using positive functions is more intuitive for teaching purposes than using negative functions.

[@nikhilkenvetil1594]

Doesn't sound like Sal.. I thought it was his other channel.

[@The_Aleph_Null]

Khanvolution

[@sachinbadge465]

He is dad of khan academy

[@c.danielpremkumar8495]

All you guys seem to use the word "convolution" wrongly. It should be "convolved with" or "convolving with".

[@fawesomekila7217]

haha

[@juniorjay001]

huh??

[@avijitsazzal4184]

Ha Ha

[@NadimC137]

khanvolution? xD

[@yashgupta3127]

This is not intution man you jus explained formula

[@shwetabhsingh5437]

khanvolution lol

[@the_sophile]

Thank you

[@dk-cn1tc]

khanvolution

[@ebisharifi5244]

Thank you

[@FacultyofKhan]

Welcome!