The Convolution of Two Functions | Definition & Properties

We can add two functions or multiply two functions pointwise. However, the convolution is a new operation on functions, a new way to take two functions and combine them. In this video we define the convolution of two functions, state and prove several of its nice algebraic properties, and see in particular how convolutions play very nicely with Inverse Laplace Transforms and give us way to deal with products. Much like the product rule for derivatives, the formula isn't quite as simple as the product of the inverse laplace transforms, instead it is given by convolutions.
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This video was created by Dr. Trefor Bazett. I'm an Assistant Teaching Professor at the University of Victoria.
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[@ejkitchen]

As many others have stated, thank you for making this very easy to understand. It's fascinating how many bad profs there are out there attempting to teach this. If you think about it, it's not that hard but for some reason, many people struggle to explain this properly because they do not spend the time like you did to make it accessible. Again, thank you for doing this.

[@nice4hat]

I like your t-shirt haha

[@vaibhavdlxit1050]

You turned convolution from a dreaded enemy to an important ally which springs up to rescue while performing inverse laplace of evil looking functions. You sir, are a hero.

[@bromarskiii]

You just explained convolution 100 times better than my textbook and my professor! Thanks!!

[@DrTrefor]

Haha thanks!

[@DrDerivative]

if a = tau and b = t - tau, then how did you get b - a = 2tau - t? shouldnt it be t - 2tau?

[@ricardobautista-garcia8492]

Wonderfully explained. Time to do my final exam. Thank you once again.

[@DrTrefor]

Good luck!

[@migo77]

Starting from 10:06 when you starting evaluating the integral & on the second line, the last part should have read: -0.5*TAU*cost t, because your're integrating w.r.t. TAU. Anyways, thanks v. much for all your efforts. God bless!

[@RubALamp]

I hope you grow in popularity. Not that it should be needed, but I think many people would benefit from your insight. Thank you for your videos.

[@josiahkolar4209]

This playlist and the ODE playlist have been getting me through this 5 week summer class

[@sobhanboss3935]

this came in clutch, you made it so simple while my professor made it super confusing so thank you Trefor

[@DrTrefor]

Haha thanks!

[@sarveshchandak]

You cleared up all my confusion. Thanks a lot. ❤️❤️❤️

[@DrTrefor]

You’re welcome 😊

[@jaydean5243]

Dr Trefor, as usual excellent videos, If there are any math/engineering students that don't "Ace" their exams it is THERE fault...Great JOB !

[@DrTrefor]

Thank you!!

[@ConceptualCalculus]

Just yesterday I was searching for a vid on convolution for my ODE class. Today this new vid popped up in my feed. I'm using it to replace the one I found yesterday, I suppose I should watch this one first, but that's not really necessary. If it is a Trefor Bazett vid, it is good.

[@BentHestad]

A quite difficult (but so essential!) concept very well explained! Thanks!

[@DrTrefor]

Glad you enjoyed!

[@1qaz1qaz15]

*Over

[@universalalgorithm3263]

Why do convolutions need to be so convoluted?

[@hsotnicam]

Little known fact: the verb for convolution is "to convolve," which means to roll something together. Although it absolutely also "convoluted" the two functions - meaning making something more difficult to understand, it is not what it intend to do :-D

[@NPCNo-xm2li]

You are single-handedly carrying all my math related subjects throughout my degree.

[@rmb706]

Believe it or not, probability brought me here. I was looking for a more rigorous definition of a convolution. I never studied the Laplace transform. Interestingly enough, two nonnegative RVs that have the same Laplace transform have the same distribution.

[@TUCSKIRTHIKR]

Everybody : let us revise for tomorrow's exam Me at 12.am :

[@PiyushPastor]

Wow. Only if the classes were this good.

[@RanbirSingh-no3mc]

Best video on KZbin till date

[@eric_welch]

Just got yourself a new sub :) thank you for the concise way you explain things

[@DrTrefor]

Thanks for subbing!

[@visualgebra]

You and Dr Peyam have best explanation of Convolution!

[@BoZhaoengineering]

Professor Trefor your T shirt is fantastic. Can you please tell me where I can buy one?

[@Tosti_bakker]

Same haha, I was looking more at his shirt than listening to his explanation

[@ProfeARios]

Great video. Thank you for sharing. Regards from Panama 🇵🇦

[@joaomattos9271]

Thank you very much!!!!!

[@m7mdarwani964]

Clear and concise.

[@eggxecution]

at 10:00 I think you made a mistake in trigo identity, it should be (a-b) also (a+b) to prevent confusion. This identity is product to sum formula if I recall correctly

[@SHAHHUSSAIN]

As usual outstanding...❤❤👏👏

[@Ayrmodros]

Thanks so much. The example my teacher gived is so complicated and confusing. Your example helped me totally understand it.

[@fernandogallardo3477]

Good stuff for my signals and systems class.

[@zulhishamtan4308]

Dr. Trefor.... First of all, I like your T-Shirt. Thank you very much for sharing the knowledge.

[@ase1362]

You are of great help. Thanks

[@JohnAbreu17]

Thank you for your great work. Probably missing a little comment about the Domain of *.

[@johanfrancis2001]

Dr you really doing a good job...appreciate ur work. ..

[@paulkanucawec9489]

What a concise explantation? Thank you Dr.

[@codyfan1097]

Your videos are so helpful doc ❤️🙏

[@abang571]

Dr, in your example, how can you get (2T - t) instead of (t - 2T) ? Help me out 🥺😭

[@edilmarlulab7079]

Use trigo. Identities from sin(T)sin(t-T) let A=T & B=t-T Then Subtract to get sinAsinB: cos(A-B) = cosAcosB + sinAsinB cos(A+B) = cosAcosB - sinAsinB cos(A-B) - cos(A+B) = 2sin(A)sin(B) sin(A)sin(B) = 1/2[cos(A-B) - cos(A+B)] Then Substitute: (A = T ; B = t-T) sin(T)sin(t-T) = 1/2{cos[(T) - (t - T)] - cos[(T) + (t - T)]} Then you get: sin(T)sin(t-T) = 1/2[cos(2T - t) - cos(t)]

[@thejohncenafan100]

Cos is an even function therfore cos(x) = cos(-x) for every x in R.

[@eggxecution]

I know what you meant fam, I saw the mistake in the video too. If you check the trigo identity he posted at 10:00 its (b-a) it should be (a-b) you'll arrive at 2T-t

[@matteodare1706]

this is what i was looking for! thank you for the amazing video.

[@sergiopallas4567]

I'm sure I can't be the only one with this question. Where can we get that t-shirt? As always great video, thanks for helping those in need!!

[@hectorgalva7495]

So good, so far. Awesome!!!😄😄

[@farukdemirel903]

Great explanation! and also I liked the t-shirt..

[@manrajmann4732]

love from India sir !

[@JOAE23]

Well explained. Thank you!

[@master_zenrade]

Whoah T shirt 👚 contains summary of one year of Mathematics.

[@uchiha_kakashi6716]

so what is the difference between multiplication and convolution

[@beophobic9653]

Great explanations!

[@sujanban686]

I looked at your T-shirt more times than I used in the class.

[@RAMKUMAR-rh4um]

Nice T shirt sir, covered all functions 😊

[@yuzou2008]

I love ur T-Shirt!

[@noureldinechaker131]

spent more time looking at the shirt than actually paying attention XD

[@MisterBinx]

I need to solved this integral and it really confuses me. I don't understand that t-tau part being in the argument.

[@philcooper279]

Simply brilliant.

[@thepie-phitalks07]

Please bring a video on generalized convolution ( reference t. M Apostol)

[@gulsumsecilsermet8583]

Thank you for the explanation. There is just one thing which is bother me. Sometimes speak rate is very fast then it becomes slow suddenly. Sometimes I couldn't understand that I needed to slow the video and it can be distructor.

[@chinnamuniyandi2023jrf]

If limit has between negative infinity to t in the convolution formula instead of negative infinite to positive infinite, should we call LTI system? Please reply.

[@raghuveerdani4046]

How did the sign of integral change? I did not get that part. What of we are integrating from -inf to inf??

[@muhammadfaizan2232]

Clear and Great

[@tjk581]

I don't know. The definition of convolution is integral from 0 to t, or from -inf to +inf? In different sources, it's defined differently. Can someone explain?

[@DrTrefor]

If f and g are zero for negative values, then they are the same and the int from negative infinity to infinity is the same as 0 to t

[@tjk581]

@@DrTrefor But for evaluating convolution of f(x)=g(x)=sin(x) you used integral form 0 to t , when sin(x) isn't 0 for negative x.

[@angelmendez-rivera351]

@@tjk581 But the Laplace transform only cares about the values of f(x) for positive x, so you can define f piecewise so that it is sin(x) only for x > 0, giving the result in the video.

[@AS-bg9lg]

Hello, thanks for the video, I appreciate it! Btw, where did you get your t-shirt from? I like it, haha.

[@vicadegboye684]

Where did you get that your shirt from? I love it. I was hoping to see it in your store. Would definitely buy it if you're selling it

[@DrTrefor]

My wife actually gave it as a present years ago and I've never been able to find the exact one again!

[@vicadegboye684]

@@DrTrefor Oh no! 😭

[@amroalatasi2415]

That was very helpful Dr. Bazzet, thank you. Can you please send me the link of the shirt you’re wearing? It’s amazing.

[@prashantjoshi5389]

6:25 but why is L[f(t) * g(t)] = F(s) . G(s) ? Is there a different derivation for this property??

[@sogol2261]

Great video. 👍👍👍

[@procerpat9223]

Nicely done

[@MrMobi007]

the only thing wrong with this video is we still don't know where to get the epic t-shirt.

[@jacobanderson5693]

Thanks so much.

[@nnamdiYT]

I heard another term called polynomial mutiplication used exchangable with the convulution.Are they the same or different?How do they relate?

[@asadshamsiev2462]

Thank you Marc Gasol

[@accuset]

I need a bigger numpad... dot product, cross product, scalar multiplication, convolution, holy crap.

[@j.o.5957]

Although I might not understand much of this yet, I'm beginning to grasp how to do the calculations

[@superdahoho]

nice shirt btw, where can I get it?

[@rtusiime]

I'm wondering the same thing

[@kenny44871]

Great video as usual

[@JuntaoChu]

May I know where can I buy your t-shirt? It's so cute! (That blue one with many common math functions. )

[@twillkickers]

At 4:00, how does dtau turn into negative du? Where did the t go?

[@tesset8828]

your shirt but y is the indicator function on a dense set :)

[@KhinThazin-nx8xd]

Thank you

[@cbkt35]

1.59 i guess it is not a function of t but tau. the way you wrote it down.

[@michaelholland5424]

very nice video.

[@SuperHovik]

How did you get -(1/2)tcost?

[@ahmedesamamer]

Where can l get that T shirt ?

[@armanfahradyan6207]

Great video, but actually I think sin function on your T-shirt is flipped, it is -sin ))

[@neptune8422]

I like your T shirt!

[@NimbusAbi]

where can i get this shirt?

[@blackspitit]

Hi, how is this related to a convolution in a Convolutional Neural Network?

[@x_ma_ryu_x]

life saver

[@MachoMandem]

I subbed for the shirt, stayed for the video

[@deyardeyar8163]

10:15 what happened to the tau in the interation

[@kexinsaw7092]

I like your shirt!

[@theblitz1687]

this was great

[@liheather2367]

where did you get this Tshirt, professor?

[@stringsam]

really well explained :)

[@Acesushii]

what about differential property? time shift and fourier ? halpp ;u;

[@felixgomez5085]

Nice !

[@josephhajj1570]

Oh thanks I'm taking electricity course and I'm using it very much

[@josephhajj1570]

@@DrTrefor yeah of course it's almost all EM based on laplace transform for signals....

[@leixiao169]

Where can we buy the T-shirt you wear ☺️?

[@epicmorphism2240]

Fun question: A(f,g)=Lf*Lg What is A?

[@jing-chingchen2333]

thanks for the lecture for explaining convolution, but maybe need review a couple of times just to catch up with your fluent explanation......haha

[@sergiomanzetti1021]

Where can I get that t-shirt?

[@ChristianBilankov]

Where do I get that shirt?