The Convolution of Two Functions | Definition & Properties

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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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Опубликовано:

10 апр 2020

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Комментарии : 180

@nice4hat 3 года назад

I like your t-shirt haha

@vaibhavdlxit1050 2 года назад

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.

@ejkitchen 3 года назад

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.

@bromarskiii 3 года назад

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

@DrTrefor 3 года назад

Haha thanks!

@ricardobautista-garcia8492 3 года назад

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

@DrTrefor 3 года назад

Good luck!

@josiahkolar4209 3 года назад

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

@DrDerivative 2 года назад

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

@migo77 4 года назад

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!

@sobhanboss3935 3 года назад

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

@DrTrefor 3 года назад

Haha thanks!

@RubALamp 4 года назад

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.

@sarveshchandak 3 года назад

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

@DrTrefor 3 года назад

You’re welcome 😊

@azz650 Год назад

You make this very easy to understand and follow, thank you.

@visualgebra 4 года назад

You and Dr Peyam have best explanation of Convolution!

@SHAHHUSSAIN 4 года назад

As usual outstanding...❤❤👏👏

@m7mdarwani964 3 года назад

Clear and concise.

@NPCNo-xm2li 6 месяцев назад

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

@codyfan1097 Год назад

Your videos are so helpful doc ❤️🙏

@BentHestad 3 года назад

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

@DrTrefor 3 года назад

Glad you enjoyed!

@1qaz1qaz15 2 года назад

*Over

@eric_welch 3 года назад

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

@DrTrefor 3 года назад

Thanks for subbing!

@johanfrancis2001 Год назад

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

@PiyushPastor 2 года назад

Wow. Only if the classes were this good.

@paulkanucawec9489 Год назад

What a concise explantation? Thank you Dr.

@joaomattos9271 Год назад

Thank you very much!!!!!

@ConceptualCalculus 4 года назад

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.

@RanbirSingh-no3mc 2 года назад

Best video on RU-vid till date

@TUCSKIRTHIKR 3 года назад

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

@matteodare1706 2 года назад

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

@hectorgalva7495 2 года назад

So good, so far. Awesome!!!😄😄

@ProfeARios 3 месяца назад

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

@jaydean5243 3 года назад

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 3 года назад

Thank you!!

@ase1362 2 года назад

You are of great help. Thanks

@philcooper279 Год назад

Simply brilliant.

@fernandogallardo3477 2 года назад

Good stuff for my signals and systems class.

@JOAE23 2 месяца назад

Well explained. Thank you!

@kenny44871 3 года назад

Great video as usual

@stringsam 4 года назад

really well explained :)

@hsotnicam 4 года назад

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

@rmb706 2 года назад

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.

@yunxiaomu_195 Год назад

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

@zulhishamtan4308 2 года назад

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

@muhammadfaizan2232 3 года назад

Clear and Great

@BoZhaoengineering 4 года назад

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

@Tosti_bakker 3 года назад

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

@sogol2261 2 года назад

Great video. 👍👍👍

@jacobanderson5693 4 года назад

Thanks so much.

@farukdemirel903 2 года назад

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

@JohnAbreu17 3 года назад

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

@universalalgorithm3263 Год назад

Why do convolutions need to be so convoluted?

@procerpat9223 2 года назад

Nicely done

@yuzou2008 3 года назад

I love ur T-Shirt!

@RAMKUMAR-rh4um 2 месяца назад

Nice T shirt sir, covered all functions 😊

@manrajmann4732 3 года назад

love from India sir !

@eggxecution 7 месяцев назад

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

@nnamdiYT Год назад

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

@sujanban686 Месяц назад

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

@michaelholland5424 Год назад

very nice video.

@master_zenrade 3 года назад

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

@j.o.5957 3 года назад

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

@AS-bg9lg Год назад

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

@amroalatasi2415 2 года назад

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

@thepie-phitalks07 2 года назад

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

@sergiopallas4567 2 года назад

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!!

@blackspitit 3 года назад

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

@noureldinechaker131 2 года назад

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

@theblitz1687 8 месяцев назад

this was great

@abang571 3 года назад

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

@edilmarlulab7079 3 года назад

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 2 года назад

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

@eggxecution 7 месяцев назад

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

@raghuveerdani4046 Год назад

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

@felixgomez5085 2 года назад

Nice !

@twillkickers Год назад

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

@KhinThazin-nx8xd Месяц назад

Thank you

@neptune8422 2 года назад

I like your T shirt!

@gulsumsecilsermet8583 Месяц назад

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.

@MisterBinx 3 года назад

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

@rajkumarray2907 4 года назад

How can we solve linear differential equations of first order derivative like - dy/dx - y/x = (2x) whole cube by Laplace transform. Please sir, plzz reply

@rajkumarray2907 4 года назад

@@DrTrefor Thanks sir, i love ur channel. You are doing a great job.

@MrMobi007 3 года назад

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

@tesset8828 4 года назад

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

@JuntaoChu Год назад

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

@MachoMandem 3 года назад

I subbed for the shirt, stayed for the video

@prashantjoshi5389 Год назад

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

@vicadegboye684 7 месяцев назад

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 7 месяцев назад

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

@vicadegboye684 7 месяцев назад

@@DrTrefor Oh no! 😭

@josephhajj1570 4 года назад

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

@josephhajj1570 4 года назад

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

@uchiha_kakashi6716 Год назад

so what is the difference between multiplication and convolution

@SuperHovik Год назад

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

@liheather2367 3 года назад

where did you get this Tshirt, professor?

@Acesushii 3 года назад

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

@tjk581 3 года назад

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 3 года назад

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 3 года назад

@@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 3 года назад

@@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.

@x_ma_ryu_x 2 года назад

life saver

@jing-chingchen2333 3 года назад

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

@sajjad213 3 года назад

Hi. in fourier analysis instead of 0 and t, we write -∞ and +∞ for bounds. why there is a difference between these two?

@rezhaadriantanuharja3389 3 года назад

Sajad in this video, the value of f(x) and g(x) is 0 when x < 0. This is the normal approach to evaluate system’s response since we assume the input function is applied to system at t = 0 and before t = 0, we do not do anything to the system.

@sajjad213 3 года назад

@@rezhaadriantanuharja3389 what about the upper bound?

@rezhaadriantanuharja3389 3 года назад

Sajad If tau > t, then (t - tau) < 0 and g(t-tau) = 0

@sajjad213 3 года назад

@@rezhaadriantanuharja3389 Thank you