The Convolution of Two Functions | Definition & Properties
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- เผยแพร่เมื่อ 14 ต.ค. 2024
- 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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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.
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.
I like your t-shirt haha
You just explained convolution 100 times better than my textbook and my professor! Thanks!!
Haha thanks!
Wonderfully explained. Time to do my final exam. Thank you once again.
Good luck!
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.
This playlist and the ODE playlist have been getting me through this 5 week summer class
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!
if a = tau and b = t - tau, then how did you get b - a = 2tau - t? shouldnt it be t - 2tau?
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.
this came in clutch, you made it so simple while my professor made it super confusing so thank you Trefor
Haha thanks!
You cleared up all my confusion. Thanks a lot. ❤️❤️❤️
You’re welcome 😊
Professor Trefor your T shirt is fantastic. Can you please tell me where I can buy one?
Same haha, I was looking more at his shirt than listening to his explanation
A quite difficult (but so essential!) concept very well explained! Thanks!
Glad you enjoyed!
*Over
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 !
Thank you!!
You are single-handedly carrying all my math related subjects throughout my degree.
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
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
Wow. Only if the classes were this good.
Best video on TH-cam till date
Just got yourself a new sub :) thank you for the concise way you explain things
Thanks for subbing!
Great video. Thank you for sharing. Regards from Panama 🇵🇦
You and Dr Peyam have best explanation of Convolution!
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.
Thanks so much. The example my teacher gived is so complicated and confusing. Your example helped me totally understand it.
Clear and concise.
Thank you for your great work. Probably missing a little comment about the Domain of *.
Everybody : let us revise for tomorrow's exam
Me at 12.am :
Dr. Trefor....
First of all, I like your T-Shirt.
Thank you very much for sharing the knowledge.
You are of great help. Thanks
Great explanation! and also I liked the t-shirt..
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
My wife actually gave it as a present years ago and I've never been able to find the exact one again!
@@DrTrefor Oh no! 😭
What a concise explantation? Thank you Dr.
Hello, thanks for the video, I appreciate it!
Btw, where did you get your t-shirt from? I like it, haha.
Thank you very much!!!!!
Good stuff for my signals and systems class.
Well explained. Thank you!
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!!
As usual outstanding...❤❤👏👏
That was very helpful Dr. Bazzet, thank you. Can you please send me the link of the shirt you’re wearing? It’s amazing.
Dr you really doing a good job...appreciate ur work.
..
Your videos are so helpful doc ❤️🙏
Great explanations!
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.
So good, so far. Awesome!!!😄😄
love from India sir !
this is what i was looking for! thank you for the amazing video.
so what is the difference between multiplication and convolution
Dr, in your example, how can you get (2T - t) instead of (t - 2T) ? Help me out 🥺😭
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)]
Cos is an even function therfore cos(x) = cos(-x) for every x in R.
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
Please bring a video on generalized convolution ( reference t. M Apostol)
I love ur T-Shirt!
Nice T shirt sir, covered all functions 😊
Simply brilliant.
May I know where can I buy your t-shirt? It's so cute! (That blue one with many common math functions. )
I heard another term called polynomial mutiplication used exchangable with the convulution.Are they the same or different?How do they relate?
Clear and Great
Why do convolutions need to be so convoluted?
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.
Whoah T shirt 👚 contains summary of one year of Mathematics.
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?
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
@@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.
@@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.
Nicely done
Great video. 👍👍👍
How did the sign of integral change? I did not get that part. What of we are integrating from -inf to inf??
Although I might not understand much of this yet, I'm beginning to grasp how to do the calculations
I looked at your T-shirt more times than I used in the class.
nice shirt btw, where can I get it?
I'm wondering the same thing
very nice video.
6:25 but why is L[f(t) * g(t)] = F(s) . G(s) ? Is there a different derivation for this property??
Great video, but actually I think sin function on your T-shirt is flipped, it is -sin ))
I need to solved this integral and it really confuses me. I don't understand that t-tau part being in the argument.
Great video as usual
the only thing wrong with this video is we still don't know where to get the epic t-shirt.
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
@@DrTrefor Thanks sir, i love ur channel. You are doing a great job.
Hi, how is this related to a convolution in a Convolutional Neural Network?
I subbed for the shirt, stayed for the video
Where can we buy the T-shirt you wear ☺️?
where did you get this Tshirt, professor?
what about differential property? time shift and fourier ? halpp ;u;
Thanks so much.
I like your T shirt!
Where can l get that T shirt ?
spent more time looking at the shirt than actually paying attention XD
At 4:00, how does dtau turn into negative du? Where did the t go?
Thank you
How did you get -(1/2)tcost?
this was great
Hi.
in fourier analysis instead of 0 and t, we write -∞ and +∞ for bounds.
why there is a difference between these two?
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.
@@rezhaadriantanuharja3389 what about the upper bound?
Sajad If tau > t, then (t - tau) < 0 and g(t-tau) = 0
@@rezhaadriantanuharja3389 Thank you
Fun question: A(f,g)=Lf*Lg
What is A?
I like your shirt!
This is a great series. Stats question. Could you do a video on using the Laplace transform to get a characteristic function of a pdf?
If possible, example 1 compares the moment generating function to the characteristic function of a pdf.
Example 2 does a pdf with an undefined moment generating function to show how the characteristic function is more broadly defined.
life saver
Nice !
really well explained :)
your shirt but y is the indicator function on a dense set :)
thanks for the lecture for explaining convolution, but maybe need review a couple of times just to catch up with your fluent explanation......haha
Thank you Marc Gasol
Oh thanks I'm taking electricity course and I'm using it very much
@@DrTrefor yeah of course it's almost all EM based on laplace transform for signals....
Where can I get that t-shirt?
1.59 i guess it is not a function of t but tau. the way you wrote it down.
where can i get this shirt?