Convolution Equation Explained ("Best explanation on YouTube") 

This is literally the "Best explanation on RU-vid". No clickbait, pure gold.

I studied signal processing but was only memorizing the mathematical rules without understanding. But after watching your lessons it seems very simple,every professor has to watch your lessons ... you are genius.

Iain, I dont know where you're from but by god, if i ever meet you - im buying you a beer. Your videos are intuitively explained and simple to follow with often very relevant examples. Really appreciate it, thank you for the effort into these videos

This is really the best explaination on youtube. No fancy graphics or bizarre examples, just a simple solid explanation given in a straightfoward way.

The title wasn't an exaggeration. This is indeed the best intuition builder on convolution!

Searched the internet for intuitive explanations of convolutions but this is the only video that made complete sense and actually 'explained' it. Great stuff Iain. Please keep these coming, perhaps beyond signals and systems as well, for broader math.

This is one of the best convolution explanations on RU-vid or anywhere. I'm an RF engineer with 10+ years of industry experience. I think it would also be helpful to point out towards the end that for causal systems (where tau > t) , h(t - tau) is zero, so the integral can go from -infinity to t , instead of -infinity to +infinity.

THE BEST EXPLANATION OF CONVOLUTION !!! Your channel is going to get a lot of traction very soon and I have no doubt about it!

I'm glad I found this video. Nobody else explains it as well as you. Well done sir! I'm glad there are people like you out there that can explain these concepts so well. I'm subscribing.

Best explanation out there. It makes much more sense when you first approach it with a discrete amount of delta functions.

That's [by far] the best explanation on Convolution. Thank you very much, Sir. I hope your channel goes 🔥🔥. I am already doing my part (liking and commenting haha) Your knowledge + a slightly more-digital approach (maybe using a tablet, where you can give colours to the drawings, pre-draw some of the pics, highlight some terms, etc...) and your channel would grow up quicker, I guess. Ah, and by doing that, you'd have a PDF version of what you do on the video. People (me too) love that thing, because they can annotate on them while watching the video, and that is very effective. PS: I am fine with pen and paper, no problem there, but I have seen people not watching these kind of videos, because there was another one using some tech, and they think they'd understand better that way. I am sure that if they gave it a try, they'd love your channel and subscribe right away. So, we just have to get them to try at least one video and that's it.) But, this is only a thought, because I want you to ⬆️⬆️. But don't forget, your videos are amazing the way they are. And I love all of them. I hope you're doing well, Sir. Greetings from Germany!

Best explanation I could find on the internet, both intuitively and mathematically.

It is, without a doubt, the best and simplest ever explanation. Although I did comprehend convolution in a different way, I have mentally retraced your graphical explanation. Therefore, keep jockeying your explanation in my memory storage whenever I need convolution.

Awesome video!!! A much-needed explanation for my Digital Signal Processing revision.

Thanks lain, this helped me visualize the purpose of Tau and where the time difference comes from! Hope you're having a fantastic day!

Best explanation of the formula indeed that i've seen on YT (and I've seen a lot!) thank you

I learned this back in university and got a real high score out of the course by pure memorizing formulas but here I am after 10+ year and now I really understand what is going on! Thanks, and this is the "Best Exlanation on RU-vid"

Love your explanations mate, thank you

I watched many videos explaining convolution, but you really nailed it! Thanks a lot!

I watched many videos of mathematicians trying to explain convolution and all were disconnected from the real life. The Teacher explained it perfectly well. These videos are must-watch for every engineer.

Wow, what a super explanation!!! The best I have seen online. Thanks

Thank you so much. Best Explanation by far.

This is the best explaination I have ever seen in giving the convolution formula intuitive sense.

Best teacher 🙂 The Google should suggest your answer whenever someone types what is convolution 👌

Really well explained, thank you so much for helping us to understand it sir!

Thank you professor, you helped me a lot in my preparation for the midterm and finals of my signals and systems course!

This is a beautiful explanation, thanks for this.

This is the best explanation I have seen . I have seen the lecture videos of Professor Oppenheim himself but your explanation is more intuitive and clear than that

Thank you sooooooooooo much! Excellent Job!!

literally...the best explanantion on yt

thank you for these videos sir, im 16 and studying this on my own and your videos make difficult concepts like these very easy to intuit and understand mathematically

Fourtunately got this video!!! hats off to you

I have an examination coming up soon and watching this video is explaining everything. Thank you so much for this playlist. Three years ago still valuable today

The way of your explanation is easy and understandable, you are such an intelligent guy.

very well explained ! thanks !

This is gold. Thanks a lot

Thank you very much, Iain. I really understand well now. It was confusing at all times. I understand why we are using Convolution 😊😊😊

Mind = Blown Thank you So much Sir 🙏🙏

Many thanks, great explanation.

great explanation GOD BLESS YOU MAN

Just amazing!

It's amazing that we can learn these things for free. Thank you for your clear explanations 😎

thank you

Yep, it is the best explanation. Thank you

Hello Iain, you are blessed with knowledge, passion to teach and intelligence. I love that you responded to almost all your viewers regardless of how long the videos have been posted. I have some confusions on the application of convolution. I understand the operation of convolution from pure math standpoint, namely finding the overlapped area of 2 signals, with 1 of them reflected/reversed and shifted from left to right on an xy plane. However, if it is applied to electrical signal over time for example, the impulse response represents the output of the system for an unit impulse as input, specifically, you mentioned that a continuous time input signal can be seen as many scaled unit impulse with infinitesimally small time interval. If we take at each time instant, the ouput is the sum of input signal multiplied by its impulse response delayed by the same amount of time, and the result could be a non-trivial positive number, even at time t slightly greater than 0. However, if we use integration, given the small range in integration interval, the area would be very very small. This is my confusion, can you help explain? Was my question clear?

you are such intelligent guy .Your explanation is awesome.

Thank you so much i get better understanding

Best explanation on RU-vid

Excellent thank you ❤

Sir very clear analysis of important operation convolution

Thank you so much. Way better explanation than UCL lecturers....

i can confirm that this is the best explanation IN THE WORLD!!!!!!!!!!!

It couldn't be better!!

One of the ways that students learn, or demonstrate that they have learnt, is when the teacher asks the student to explain certain features. So here we are in the lecture theatre, Prof Iain has just explained the convolution equation. Iain asks "have you all followed the reasoning and the equation?" We students all dutifully and appreciatively nod an acknowledgement that yes we understand. Ok says Iain, I would like one of you to come out to the board and explain to the class why the convolution equation is a function of time written as y(t) with an integral that is written not with respect to dt, but to d(Tau). Often integrals I have seen written as a function of some variable such as t are determined with respect to dt. I get picked out to try and explain.... Ok I checked my ancient copy of Taub and Schilling "Principles of Communication Systems" and can confirm that within very early pages, the convolution equation as written by Prof Iain is quite correct. To try and explain then - it is to be noted that y(t) is a function of time and there is an expression h(t- Tau) within the integral. With t on both sides of the equation, we can have dimensional stability on both sides of the equation. y(t) will yield a value at any time t. Whilst not written on the vertical axis, y(t) was sketched as the 4th diagram in the lecture. In order to evaluate y(t) at some specific time t, we need to evaluate the sum of all the impulses (delta functions) and their impact on the output to give us a response at that particular time t. But the output at time t depends on many (an infinite number of) delta functions each causing a response which have occurred previously in time and are possibly still continuing to affect the system. So we start at -infinity to account all values of Tau that have occurred (almost like saying time, but not quite) and we look at the impact that an infinite number of delta functions occurring at all Tau's have by multiplying the system response at each Tau as seen at a specific time t. Each value of Tau is infinitesimally close and so are separated by an interval d(Tau). Repeating a little, we take all instances, an infinite number of instances, of delta functions multiplied by the system response which we sum. This continuous sum gives us our integral. I think I passed, and am about to put the pen down and look over to Prof Iain to see whether he approves, but then one of you students calls out and says "What about that infinity upper limit in the integral? How can you sum something or say something that is yet to happen has to be accounted in the output at a specific time t? Is that what you are saying? Shouldn't the limits of integration be -infinity to t rather than -infinity to infinity?" (sigh...) Just when I thought I had passed, this one comes like a curved ball. With my pass now looking quite shaky, I bite into my bottom lip and shake my head. No, sorry, you have me on that one. Can anyone help?

I give like before watching

quick question, when summing the output responses at 4:25 and we say it begins decreasing, can we say it decreases twice as fast after summing the first two responses, and three times as fast when summing the third response, as in the decreasing section of the response, it would be the sum of 3 functions which are all decreasing? Basically im asking if the rate of decrease in the summed functions remains the same in each section. Thanks!

Thank you lain

Superbe.merci

Nice explanation

I think you should say more about x(t) being "identical in its effect on a system" to an "infinite series of short rectangular pulses" with area x(ti)•Dti each. Each of these pulses generates a response very similar to the system's impulse response multiplied by x(ti)•Dti or h(tt-ti)•x(ti)•Dti... Evaluating all these responses per all ti at tt=t and summing these evaluations leads do the total response of the system to x at time t. Taking the limit of the sum as Dti goes to zero results in expressing the sum as an integral - the convolution integral... This is possible if the system for which h(tt) is given is linear and time independent, i.e., LTI.

hey ian! great video. one question, why do we mutiply the input with the IR of a system? is there a reason for us not using sum instead of multiplication for example?

Thanks, Professor for giving these wonderful presentations. I really lacked this during my studies. I have a doubt: When we find the output of an LTI sustem at the particular time t, as you said, it is the sum of all the values of system responses at that particular instant, but y is the limit of convolution from -infinity to infinity? shouldn’t it be -infinity to t logically? Because the future excitations might not have effect on the current output.

man this is like magic

@ Iain Explains Signals, Systems, and Digital Comms - Is the result of product some function with impulse function value of impulse function times value of that function in the interval of impulse function? I draw on Desmos product of these two functions and got a different result than you in 3rd figure.

I finally understood what's going on here, and why it has to be that the 2nd function is inverted and shifted.

I got it, finally! It's the ONLY way of accounting for the "blast from the past" into the output signal. The "flip and delay" story does not work for me. Thank you, Iain!

I wonder if the system is not an LTI, do we also have a way to cover that ?, like say if the spring of your cycle (as mentioned in your intution video), fatigues, my thoughts are, that it can still be tranformed into an LTI system, with a transfer function that is somehow obtained from the fatigue fn and non fatigued transfer function. Btw you just gave me a picture that would help me remember convolution for rest of my life i keep on forgetting it, being out of touch..., thanks a lot...

Best!

Ian a question, why do we do multiplication of x and h? Is it convention? Why not addition?

Which text book to refer for signal and systems

In case if the system is not an LTI, then what can we do to find the output? In a NON LTI system, The convolution with it's impulse response won't work right?

I understand that it must be a continuous sum, but why multiply by d-tau? if the input signal is simply a unit impulse wouldn't the output simply be the product of the impulse and the impulse response?

💯

how does x(0)*h(0), x(1)*h(0), and x(2)*h(0) give you some finite height, doesn't x(0), x(1), and x(2) evaluate to infinite since they are unit delta functions signified by the upward pointing arrow?

I'm missing something obvious - why (at 3:-00) is the result x(0) TIMES h(0) rather than x(0) PLUS h(0)?

Where was this video till now?

explanation is nice but the audio is lagging behind the video. correct this please

I do not what to say! God bless you from here all the way to paradise!