Convolution Equation Explained ("Best explanation on YouTube")
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- Опубліковано 26 сер 2019
- Explains the equation for Convolution in a graphical way.
Related videos: (see iaincollings.com)
• Intuitive Explanation of Convolution • How to Understand Conv...
• Convolution in 5 Easy Steps • Convolution in 5 Easy ...
• What is Convolution? And Two Examples where it arises • What is Convolution? A...
• Convolution of two square functions: • Convolution of Square ...
• Convolution of square and exponential functions: • How to do a Convolutio...
• What is an Impulse Response? • What is an Impulse Res...
• How are Correlation and Convolution Related in Digital Communications? • How are Correlation an...
For a full list of Videos and accompanying Summary Sheets, see the associated website: www.iaincollings.com
This is literally the "Best explanation on UA-cam".
No clickbait, pure gold.
Thanks for your endorsement. I'm glad you like the video.
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
Thanks Trent. If there are any other topics you'd like me to cover, let me know and I'll see what I can come up with.
I think … it would need to be a Foster’s Lager. Because, Down Under … Foster’s is Australian for beer. LOL
He is my professor, I will get him one for you
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.
Thanks for your nice comment. I'm glad you have found the videos helpful.
This is really the best explaination on youtube. No fancy graphics or bizarre examples, just a simple solid explanation given in a straightfoward way.
I'm so glad you think so. I especially agree with your comment about other "bizarre examples" seen elsewhere. The ones I'm thinking of just are _not_ examples of convolution. Having said that, you might like to check out my video where I have used a genuine "example" from real life (it's taken me 20 years to think of the ideal "real life" example that really demonstrates the convolution equation, so I understand why others resort to "bizarre examples"): "How to Understand Convolution" ua-cam.com/video/x3Fdd6V_Hok/v-deo.html
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.
Glad it was helpful! That's great to hear. The topic of Convolution is what got me started making videos in the first place. Students always find it difficult, and it's so fundamental and important (and it's usually badly explained).
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.
I'm glad the video gave you a different perspective.
THE BEST EXPLANATION OF CONVOLUTION !!! Your channel is going to get a lot of traction very soon and I have no doubt about it!
Thanks. Glad you like the videos.
This is one of the best convolution explanations on UA-cam 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.
Good point. Thanks. And I'm glad you like the explanation.
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.
Thanks so much for your comment. I'm glad you've found the videos to be helpful.
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!
Thanks for your comment and suggestions. I'm glad you like the videos and thanks for supporting the channel. I hear what you're saying about some people liking to see tech in the presentations. Sometimes that can be helpful, no doubt, but for basic explanations I find it most often distracts from the message. For most of my videos you can find a pdf of the summary sheet at iaincollings.com
@@iain_explains Totally agree with you, Iain. "Teach from home" has added to the number of instructors struggling with graphics tablets, cursors and stylus pens, seriously distracting from teaching. It'll take some time and practice for everyone to be familiar with all the tools new tech offers. On the other hand, there are professors who painfully slowly write by ball pen on paper, and take hours to draw a simple sketch, while the target audience yawns and goes off to sleep.
Very few have found the right mix. You certainly have.
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.
I'm glad you liked the video. You might also like this video which gives even more intuition (if you haven't seen it already): "How to Understand Convolution" ua-cam.com/video/x3Fdd6V_Hok/v-deo.html
Best explanation out there. It makes much more sense when you first approach it with a discrete amount of delta functions.
Thanks. I'm glad you liked the explanation.
Best explanation of the formula indeed that i've seen on YT (and I've seen a lot!) thank you
Glad it was helpful!
Best explanation I could find on the internet, both intuitively and mathematically.
Glad you liked it.
I watched many videos explaining convolution, but you really nailed it! Thanks a lot!
Glad it was helpful!
Thank you so much.
Best Explanation by far.
Glad it was helpful!
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
Glad it was helpful! For an even more intuitive video, you might like: "How to Understand Convolution" ua-cam.com/video/x3Fdd6V_Hok/v-deo.html
Thanks lain, this helped me visualize the purpose of Tau and where the time difference comes from! Hope you're having a fantastic day!
That's great to hear. I'm glad the video helped.
This is the best explaination I have ever seen in giving the convolution formula intuitive sense.
I'm so glad you think so. You might also like this video: "How to Understand Convolution" ua-cam.com/video/x3Fdd6V_Hok/v-deo.html
Love your explanations mate, thank you
Glad you like them!
Thank you professor, you helped me a lot in my preparation for the midterm and finals of my signals and systems course!
That's great to hear. Glad I could help!
Awesome video!!! A much-needed explanation for my Digital Signal Processing revision.
Glad it was helpful!
Thank you sooooooooooo much! Excellent Job!!
I'm glad you liked the video. You might like to check out the other related videos on the channel, which give more insights into convolution, including: "How to Understand Convolution" ua-cam.com/video/x3Fdd6V_Hok/v-deo.html
Wow, what a super explanation!!! The best I have seen online. Thanks
Glad it was helpful!
Best teacher 🙂
The Google should suggest your answer whenever someone types what is convolution 👌
Thanks for the endorsement. I'm glad you are finding the videos useful.
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 UA-cam"
Thanks, I'm glad you like the video. It's always great to hear from people who have studied the topic in the past, and like my videos now.
literally...the best explanantion on yt
Thanks. I'm glad it helped.
Really well explained, thank you so much for helping us to understand it sir!
Glad it was helpful!
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?
I’m so glad you like the video. Sorry, but I don’t understand the point you are trying to make in your question.
@@iain_explains Hi professor Iain, let's assume a unit impulse response of a rectangular pulse of 2V from time t=0 to t=2, and an input signal of rectangular pulse of 1V from time t=0 to t=1. If I understand you correctly, you were saying that a continuous input signal could be considered as many unit impulse (in this case with weight = 1) infinitesimally close to each other. If we measure the output of the system at t=0.1, if I understand you correctly, then it's the sum of x(n)h(t-n), where n starts from 0, and increases infinitesimally small to 0.1, while t=0.1. In this particular case, since x and h are both rectangular pulse of 1v and 2v respectively, so the output would be adding many 1v * 2v together, which would sum up to a large positive value, since there are infinitesimally many of them. For discussion sake, even if we just sample 3 unit impulses at t = 0, 0.01, and 0.02, the output would have been 3 * (1v * 2v) = 6v. However, if we use integration, the integral part of the overlapped signal between x and h is just (1v * 2v) (i.e the height of x * h) multiplied by 0.1 (i,e the width of the overlapped area of x and h, since t=0.1), which integrates to a value of 2 * 0.1 = 0.2. I think my understanding of the relation between unit impulse and the continuous signal is wrong. Could you help clarify my confusion ? I hope my example isn't adding more confusion to my question.
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
I'm glad you've found it useful. Three years is almost no time at all, for such a fundamental operation that describes the operation that happens in all linear systems on the planet. It will still be valuable for many many years to come.
This is a beautiful explanation, thanks for this.
Glad it was helpful!
Fourtunately got this video!!!
hats off to you
Great. Thanks.
The way of your explanation is easy and understandable, you are such an intelligent guy.
Thank you for your very nice comment. Glad you like the videos.
you are such intelligent guy .Your explanation is awesome.
I'm glad you found the video helpful.
Thank you so much i get better understanding
I'm so glad!
very well explained ! thanks !
Glad it was helpful!
great explanation
GOD BLESS YOU MAN
Glad it helped
Yep, it is the best explanation. Thank you
Glad it was helpful!
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
That's great to hear! I'm so glad you're interested to self study these topics. Let me know if there are specific things that you'd like me to make a video on, that aren't already covered on the channel.
Sir very clear analysis of important operation convolution
Thanks. I'm glad you liked the video.
Mind = Blown
Thank you So much Sir 🙏🙏
Glad you found it useful.
Many thanks, great explanation.
Glad it was helpful!
It's amazing that we can learn these things for free. Thank you for your clear explanations 😎
Glad it was helpful!
@@iain_explains Actually you are amazing!!
thank you
Thank you so much. Way better explanation than UCL lecturers....
I'm so glad you found it helpful.
Thank you very much, Iain. I really understand well now. It was confusing at all times. I understand why we are using Convolution 😊😊😊
Glad it was helpful!
Excellent thank you ❤
Glad you liked it.
i can confirm that this is the best explanation IN THE WORLD!!!!!!!!!!!
Thanks so much. I'm really glad you liked it.
Just amazing!
Thanks. I'm glad you liked it. Have you seen my other video with a practical example: "How to Understand Convolution" ua-cam.com/video/x3Fdd6V_Hok/v-deo.html
This is gold. Thanks a lot
Glad you enjoyed it!
Best explanation on UA-cam
Glad you think so!
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?
The equation for convolution does not _only_ apply to causal LTI systems. When you're dealing with a causal LTI system, the impulse response of the system will be zero for negative time, and this will limit the range of the integral in the convolution in the way you have said you expect it to be (ie. not going all the way to infinity). But the convolution equation is more general, and applies for non-causal systems too (eg. for images, where you're not dealing with _time_ but with _pixels_ which can be smoothed in two directions - forwards and backwards). It also applies for probability density functions. See this video for more on that example: "What is Convolution? And Two Examples where it arises" ua-cam.com/video/X2cJ8vAc0MU/v-deo.html
Nice explanation
Thanks for liking
Thank you lain
You're welcome 😊
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!
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?
Perhaps this video will help: "How to Understand Convolution" ua-cam.com/video/x3Fdd6V_Hok/v-deo.html
I give like before watching
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.
Here's a video where I give a bit more intuition along those lines: "How to Understand Convolution" ua-cam.com/video/x3Fdd6V_Hok/v-deo.html
man this is like magic
Glad you've found it useful.
It couldn't be better!!
Thanks. I'm glad you found it helpful.
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.
If you're only considering causal real-time systems, then yes. But the convolution equation holds for non-causal systems too. For example, when filtering an image, the "impulse response" of the filter affects the pixels that are on both sides of the "impulse".
I finally understood what's going on here, and why it has to be that the 2nd function is inverted and shifted.
That's great to hear. Have you seen my other videos on Convolution? This one in particular might help with visualising it: "How to Understand Convolution" ua-cam.com/video/x3Fdd6V_Hok/v-deo.html
@
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.
This video might help: "What is an Impulse Response?" ua-cam.com/video/WTmelRV_Yyo/v-deo.html
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!
Glad it helped.
Superbe.merci
Glad you liked it.
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...
Yes, it can be extended to cover time-varying linear systems. One example of this is with time-varying mobile communication channels. I've just added this topic to my "to do" list.
Ian a question, why do we do multiplication of x and h? Is it convention? Why not addition?
Convolution is not just a maths equation that's part of a convention. It is what really happens in real life. For example, when a signal goes into a linear time invariant system. So we don't "choose" to do multiplication - it's what really happens. These videos might help: "What is Convolution? And Two Examples where it arises" ua-cam.com/video/X2cJ8vAc0MU/v-deo.html and "How to Understand Convolution" ua-cam.com/video/x3Fdd6V_Hok/v-deo.html
Best!
Glad you liked it.
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?
For time varying linear systems, you can still take an impulse response approach to modelling the system, but the impulse response will be a function of time. eg. h(t, tau) where "t" is the time that the impulse is applied to the system, and "tau" is the variable that indexes the response to that impulse, over time.
💯
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?
Excellent point. I didn't want to dwell on this in the video, since it's a mathematical technicality that detracts from the intuitive explanation of the video. However, you're correct, the delta functions are infinite height. But they are also infinitely narrow. So, what does it mean to "multiply" by a delta function? Well it's not really an _instantaneous_ multiplication, since delta functions don't really exist. And a system cannot be _instantaneously_ impacted by an input, because nothing happens exactly _instantaneously_ in the real world. But the resulting function is correct (ie. x(0)h(t) ), and its value is x(0)h(0) at time 0. The number that's written next to the delta function (ie. x(0) for the delta function at time 0) represents the area under the delta function (... it's infinitely high and infinitely narrow, but its area is x(0) ). So it is a measure of the "size" of the delta function (or more intuitively, the "size" of the impact/impulse that is being put into the system, in this case).
@@iain_explains Oh I see thank you that makes more sense. Follow-up questions wouldn't x(t) always equal 1 then if x(t) represents the area under the delta function and a delta function has infinite height and infinitely small width?
There's no reason why it should always equal 1. It is just a mathematically defined function. It can equal any value you define it to be. Infinity (height) times infinitesimal (width) is not well defined as an area. So we can define it to be whatever we like. In general it makes sense to define the impulse response of a system, h(t), to be a function with unit energy ( integral of h(t)^2 over all t = 1 ) and define the corresponding unit input impulse, to be a delta function with area/value =1. This captures the situation where all the energy from the input impulse signal is transferred to the system's output signal.
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?
Hopefully this video will help with the intuition: "How to Understand Convolution" ua-cam.com/video/x3Fdd6V_Hok/v-deo.html
Which text book to refer for signal and systems
I like Alan V. Oppenheim and Allen S. Willsky, “Signals and Systems”. Also, this book is good too: Haykin and van Veen, “Signals and Systems”
I'm missing something obvious - why (at 3:-00) is the result x(0) TIMES h(0) rather than x(0) PLUS h(0)?
The output of a linear system doubles, if the input doubles, ... and triples, if the input triples, etc. It is a multiplicative relationship.
Where was this video till now?
It was waiting patiently 😁 If you're looking for more videos that are waiting patiently to be found, then there's a full list here: iaincollings.com
I do not what to say! God bless you from here all the way to paradise!
I'm so glad you liked the video. Have you seen my new one on the topic? "Intuitive Explanation of Convolution" ua-cam.com/video/x3Fdd6V_Hok/v-deo.html
@@iain_explains Can please some examples how to make calculations using MATLAB.
Thanks for the suggestion. I'll put it on my "to do" list.
@@iain_explains Thank you :)
@@iain_explains In which country are you living?
explanation is nice but the audio is lagging behind the video. correct this please
The video is fine. It must have been your operating system causing the issue.