Convolution Integral Example 03 - Convolution of Two Triangles 

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I study the same thing, good luck bro x) #EngineeringPower

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For (1+ t) consider x(t) = t : it's a straight line, we time shift this to the left by 1 to get the line x(t) = t + 1, or = 1 + t. Similarly for the 1 - t line; x(t) = -t, but time shifted to the right: x(t) = -(t - 1) = 1 - t. When we convolve the triangles together we replace t with T (can't write tau here so I'll use capital T), giving 1 + T, 1 - T. The other triangle we are time shifting about it's centre t: consider x(T) = 1 + T: here t is 0 because it's simply where it started on the origin. If we change this t to 'slide' them together we also change the equations in time. So we end up with x(t - T) = 1 + T - t and x(t - T) = 1 - T + t. I hope I've managed to explain that in a way you can follow.

right at 2:34, were the intercept is a function of t. Let's draw pictures, triangle with t to left of zero, let's say 1/2, so the peak is at 1, the slope is -1 and the intercept is 1/2. I can't see this analytically but as he says, when t is - 1/2, the intercept will be 1/2, which is 1 PLUS (-1/2). So the downward side is - tau + 1 PLUS t. On the upward side: at t=0, it's tau +1. Let's shift is right, make the peak t=1/2. Now the peak is 1 and the intercept is 1/2. We moved it right, from t=0 to t=1/2 and the intercept went down. So the left side is tau + 1 MINUS t. This has been driving me nuts.

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Sure thing, that would be totally fine. I was mainly focusing on the convolution steps in this video, and not so much on how to simplify the resulting terms. Thanks for watching!

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yes please, if you solved can you show it

You should definitely know how to integrate if you are doing convolutions though

As the shapes slide by each other, anytime a new boundary is encountered the product of the equations changes. So, you have to change cases every time a new product is encountered. Definitely takes some practice!

Adam Panagos But how do you decide t+1

After you've sketched the time-reversed and shifted signal, you "slide" that signal until it collides with the original signal. The front edge of the time-reversed and shifted signal is always at t+1. The back edge of the original signal is at -1. So, the first case occurs for all values of t such that t+1 < -1. It's all about sketching those pictures and keeping track of when the signals "collide" with each other. Hope that helps, Adam

Adam Panagos Ok I understand thank you so much!

No, the equation of the line on the left side of the triangle is still a line with a positive slope, the labels are correct.

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Convolution in time domain, and normal multiplication in Frequency domain for your question. And it is also reciprocal, suppose you have to convolve between two functions in frequency domain, then it is multiplication in time domain.

Aha okay well that's understandable thanx, well now I remember that the handbook also has that principle written down

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+allan rocha Thank you! I appreciate the nice feedback.

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Yes, that's handled in case 4 when t = 0. At this point, they completely overlap.

+Mohammad Nadeem That's definitely a completely valid way to approach the problem. I was just trying to work through each part of the computation to demonstrate how to explicitly evaluate (and understand) the convolution integral. Hopefully I ended up with the correct final answer! Adam

+Gustavo Alejandro Malvaez Olguin Unfortunately, that's just one of those things that takes practice to figure out. In general, anytime the part of the signal you're dealing with changes equation (e.g. positive slope line to negative slope line for example) or the signal you're multiplying against changes equation then you'll need a new case.

+Adam Panagos Thanks a lot! then, I need more practice! :D

+Gustavo Alejandro Malvaez Olguin Yup, that's just the way it goes sometimes. Definitely some tricky stuff. Hang in there!

Can you recommend some books or articles that will help me to practice convolution? :)

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+Mete Yildirim I use an iPad app called Doceri (www.doceri.com) for most of my videos. This app lets you record all your handwriting ahead of time and use "breakpoints" to pause as needed. Once all the writing is down you can "play" the handwriting back while recording audio over it. I find this works much better than trying to write and talk at the same time. I'd definitely recommend checking out the app, I've found it very useful. Hope that helps!

Mainly just the name; great-grandfather came to the US from Greece so 1/8th Greek here. Thanks for watching and good luck with your studies! Best, Adam

+Kai Sheng Wang You're welcome, glad to help.

At 2:23 we had the line 1+tau. The substitution we performed was replacing -tau with t-tau. So, 1+tau = 1-(-tau), and we replace -tau with t-tau to yield 1-(t-tau) = 1-t+tau = 1+tau-t as we had in the video. Same thing on the other line. We had 1-tau and we replace -tau with t-tau, so we have 1+(t-tau) = 1-tau + t. Hope that helps. Adam

@@AdamPanagos I really apreciate your care to answer , your answer makes sense now , so i presume that what i did is wrong ( i took the original equation of that ramp where the varibale was just " t " and since what we did was fliping and shifting so, t => t - tau , and the equation was 1+t from the left ) i watched another video just because of that part where finding the new equations and a guy just substitued in the original equation "t" by "t - tau" , i got your answer, but it's still not clear if i try some other examples , thank you .

In the video titled " convolution example 02" at 2:05 you have substituted "t" by " t - tau" in the original equation , to get the new equation of the " flipshifted " signal , which is confusing me a lot .

@@ghilashamaili2733 No, the video is correct. When doing convolution, we replace "t" with "tau-t". When doing this, I feel it is helpful it draw each step of this transformation so one can keep track of where the bounds of the final integral will be. That's why I first sketch x(tau), then x(-tau), then x(t-tau). Sketching x(-tau) is just the time-reversed signal on the tau axis. Sketching x(t-tau) next just takes our time-reversed signal x(-tau) and shifts it by an amount t. So, in this case, I just keep track of where the time-origin tau = 0 gets shifted to. It will always get shifted to t. The thing to be careful with is that since we've already done a time reversal, shifting by "t" moves the signal in the opposite direction "from normal". For example, if we have the signal x(t), and then plot x(t-4), this shifts us 4 units to the right on the t axis. The signals x(t+5) shifts us 5 units to the left. However, if we're working with x(-t), shifting 4 units to the right means replace t with t-4, which would be x(-(t-4)) = x(-t+4). So, when comparing x(-t) to the signal x(-t+4), the signal x(-t+4) has actually been shifted 4 units to the right. The plus sign "+" on the 4 might make you think it's been to the left, but there is a minus sign on the t. You just have to be careful to keep track of whether we're dealing with a "normal" signal, or a "time-reversed" signal. If we've time-reversed already, that reverses the direction that we shift for some given + or - value of the shift. Hope that helps, Adam

@@AdamPanagos thanks a lot , shifting a reversed signal was the issue , thanks again.

I use an iPad app called Doceri (www.doceri.com) for most of my videos. This app lets you record all your handwriting ahead of time and use "breakpoints" to pause as needed. Once all the writing is down you can "play" the handwriting back while recording audio over it. I find this works much better than trying to write and talk at the same time. I'd definitely recommend checking out the app, I've found it very useful. Hope that helps!

Convolution is the fundamental relationship between the input signal and output signal of a linear time-invariant system. It describes how to compute the output of a system, given the input signal and the impulse response of the system. It's used everywhere. Watch the other videos on my channel to see more details.

@@AdamPanagos Thanks a lot.