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### Course: Differential equations>Unit 3

Lesson 4: The convolution integral

# Introduction to the convolution

Introduction to the Convolution. Created by Sal Khan.

## Want to join the conversation?

• why at 10.10, can Sal change the integral with respect to 'tau' to an integral with respect to 'u', but not change his limits?
• Because the substitution was only temporary. He switched back from u to tau at after the integral was done, and then evaluated them with tau-related limits ;)
• i would like to know more about the convolution on discrete data rather than the continuous functions
• In the video f(t) = sin(t) and the convolution of f(t) is Integral[sin(t-tau)].
My question is this: If I have f(t) = sin3(t) would the the integral of then be Integral[sin3(t-tau)]
• Yes, You introduce (t-tau) where t is:
Right: cos (3t+2) ->>cos (3*(t-tau)+2)
Wrong: cos (3t+2-tau) Don't do this!
• So, convolution is commutative? It'd would be helpful if you would go through the properties of the operation :)
• Yes

Proof: (note - I'm using S(x=A,x=B) f(x) dx to represent the integral of f(x) from A to B.)
let u=t-τ, so τ=t-u and du=-τ, then
(f*g)(t) = S(τ=-∞,τ=∞) f(t-τ)g(τ) dτ
= S(t-u=-∞,t-u=∞) f(u)g(t-u) -du
= -S(u=∞,u=-∞) g(t-u)f(u) du
= S(u=-∞,u=∞) g(t-u)f(u) du
= (g*f)(t)
(1 vote)
• Are tau and t both a variable? I think Sal need to define this the very first time he introduce the formula so that we will not be confuse the denotations used in the formula.
• They both are variables but we are interested in taking the integral only for tau so we assume that t is a constant. In fact, t is our independent variable. It should be obvious to see which are variables. I think Sal is clear about it.
• I have a question about the definition of convolution. Why would that integral be chosen as the definition of convolution? What's so special about that integral? I can follow the algebraic computation, but it's like someone tells me that a piece of paper falls from the sky and the definition of convolution was written on the paper; therefore, we need to just accept it. I dislike learning something without understanding the reason behind it. Thanks in advance.
• Computationally fantastic....but visually....what is a convolution?

Does this have anything to do with the fact that tau = 2pi = a full circle?
• Very unlikely. The use of the symbol tau for 2pi, I believe, is only a relatively recent idea; in 1958, Albert Eagle suggested tau=1/2 pi , and later the concept of tau=2pi became popular. The convolution has existed since long before that. Also, in your journey in mathematics, you will see that greek letters are used to denote entirely different things, so its more likely a coincidence.
• On Wikipedia (and in my textbook), the convolution integral is defined somewhat differently - it has minus infinity and plus infinity as integration limits. Of course, if the integrand is zero when tao is not in [0, t] the integration limits are reduced to 0 and t. But here, this is not the case so why does Sal define convolution in this way?
• Sal said that f(t) = sin(t) and g(t) = cos(t). However, this wasn't quite right.
What he meant to say is
f(t) = sin(t) for t>=0 and 0 for t<0
g(t) = cos(t) for t>=0 and 0 for t<0
Thus, the integrand is zero when 0>τ>t and the limits of integration reduce to 0 and t.
(1 vote)
• How can you show that:
[f1(t) e^-i2(Pi)f0t]* [f2(t) e^-i2(Pi)f0t] = [ f1(t)*f2(t)] e^-i2(Pi)f0t ?? I dont even know what functions are f1(t) and f2(t). Thanks
• nice video, but do anybody knows What’s a Convolution Reverb? Is the topic related?
• Convolution reverb does indeed use mathematical convolution as seen here!

First, an impulse, which is just one tiny blip, is played through a speaker into a space (like a cathedral or concert hall) so it echoes. (In fact, an impulse is pretty much just the Dirac delta equation through a speaker!) Then, this echoed impulse is recorded to create a digital "picture" of the way that space echoes. Let's call this echoed impulse signal g(x).

When we have a "dry" audio signal (which we will call f(x)) and put it into the convolution reverb device or plugin, it mathematically convolves the two signals into a "wet" audio signal with the reverb in effect. That's f ∗ g of x! And these devices and plugins often come with pre-recorded echoed impulses, so half the work is done for you. it's as if you can take your dry audio to a cathedral, concert hall, or echo chamber and back without ever setting foot there or even knowing where it is and whether it exists! Yeah, books can take your imagination anywhere, but they've got nothing on convolution 'verb.

So you would absolutely need to know how convolution works if, say, you were working with an audio electronics company on creating a convolution reverb pedal or plugin. That's the theoretical muscle behind it all.

https://en.wikipedia.org/wiki/Convolution
https://en.wikipedia.org/wiki/Convolution_reverb
(1 vote)