Convolution Filters

Convolution filters

So the convolution theorem well, actually, before I even go to the convolution theorem, let me define what a convolution is.

What's the convolution of these two things?

Remember, I was doing this in the context of convolution, so

So we just showed you that the convolution if I define

We're taking the convolution of sine of t with cosine of t.

And to kind of give you that comfort, let's actually compute a convolution.

But anyway, let's just try to get this in terms of a convolution integral.

The Gaussian minus exponential distribution is a convolution of a normal distribution with (minus) an exponential distribution.

So the convolution of sine of t with cosine of t is 1 2t sine of t.

In mathematics, the Dirichlet convolution is a binary operation defined for arithmetic functions it is important in number theory.

So hopefully, this second example with the convolution to solve an inverse transform clarified things up a little bit.

So the convolution of f with g, and this is going to be a function of t, it equals this.

And you saw Ty's little evolutionary example where he sort of did a little bit of evolution on the Convolution program right before your eyes.

So the inverse Laplace transform of that first term, alpha over s squared, plus alpha squared, convoluted with I'll do a little convolution sign there.

But in this video I just want to make you comfortable with the idea of a convolution, especially in the context of taking Laplace transforms.

Of this let me pick a nice color of this blue thing in the blue box, and then I can express it as a convolution integral.

And now the convolution theorem tells us that this is going to be equal to the inverse Laplace transform of this first term in the product.

And in this video, I'm not going to dive into the intuition of the convolution, because there's a lot of different ways you can look at it.

In this video, I'm going to introduce you to the concept of the convolution, one of the first times a mathematician's actually named something similar to what it's actually doing.

I can independently take the inverse of each of these things, so the inverse Laplace transform of their products is going to be the convolution of each of their inverse transforms.

So if I convolute f with g so this means that I'm going to take the convolution of f and g, and this is going to be a function of t.

So, hopefully, you have a little of intuition of well, not intuition, but you at least have a little bit of hands on understanding of how the convolution can be calculated.

Now that we know a little bit about the convolution integral and how it applies to the Laplace transform, let's actually try to solve an actual differential equation using what we know.

So if I can figure out the inverse Laplace transform of this, I could at least express our function y of t at least as a convolution integral, even if I don't necessarily solve the integral.

It has a lot of different applications, and if you become an engineer really of any kind, you're going to see the convolution in kind of a discreet form and a continuous form, and a bunch of different ways.

This is the radius of the circular convolution. It is the most important parameter for using this plugin. For most images the default value of 1.0 should give good results. Select a higher value when your image is very blurred.

If I have the product of two Laplace transforms, and I can take each of them independently and I can invert them, the inverse Laplace transform of their product is going to be the convolution of the inverse Laplace transforms of each of them, each of the terms.

And the whole reason why I went through all of this mess and kind of bringing out the neurons that had the trig identities memorized or having to reproof them or whatever else is to just show you that this convolution, it is convoluted and it seems a little bit bizarre, but you really can take the convolutions of actual functions and get an actual answer.

And the definition of a convolution, we're going to do it over a well, there's several definitions you'll see, but the definition we're going to use in this, context there's actually one other definition you'll see in the continuous case, is the integral from 0 to t of f of t minus tau, times g of t let me just write it sorry, it's times g of tau d tau.

So if we make this subsitution, this you'll find on the inside cover of any trigonometry or calculus book, you get the convolution of f and g is equal to I'll just write that f star g I'll just write it with that is equal to the integral from 0 to t of, instead of sine of t minus tau, I'm going to write this thing right there.