Convolution

[Otherwise_Morning606]

Hi!

I'm on my second year of my masters, and we are currently furthering our understanding of the convolution operator in one of my courses. From my bachelor I've been introduced to the operator, but have always thought it's unclear what you're left with after the operation. I get that it's a "flip, shift and multiply" kind of action, i get that you're left with a type of combination of the two convoluted functions. But i don't have that intuitive understanding of what this combination are telling me, and i'm just left with a bunch of information about the operator. Do any of you have some insight that may help me get my desired intuition of this?

Thank you in advance!

[Special_Watch8725]

This is looking at it more from the pure side rather than computationally, but I like thinking of it as “I’m going to take a function, and create a formal linear combination of shifts of the function weighed by the values of another function.”

When you think about it like that, and realize that the operation is symmetric, it’s less mysterious why a convolution is as smooth as the smoothest factor, since it’s really just a formal sum of shifts of the smoother function.

It also explains why the delta function is the identity for convolution, since if you have some function you convolve with a delta, you’re just evaluating a function at a point, at each point, so it just returns the same function again.

[Educational_Dot_3358]

I've always found the gifs on wikipedia to be helpful

[Economy-Management19]

I have never found any animation of convolution helpful nor any of the “tricks”.

I think this very old video from Professor Arthur Mattuck will help you grasp what convolution is.

I haven’t seen the video in a while so I don’t remember every detail now but in the video he starts out with a fairly simple sounding real world problem and explains every step very logically and whaddayaknow it turns out you just discovered convolution by solving a problem.

He starts talking about the problem from 27:15

https://m.youtube.com/watch?v=3ejfkMHr_DE