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Multiplying Apples

“So how do you go about teaching them something new? By mixing what they know with what they don’t know. Then, when they see in their fog something they recognize they think, “Ah, I know that!” And then it’s just one more step to “Ah, I know the whole thing.” And their mind thrusts forward into the unknown and they begin to recognize what they didn’t know before and they increase their powers of understanding.”
-Picasso

Preface

This article is the first part in a series of articles exploring convolution, and its application in Mathematics and Engineering.

Introduction

I was first exposed to convolution as a second-year undergraduate in a Linear Circuit Analysis class. I use the term expose because the experience was akin to an physical assault. I was presented with an incomprehensible concept with a name that seemed to mock me as I struggled to understand it. However, as with most things, the barrier of impenetrability was pierced, and now convolution sits as one of the most useful and elegant tools in my engineering toolkit.

Eq. 1: Continuous-time Convolution

My motivation to write this piece is to distill the knowledge I’ve acquired over the years. I’ve seen convolution pop up and out of nearly every field of engineering that I have studied. The concept that it presents is suffused into the very foundations of mathematical thought.

Eq. 2: Discrete-time Convolution

In fact, I can safely bet that you already know convolution. You’ve used it hundreds of times since it was introduced to you in middle- or high-school; an unnamed piece of a larger framework. I’ll even go so far to say that it comes as easily to you as adding 2 + 2. Yes, with that statement, we begin.

The very beginning

Convolution in signal processing and circuit theory

Convolution in probability theory

Convolution in Photoshop

Convolution meets Owsley

Correlation