 Convolution is a mathematical way of combining two signals to form a third. It's how the Fourier transform does what it does. But to demonstrate what convolution is and why the Fourier transform uses it, here's a more visual example. Imagine you were looking for me in a crowd of people. How would you find me? Assuming you know what I look like, you could slide the image of me you have in your memory over all the people in the crowd and compare the two images. When they converge, you will have found me. You've just performed a convolution. Now it's easy for you to see with your eyes when the two images converge. But how would a computer do it? And what does convolution got to do with the Fourier transform? Hi, I'm Mark Newman and this channel is all about helping you understand the fascinating world of signals and systems. If, like me, you find all the maths involved just a load of Greek, then I'm here to help you break through that barrier so you can stop worrying about the equations and start dreaming about your next invention. When I was at university, my signals and systems lecturer began the course by teaching us about the Fourier transform. He explained that the Fourier transform did what it did by convolving one signal with another. He went on to explain convolution as the integral of the product of two functions after one is reversed and shifted and attempted to clarify this with the equation Well, that. As if that would explain everything. It didn't. Convolution and the Fourier transform would remain a mystery to me for the rest of my university career. In these videos, I want to make sure that you don't have to go through the frustration I had to for I firmly believe that complicated mathematical concepts can be explained simply in a visual and intuitive way. So let's try to visualize what's happening in this equation and relate it to the Fourier transform. f of tau is an arbitrary function. A function is another word for a signal. So let's generate a signal. Here's my signal f of tau. g of tau is another arbitrary signal. But what signal should we choose? The Fourier transform assumes that all signals are built out of sinusoids. So why do we make g of tau a sinusoid 2 and see what happens? I'll plot g of tau on the same graph so you can see the two signals together. So what does tau mean? tau is what is known as the independent variable. In other words, it is tau that makes the signal do what it does. So we plot it on the x-axis. Time is an example of an independent variable. So let's make the x-axis the time axis. So tau is time in seconds. But in the convolution equation, we don't have g of tau. We have g of t minus tau. As we said, the independent variable in this signal is time. This means that for this signal, t is some offset in time. As t changes, we are sliding the signal g of tau along the x-axis. Why is it g of t minus tau and not g of tau minus t? Because for reasons I don't want to go into here, the process of convolution requires that the signal g of tau is reversed, becoming g of minus tau. But our g of tau is a sinusoid. If we reverse it, it looks exactly the same, because sinusoid is symmetrical. f of tau stays still while g of tau slides over it. As this happens, the two signals are multiplied together. Let's plot this on a second graph. As t changes, the graph of the product of the two signals moves up and down over the x-axis. The next part of the equation tells us to integrate. Integration means finding the area under the graph. Let's plot the result of the integration on a third graph. As t changes and the two signals move closer together, the area under the multiplied graph increases, meaning that the result of the integral increases too. When g and tau and f of tau converge, the result of the convolution reaches a maximum value. We could use this value as a sort of score to tell us how well matched g of tau and f of tau are. Now here's why convolution is such an integral part of the Fourier transform. Remember that the Fourier transform transforms a signal from the time domain into the frequency domain. In other words, assuming, as it does, that any signal can be built out of sinusoids with different frequencies, the Fourier transform tells us which frequencies are present in the signal. In this example, g of tau just happened to be a sinusoid with the same frequency as the fundamental frequency in f of tau, 10 hertz. But what if we tested a sinusoid with a slightly lower frequency, say 9 hertz? This time, the result of the integral reaches a slightly smaller peak sooner than it did before. How about if we test a very low frequency, say 1 hertz? The result of the integral is always 0, no matter how much I slide it over the signal. So if we were to keep changing the frequency of the sinusoid we called g of tau, then we could find the maximum score for each frequency and plot a graph with the frequency on the x-axis and the maximum score on the y-axis. This is the Fourier transform for this signal. However, convolution gives us more information than just which frequencies are present in the signal. While the maximum score for each sinusoidal component is proportional to its magnitude within the signal, the value of t when the score is at its maximum is proportional to its phase. With convolution being the method that the Fourier transform uses to tell you which sinusoid are present in your signal, it's hardly surprising that the convolution equation and the Fourier transform equation share certain similarities. So here's your challenge for this video. Which term in the Fourier transform equation is equivalent to the f of tau term? And which term is equivalent to the g of tau term? You'll find the answer in the next video. When we explore how the imaginary number i helps the Fourier transform take a convolution shortcut.