 The Fourier transform. What's it for? What does it do? And how does it work? We use the Fourier transform to tell us what a signal is made out of. Just as this car can be built out of individual Lego bricks, each brick with its color, its shape, and its position within the car. So signals can be built out of individual sinusoids with properties like their frequency, amplitude, and position in the signal, or phase. But can a complicated signal like the sound of my voice really be built out of something so basic as a collection of sinusoids? Let's begin with a single sinusoid. Here it is in the time domain, and here it is in the frequency domain represented as a single line. The height of the line on the y-axis tells us its magnitude, and the position of the line on the x-axis tells us its frequency. Doesn't sound much like my voice, does it? Adding in more sinusoids at higher frequencies changes the sound, but it still doesn't sound much like my voice. Let's add in some more sinusoids around the frequencies already playing. Now it's starting to sound interesting. Let's add some more. And some more. As I add more and more sinusoids, it begins to sound more like my voice, until Fourier transform. I've just built my voice out of individual sinusoids. I've performed an inverse Fourier transform. So why is the Fourier transform so useful? Once you know which sinusoids a signal is built out of, there is no end to what you can do with that information. For example, you can compress it, making it easy to store or send over a network. You could filter it and get rid of frequencies you don't want. You could process images with it, predict future events with it, measure distances with it. Every day, new ways are being found to make use of the information the Fourier transform gives us. In this series of books, we study each of the core concepts of the Fourier transform visually and in detail, until you have an intuitive and comprehensive understanding of how the Fourier transform works. In how the Fourier series works, we study the fundamentals of Fourier's original theory and how he broke down repeating signals into individual sinusoids. In how the Fourier transform works, we study how Dirichlet and others turn the Fourier series into the Fourier transform and beyond, expanding its capabilities to model non-repeating signals, such as the diverse and practical signals we encounter in our daily lives. By the time you finish reading these books, not only will this rather strange-looking equation, the Fourier transform equation, actually make sense, but you'll also understand how the fast Fourier transform works and how to use it to analyse almost any signal you'll ever meet.