 So you feed your signal into FFT black box, what you get as your FFT output, a list of numbers looking something like this. Where's my frequency information? How am I to understand this gobbledygook? The information you want is actually here, contained within this strange list of numbers. The position of each item in the list tells us its frequency. Let's find out how. Hi, I'm Mark Newman and this channel is all about helping you understand the fascinating world of signals and systems. The FFT or Fast Fourier Transform is simply a more efficient way of calculating the DFT or Discrete Fourier Transform. The DFT is a Fourier Transform for digital signals, and the Fourier Transform itself is a development of the Fourier series, enabling it to model non-repeating signals. The principle behind the Fourier Transform in all its forms is always the same. It assumes that your signal is built out of sinusoids. Apart from a couple of theoretical exceptions, in the real world this is a pretty good assumption. Before it tests your signal with a collection of different sinusoids until it finds which sinusoids fit your signal. It's a bit like trying out a lot of keys in a lock until you find the one that will open the door. However, unlike my key analogy, your signal will most likely be built out of more than one sinusoid, so more than one wave will fit into your signal. This is why you get a list of results. Each line in that list represents a different sinusoid. Sinusoids have three properties, frequency, magnitude and phase. If the Fourier Transform is to find out which sinusoid your signal is made out of, it needs to test sinusoids with similar properties to the ones in your signal. If a sinusoid with a certain frequency does exist in your signal and you plot the output of the FFT as a graph of frequency against magnitude, this sinusoid will be represented by a line on the graph, whose height represents the magnitude of the sinusoid at this frequency. If a sinusoid with a certain frequency doesn't exist in your signal, then the magnitude will be zero for this frequency. When it starts out, the Fourier Transform has no idea which sinusoid your signal is built out of, so theoretically it needs to test an infinite number of different sinusoids to cover all the different possibilities. Obviously, we cannot do this in the real world, especially if our Fourier Transform is going to be fast. Therefore, we have to be a little more selective and only test a limited number of sinusoids. How does the Fast Fourier Transform know which sinusoid is to test? As we are working with digital signals, this has to do with how often you sample your signal, known as the sampling rate, and also how much of your signal you are feeding into your FFT, known as the FFT size. If you've ever run a Fast Fourier Transform, you'll notice that the number of items in the list of results is exactly the same as the FFT size. Let's call this number N. So it's only going to test N different sinusoids. The first sinusoid it will test is one with a frequency of zero. The last sinusoid it will test will have a frequency just one jump less than the sampling rate. I'm going to give the sampling rate the symbol R. The rest of the frequencies are divided up into equal jumps. Each test frequency is given an index K, where K equals 0, 1, 2, 3, etc. This index is the position of the result in the FFT output list. So the position of each item in the FFT output tells us its frequency. To work out what frequency this is, we divide this position in the list K by the total number of items in the list N and multiply that by the sampling rate R. Let's do an example. Say I wanted to find the frequency of the second item in the list. This has an index K of 1. Remember, K starts from 0. I sampled the signal at a sampling rate R of 50 kilohertz and there are 1024 samples in my signal. So N equals 1024. Plugging these values into the equation, we find that the frequency of this sinusoid in the list is 48.83 hertz. We now know that the position of each item in the list tells us its frequency. But sinusoids have three properties, frequency, magnitude and phase. So here's your challenge for this video. If the output of the FFT gives you a list of all the sinusoids present in your signal, how do we work out the magnitude and phase for each sinusoid? We'll find out in the next video. As I mentioned before, the fast Fourier transform is a development of the Fourier series and makes use of many of the same principles. Check out my new book for a step-by-step intuitive and visual explanation of how the Fourier series works. The link is in the description below.