 so you've just run your FFT and here is a graph of the magnitudes of each sinusoidal signal. The first thing you notice is that it's symmetrical about this center frequency. Is there something wrong? How comes your signal contains this symmetrical pattern of frequencies? This isn't the fault of your signal, this is a consequence of the FFT sampling your signal. Your signal is now discreet. That is to say we only know where your signal is at certain discrete points in time. Now you see it, now you don't. What is more? This graph tells us something very important about how we must sample the signal if we're going to be able to reproduce it accurately. Let's find out why this symmetry happens and what it means. Hi I'm Mark Newman and this channel is all about helping you understand the fascinating world of signals and systems. When I started using the FFT I got really confused about its output. No matter which signal I tried the magnitude graph was always symmetrical. This can't be I said to myself. I must be doing something wrong. None of my signals can be this tidy. To answer why this happened we need to think about the nature of our signal and what has happened to it from the moment we recorded it. The important thing to remember about the fast Fourier transform is it is simply a more efficient version of the discrete Fourier transform. The discrete Fourier transform is exactly that. Discrete. The signals it represents are sampled. This means that the information we have about the signal only exists at certain discrete points in time. Between those points anything could happen. We have absolutely no data about what is going on. In the time between the samples as the DFT and the FFT work in a similar way what is true for the DFT must also be true for the FFT. Throughout this series on the output of the FFT we've seen how the FFT tests the signal with sinusoids of different frequencies. There's a link to the playlist in the description below. To understand what effect a discrete signal has on this process let's first remind ourselves of how the DFT works. Let's experiment by taking the DFT of a simple 2 Hertz cosine wave. I'm keeping the signal simple for the purposes of this demonstration but the same is true for any signal. This particular signal was sampled at 16 Hertz and there are 16 samples in the signal. The DFT will therefore test 16 different sinusoids at intervals of 1 Hertz. The first sinusoid it will test will have a frequency of 0 Hertz. The second will have a frequency of 1 Hertz. The third 2 Hertz and so on all the way up to 15 Hertz. For each frequency it will test the signal with a cosine wave and an inverted sine wave. Everything we're about to do we do for both these waves so I've divided the screen in two with the left half of the screen showing the cosine component and the right half of the screen showing the sine component. Testing the signal means that the DFT will multiply the signal by the test wave. Here's the result for each component. Now it adds together all the points on the multiplied graph to get a score which indicates how much each component contributes to the signal at this frequency. Combining these scores using Pythagoras theorem tells us the magnitude of this frequency in the signal as we learned in the last video. I've linked to the video in the description. Now for this first frequency 0 Hertz we get a score of 0 for both the cosine component and the sine component. This produces a magnitude of 0 telling us there is no 0 Hertz component in this signal. Repeating the process for 1 Hertz produces another 0 result. However when we test a 2 Hertz frequency the cosine component is not 0. You can see how all the multiplied points are positive producing a positive score. This means that 2 Hertz is a frequency present in our signal. The sine component is till 0 though. This tells us that cosine component is contributing all the energy to the signal at this frequency. Testing 3 Hertz, 4 Hertz, 5 Hertz etc gives us 0 again as these frequencies are not present in the signal. This is fine for frequencies up to 8 Hertz but then something strange happens the moment the frequencies we test get higher than 8 Hertz. I'm going to demonstrate what I'm talking about with a 14 Hertz test wave. You'll see why in a second. When I test the signal with a 14 Hertz wave the cosine component gives exactly the same score as it did when I tested a 2 Hertz frequency. Looking at the multiplied graph we can see why. There's no difference between the 2 Hertz frequency and the 14 Hertz frequency. The points are all the same. To find out why this is we need to look at the signal and the 14 Hertz test wave. Remember the signal is sampled and we only perform the multiplication at the points where information about the signal exists. At 14 Hertz the test wave occupies exactly the same sampled values as the 2 Hertz test wave. The fact that the 14 Hertz test wave has oscillated between the samples makes no difference to the discrete Fourier transform as anything can happen between the samples and the DFT wouldn't know about it. The information about both waves only exists at the points where they are sampled. The same thing happens for each pair of frequencies around the center frequency of 8 Hertz. So it is true for the 7 Hertz and 9 Hertz frequency, the 6 Hertz and 10 Hertz frequency, the 5 Hertz and 11 Hertz frequency and so on. All the frequency pairs where the samples for each frequency in the pair occupy the same values. Our signal doesn't contain any frequency component at these values so they come out as zero which is why I jumped directly to 14 Hertz for this demonstration. This is the reason why the magnitude output of the FFT is symmetrical about this center frequency which is a very special frequency called the Nyquist rate. The Nyquist rate is always half the sampling rate, 8 Hertz for our signal which was sampled at 16 Hertz. When reconstructing the signal from the samples like a digital to analog converter does any frequencies in the signal that exceed the Nyquist rate will get reflected back into the lower half of the signal's frequency spectrum in a phenomenon known as aliasing distorting the signal. This is especially noticeable in audio recordings. Here's what it sounds like. Aliasing occurs because there aren't enough samples in the signal to represent the higher frequency just like our 14 Hertz and 2 Hertz example from before. Once sampled both frequencies look exactly the same therefore the higher frequency gets aliased given the value of the lower frequency of the pair. So if you want to be able to reconstruct that signal accurately you have to sample it at a rate that is at least twice the highest frequency in the signal. This is why for example CD quality audio is sampled at 44.1 kilohertz. The highest frequency we humans can hear is about 20 kilohertz therefore CD quality audio is sampled a little above 40 kilohertz so that any 20 kilohertz sound can be faithfully reproduced. This means that before sampling a signal you have to make sure to set your sampling rate high enough to capture the entire range of frequencies you are interested in and filter out any frequencies higher than the Nyquist rate so that they don't appear in the sampled signal. While we've spent this video talking about the symmetry in the output of the FFT above zero Hertz we've neglected to mention another symmetry inherent in the Fourier transform. You're probably used to seeing Fourier transforms for a signal represented like this. This is a representation of the 2 Hertz cosine wave we've been using as the signal throughout this video. You'll notice there is a negative frequency component at minus 2 Hertz as well as the 2 Hertz component we already calculated. It is if the upper half of the FFT output has been ripped off the graph and stuck on the other side of it. But frequency means the number of oscillations per second. How can you have negative frequency? The number of oscillations cannot be negative and time can't flow backwards either, can it? So here's your challenge for this video. Why does the Fourier transform of a signal look like this? And why this negative frequency? We'll find out in the next video. I'll just leave you with one clue though. It has to do with the complex conjugate. You'll find more on imaginary numbers and the complex conjugate in chapters 4 and 5 of my book How the Fourier Series Works. The link is in the description below.