 Welcome to the preview of my next video, Negative Frequency Imagining Numbers and the Complex Conjugate. The full version of the video is currently in production. Ask any mathematician why the Fourier transform of a signal produces a magnitude spectrum that is symmetrical about 0 hertz, and they'll probably answer, because your signal is real. Well of course my signal's real, you might be thinking, I didn't imagine it. The thing is, what you mean by real, and what a mathematician means by real, might not necessarily be the same thing. I'm willing to bet that all the signals you've ever met in your life are real signals. That is to say, their amplitude at any moment in time can be perfectly described by a single number. That number might be in decibels, volts, meters or any number of measuring systems, but it will be a single number that perfectly describes the amplitude of the signal at that moment in time. But the Fourier transform describes how much each frequency contributes to your signal, in terms of the cosine and sine components at that frequency. Take this simple 5 hertz sinusoid as an example. If I break it down into its component cosine and sine waves, they look like this. They both have a frequency of 5 hertz, but their amplitudes are such that when added together, they produce the 5 hertz sinusoid we started with. If I perform a Fourier transform on the signal to transfer it into the frequency domain, the cosine component looks like this. Here's the peak at 5 hertz. But where is the amplitude of the cosine wave in the time domain is 0.8? The amplitude in the frequency domain is only 0.4. And there's another similar peak at minus 5 hertz. Looking at the frequency domain representation of the sine component, something similar happens. At the 5 hertz frequency, the amplitude is half of that shown in the time domain graph. But at minus 5 hertz, the amplitude is minus a half of the time domain amplitude. What's going on here? And where did this negative frequency come from? In the frequency domain, the cosine and sine components at each frequency are represented by a complex number. The cosine component is represented by the real part of the complex number. And the sine component is represented by the imaginary part of the complex number. There are two non-zero frequencies in this signal. 5 hertz and minus 5 hertz. So this signal could actually be represented by just two complex numbers. 0.4 plus 0.3i for the 5 hertz frequency. And 0.4 minus 0.3i for the minus 5 hertz frequency. In order to recover the signal and move from the frequency domain back into the time domain, we need to perform an inverse Fourier transform. When we do this, in the final stage of the calculation, we'll end up with a list of complex numbers that need to be added together. Each complex number represents the contribution of one particular frequency to the signal at a specific moment in time. Most of these numbers will be zero, say for the plus and minus 5 hertz frequencies, as they're the only frequencies that exist in our signal. What is interesting about the complex number representing the contribution of the minus 5 hertz frequency is that it is the complex conjugate of the number representing the plus 5 hertz frequency. When we add these two numbers together, just look at what happens to the imaginary terms. They cancel out. This means that the amplitude of the signal at any moment in time can be described by a single real number. This signal is real, which is why the magnitude spectrum of the Fourier transform for any real signal is always reflected in the 0 hertz line, and why the phase spectrum of any real signal is always reflected in the 0 hertz line and reversed. If this wasn't the case, when added together, the imaginary parts wouldn't cancel out, and the time domain signal would not be a real signal. In the full version of this video, we'll be looking in more detail at how to perform the inverse Fourier transform, what negative frequency actually means, and I'll also be showing you a physical effect of negative frequency in the real world, just in case you thought that it was just a quirk of the maths. The full video is currently in production and will be available in a few weeks' time. Hit the subscribe button and set the bell notification so that you hear the moment it goes live.