 This is a 440 Hz tone. The A above middle C for any musicians amongst us. This is a minus 440 Hz tone. They sound the same, don't they? That's because they are. At least when you think of frequency as how many times an event, like a vibration in the air, occurs every second. However, if you look at frequency like this, then the concept of negative frequency becomes a little easier to understand. In this video, we're going to find out what negative frequency is, why negative frequency imaginary numbers and the complex conjugate are so important to the magnitude spectra of signals, and what effect negative frequency has in the real world, just in case you thought it was only a quirk of the maths. Ooh, we'll be taking a quick look at the inverse Fourier transform too. Hi, I'm Mark Newman, and this channel is all about helping you understand the fascinating world of signals and systems. Ask any mathematician why the Fourier transform of a signal produces a magnitude spectrum that is symmetrical about 0 Hz, and they'll probably answer, because your signal is real. Well, of course it'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, may not 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 given moment in time can be 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. But the Fourier transform describes how much each frequency contributes to an overall signal in terms of the cosine and sine components at that frequency. Take this simple 5 Hz 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 Hz, but their amplitudes are such that when added together, they produce the 5 Hz 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 Hz. 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 Hz. Looking at the frequency domain representation of the sine component, something similar happens. At the 5 Hz frequency, the amplitude is half of that shown in the time domain graph. But at minus 5 Hz, the amplitude is minus a half of the time domain amplitude. What's going on here? And what's this negative frequency? In order to answer this question, we need to look at frequency in a slightly different way. Instead of calling it frequency, you may have heard the term angular velocity. It's often represented by the symbol omega. This is especially useful when we consider the complex plane. The complex plane combines the cosine and sine components of a signal into one image. The x-axis represents the amplitude of the cosine component at any moment in time, and the y-axis represents the amplitude of the sine component at any moment in time. The frequency of the sine and cosine waves can be thought of as the speed of rotation of a line in the middle of the circle. This is the angular velocity for this line. If we use the z-axis to plot the angle of the line, then our circle becomes a spiral. The speed the line moves up the z-axis is its angular velocity. However, just as the line can move up the z-axis, it can also move in the opposite direction down the z-axis. If we look at this in two dimensions, this is equivalent to rotating the other way. When the line moves up the z-axis, its angular velocity or its frequency is positive. When the line moves down the z-axis, its angular velocity or frequency is negative. This is one way of understanding negative frequency. Let's go back to looking at this idea in two dimensions. I'm going to make the rotating line into a triangle and plot how the height of the triangle changes with the angle. Look, it makes a sine wave. Now I'm going to do the same with the width of the triangle. See how it makes a cosine wave. But what if the triangle was to start rotating in the opposite direction? A rotation in the opposite direction is the way we just conceptualized negative frequency. The cosine graph looks the same, but the sine graph is a mirror image of itself. Any positive value on the sine graph during a positive rotation is now negative during a negative rotation and vice versa. Understanding negative frequency in this way explains why the amplitude of the cosine component at negative 5 hertz is positive and why the amplitude of the cosine component at positive 5 hertz and why the amplitude of the sine component at negative 5 hertz is negative and the amplitude of the sine component at positive 5 hertz is positive. 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 this signal and move from the frequency domain back into the time domain, we need to perform an inverse Fourier transform. What is the inverse Fourier transform and how does it work? We can answer this question using a special diagram which I'm going to call a Fourier cube. Here's my signal. The Fourier transform breaks the signal down into its constituent sinusoids giving us a representation of the signal in the frequency domain. The inverse Fourier transform as its name suggests does exactly the opposite, putting all the sinusoids back together again to recover the original time domain signal. But as we saw before, the Fourier transform does more than just break the signal down into sinusoids. It breaks each sinusoid apart further into the constituent cosine and sine waves making it up. Therefore, before the inverse Fourier transform can add all the sinusoids at the different frequencies together to recover the signal, it needs to first recover the sinusoids themselves from the cosine and sine components. This is what is happening inside the integral of the inverse Fourier transform. X of f is the list of complex numbers representing the amplitude of the cosine and sine components at each frequency. Each complex number is multiplied by a cosine and sine wave at the frequency we are currently interested in. But multiplying two complex numbers in this way at a single frequency doesn't recover the two-dimensional sinusoid we saw in the Fourier cube. This is because the result of the multiplication still contains an imaginary component. It's complex. It looks like a three-dimensional spiral. Now you may have noticed that the frequency dimension has disappeared from my graph. This is due to the fact that in order to draw the complex sinusoids for each frequency I've had to borrow a dimension. We live in a world with only three spatial dimensions in it, but in the abstract world of mathematics there are an infinite number of dimensions. However, if we want to represent these dimensions visually it can become a little difficult to draw more than three dimensions at once. On the Fourier cube I've already assigned all the dimensions I can draw. The x-axis represents time, the y-axis represents the amplitude in the real dimension, and the z-axis represents frequency. Unless I borrow a dimension I cannot represent the amplitude of the complex sinusoid in the imaginary dimension. Therefore, I've borrowed the z-axis, the one I used for frequency, to represent the imaginary dimension. Our signal is real. This means it only has an amplitude in the real dimension. The amplitude in the imaginary dimension must be zero. All the sinusoids making up the signal must similarly be real. But the sinusoids which the Fourier transform calculates for us are complex sinusoids. They're all spirals. How can we turn these spirals back into the real sinusoids that make up our signal? This is where the negative frequency component comes in. This graph is only showing us half of the story where the Fourier transform is concerned. For every positive frequency component there exists a negative frequency component containing a complex sinusoid that is equal and opposite to the sinusoid at the positive frequency component. The final stage of the inverse Fourier transform requires us to add all these complex sinusoid together. When we do this, look at what happens. The sine component of the negative frequency in each frequency pair is the exact opposite of the sine component at the positive frequency in the pair. These two spirals are complex conjugates. When we add them together, the sine component in the imaginary dimension cancels out, leaving us with a real sinusoid. So now we know what negative frequency is and why it is present in our signal. It's there to ensure that the imaginary component which the Fourier transform left us with cancels out when we reconstruct the signal. In other words, it's just a trick of the maths. Isn't it? Well, no it's not. I'm going to show you a real-life example of the effects of negative frequency and what it means for the bandwidth of a signal. Whenever you tune your radio to pick up your favourite radio station, what you're actually doing is tuning a filter to a certain frequency called the carrier frequency. The carrier frequency is a sine wave onto which the audio signal of the station you want to listen to is modulated. The most well-known modulation techniques are AM and FM. In AM, the audio signal is multiplied by the carrier frequency to produce a modulated wave whose amplitude changes in proportion to the amplitude of the audio signal, hence amplitude modulation. You can see how the amplitude of the carrier frequency changes with my voice. However, if we look at it in the frequency domain, it's like this. This is a frequency representation of the modulated audio signal I just recorded. What has the modulation process done? You can see the modulating frequency represented by this line here. Either side of it are these sidebands. This is the frequency representation of my voice. What the modulation process has done is to shift the frequency spectrum of my voice from its baseband range the range of frequencies you hear when I speak all the way up to this new frequency range around the carrier. But just look at this. The baseband frequency spectrum of my voice contains negative frequency. If it didn't, then the frequency spectrum of the modulated signal would look like this with the only frequencies present in the signal being above the carrier frequency. However, this is clearly not the case. The sidebands are symmetrical, meaning that this negative frequency is far more than just a quirk of the maths. It's a totally real physical phenomenon. A phenomenon we have to be careful about when assigning frequencies to different radio stations. As the use of radio communications is increased over the years, the airwaves have become more and more crowded, meaning that the distance between different radio stations and the frequency dial has become smaller. However, if one radio station is assigned a carrier frequency that falls within the range of another's sidebands they will interfere with each other. Therefore care must be taken when assigning these carrier frequencies so that enough bandwidth is left between stations to avoid interference. This has led to the development of techniques such as single sideband modulation which manages to modulate the audio signal using only one of the sidebands, meaning there is more bandwidth available for additional stations. So negative frequency is far more than just a quirk of the maths. It's a real phenomenon which impacts on our world. We've covered quite a few concepts in this video. Negative frequency, imaginary numbers, the complex conjugate, the Fourier transform and the inverse Fourier transform. All of these concepts are covered in more detail in other videos on my channel. See the links at the end of this video. If you would like to learn more about Fourier and his theories and how they led to what we now know as the Fourier transform then take a look at my book on how the Fourier series works. In the book I take all the basic concepts contained within the Fourier series, look at each idea visually and then put them together just like Fourier did in 1822. The book is available to buy in paperback and Kindle e-book formats however you can download the first few chapters for free by signing up to the mailing list. Links to the full book and the mailing list are in the description below.