Consider for a moment the Fourier Transform, which I discussed back in Copper # 18. Hands up if you think they’re tough to understand! I want to open your eyes to the world of magic that lies behind it, where the Fourier Transform itself is but one single – but very productive – play in the grand game of transform mathematics.
This column is going to kick off with Imaginary Numbers, and get steadily more head-splitting from there. If you’re up for the ride, good for you! If not, this would be a good place to get off.
Imaginary numbers is the term given to the mathematical representation of conceptual quantities which are square roots of negative numbers. In the real world we only see real numbers. Count the number of spoons in your cutlery drawer, and the answer will always be a real number. There are no real numbers which, when squared, result in a negative number. However, both mathematicians and engineers have long realized that square roots of negative numbers – the so-called ‘Imaginary’ numbers – actually have a fundamentally important relationship with the real world. We can think of all numbers as comprising a juxtaposition of two parts, a Real part and an Imaginary part. We refer to such numbers as Complex numbers. For all quantities that we encounter in our everyday lives – like the number of spoons in the cutlery drawer – the Imaginary parts of those numbers are all zero. We therefore describe them as purely Real numbers. But mathematically speaking, pretty much everything that we can do with Real numbers can also be done with Complex numbers.
So now we dive back into the Fourier Transform. An audio signal, mathematically speaking, is a Real function of time. By this, we mean that the signal itself describes the evolution of a certain property with time. It can be air pressure, it can be an electrical voltage, but whatever it is the key thing here is that it varies with time. And both the thing we are describing (the voltage, or whatever), and time itself, are Real quantities in the sense that neither have an Imaginary component. Now, we all understand that a Fourier Transform takes this data and converts it into a different representation, one which is a function of frequency. That is to say, the transformed signal now describes the relationship of a different property with frequency. Let’s take a closer look at that property.
Like time, frequency itself is purely Real. Just as there is no such thing as Imaginary time (although I have worked for bosses who were not always clear on that concept), there is no such thing as Imaginary frequency. However, in our Fourier Transform, the property of frequency that the transform has spat out is actually a Complex number. That is to say it has both Real and Imaginary parts. This is our first opportunity to deal with Complex numbers … so how are we to interpret them? In this case, the answer turns out to be quite simple. Complex numbers can be rearranged so that instead of having Real and Imaginary parts, they have a Magnitude and a Phase Angle. If we do this, the Magnitude of the property can be seen to represent the amount of that frequency present in the original signal, and its Phase Angle represents the phase of that frequency component. Usually when we deal with Fourier Transforms we ignore the phase and just display the magnitude – which is a Real number – so the Complex number aspect ends up hidden from view.
I wrote above that, like time, frequency is purely Real. But the transform itself is not constrained to working with purely real bases. After all, it did spit out a bunch of Complex numbers. In the equations governing the Fourier Transform we can easily replace the Real numbers which correspond to frequency with Complex numbers. This modified transform is normally referred to as a ‘z-Transform’, and it turns out to be even more fundamental than the Fourier Transform. Indeed, the Fourier Transform is but a subset of the z-Transform. Where the Fourier Transform’s output is in frequency space, the z-Transform’s output is in an expanded version of frequency space called z-space.
It turns out that z-space is a magical kingdom in which wonderful things happen. We can design almost any signal processing operation imaginable in z-space. In particular we can design filters that have all but limitless properties. But there’s a catch. Having designed a filter in z-space, it doesn’t tell us how to implement it in the real world. Only a certain highly specialized set of properties appear in both z-space and in the real world, and so we can only design real-world filters which are based on those specialized properties. Even so, z-space really is a magic kingdom inhabited by wizards who design signal processing algorithms.
At this point, you can think of Fourier Transform space (or Fourier Space) as a place where advanced amateurs like me – and maybe you too – can feel comfortable. z-Space then can be thought of as a place where only serious professionals will really feel at home. So, extending that metaphor, where might the real bad-@ss, serious-sh^t, PhD-and-bar experts hang out? Our Final Frontier, if you like … ?
Let’s go back to that notion of time, like frequency, being purely real. A couple of paragraphs up I replaced the Real frequency number with a Complex number and ended up in z-space. Can I do the same thing with the Real ‘time’ number, and replace it with a Complex number? What happens then? Well … something quite unexpected happens.
If you have got this far you have probably heard of Einstein’s concept of Space-Time, where space and time are melded into one entity, and all the weird stuff we observe in the universe, but couldn’t account for, seems to drop straight out into our laps. Well that’s what happens here. It turns out that both time and frequency are in fact one single entity called Time-Frequency. They’re not independent at all. In fact, we can view the Time-Frequency domain as a Complex number space, where the Real part is Time-like, and the Imaginary part is Frequency-like.
An ‘Argand’ Diagram is commonly used to represent Complex numbers. It’s an x-y plot with the Real part on the x-axis and Imaginary part on the y-axis. On the Argand Diagram, the time-based view of our music signal – the one we started out with – lives at the 3 o’clock position, along the x-axis. The frequency-based view – the Fourier Transform – lives along the y-axis at the 12 o’clock position. From this perspective, the Fourier Transform can be seen as a rotation by 90 degrees counter-clockwise. A further rotation of 90 degrees takes us to the 9 o’clock position which turns out to represent a time-reversed version of the original music signal, and a further 90 degree rotation to the 6 o’clock position actually corresponds to the inverse Fourier Transform.
This place we have arrived at has several names, and is actually at the very forefront of pure research into signal processing. It is mind-blowingly difficult to master, and I would say that those who are able to do so probably already enjoy an international reputation in their fields, and sport long beards. I for one am so not among them. In the signal-processing field the most commonly used terminology is Fractional Fourier Transform (FrFT, or FRFT). This is because, understanding the picture in the terms I just described, it becomes possible to design transformations which transform the signal into some intermediate position in the Time-Frequency domain, where the datum is neither one thing nor the other. In other words, a rotation of some arbitrary angle somewhere between zero and 90 degrees, hence the term ‘Fractional’ Fourier Transform.
This really is the cutting edge. Researchers are only just beginning to uncover what can be done using these theorems and representations. For example, I have read (but am sadly unable to fully comprehend) a paper claiming to show how FRFT can be used to design mathematically perfect sample rate conversion algorithms, something that practically escapes the current state-of-the-art. Furthermore, FRFT shows that the Shannon-Nyquist sampling theory is but a specific case of a more general theory by which sampling is not just constrained to be band-limited in the frequency domain, but in a more complex sense in the entire Time-Frequency domain. There are, I imagine, things one can do with such knowledge!
So … hands up if you thought Fourier Transforms were tough!
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