## Quibbles and Bits

# The Final Frontier

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 [email protected], 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!

It’s been over 50 years since I first studied this. This is a rather strange presentation you’ve given. Prerequisites for understanding and computing Fourier transform functions for me were, one year of integral calculus, study of Taylor and MacLauren series, study of Laplace transforms, study of Fourier Series. Fourier series deals with periodic functions. By reducing the period to nearly zero in the limit (dt) you arrive at an infinite number of frequencies between any two arbitrary frequencies for non periodic functions. The value of the Fourier Transform cannot be overstated. By converting an input waveform to the frequency domain, you simply multiply it by the transfer function of a process in del operator notation to get the output in the frequency domain. The inverse transform returns the output to the time domain if desired.

The amplitude frequency response of a system is one element of a Bode plot. The other is phase response as a function of frequency. In what has come to be known as a zero phase system such as an amplifier, there is a direct correlation between amplitude and phase response. In non zero phase systems such as a multiway loudspeaker systems there is no such correlation.

One of the “tricks” in developing Acoustic Energy Field Transfer Theory which led directly to Electronic Environmental Acoustic Simulation was ignoring the usual choice between time domain and frequency domain. Instead from each direction of arrival at a point, using the first arriving sound as “normalized to T=0 and FR flat 0 to infinity, the functions puts each arriving reflection in the time domain but describes their amplitude and spectral content in the frequency domain. This allows every single arriving reflection to be accounted for and compared to the corresponding first arriving sound as a reference. This describes the arriving field at one point from one source at another point. The function obeys the law of superposition for more than one source and can be described for multiple points in space on a surface.

Wave Field Synthesis arrives at the same conclusion from an entirely different direction. “WFS is based on the Huygens–Fresnel principle, which states that any wave front can be regarded as a superposition of elementary spherical waves. Therefore, any wave front can be synthesized from such elementary waves.” “Its mathematical basis is the Kirchhoff-Helmholtz integral. It states that the sound pressure is completely determined within a volume free of sources, if sound pressure and velocity are determined in all points on its surface.” Source Wikipedia.

The problem WFS has great difficulty with but AEFTF has solved (at least on paper) is how to measure it. Had I not figured out how, I’d have thrown my hands up thinking it was hopeless. And I almost did. If you think solving the Fourier transform is easy, consider this;

https://en.wikipedia.org/wiki/Fourier_transform

Now combine it with the feedback amplifier transform function based on the deceptively simple equation

G’ = G/(1 + GH) and you have quite a lot of math to do on your hands. BTW, where I went to school, Systems theory which introduced Fourier Transform Functions and del operator notation was a sophomore while feedback control theory was a senior course. In a full understanding no corners are cut, no shortcuts taken. The FFT method Atkinson uses to measure loudspeakers is awful by his own admission. See his lecture on how he measures loudspeakers…..on his backyard patio. What a joke. Who could believe one word they read in his magazine?

Something tells me that only electrical engineers get intensive courses in this and related areas of mathematics. That’s a shame because it is a tool which has applications in virtually every area of science and engineering. For example, the suspension in your car was designed using this tool. Mathematicians probably also get more than a smattering of it. I really don’t like the shortcuts and symbols that combine variables into one because they detach from the basic underlying parameters. This is true in other areas as well.

The square root of minus one which of course is a pure mathematical abstraction that was cooked up because it is so useful is a trick for describing phase rotation. So an electrical circuit that has an inductor and or capacitor where voltage and current waveforms don’t coincide have phase angles above and below the “real” axis which is resistance where they do coincide. Electrical engineers use the mnemonic device “ELI the ICE man” to remember in which circuit current lags and which circuit current leads voltage.

Electrical engineers have many peculiar exclusive inventions. One is the use of the letter j to express the square root of minus 1 when the rest of the world uses i. This is probably because they used up i for current. So the full expression of the Fourier transform is” F(j*omega) = 1/2pi*integral from minus infinity to plus infinity of f(t)* e Exponent -j*omega*t dt. omega = 2 pi f and e is the base of natural logrithms. Can YOU solve that integral equation for actual functions? Just how good is your math? You want to understand it? Try it a few hundred times. You’ll catch on….if you get the right answers. Now multiply the del operator form of the transfer function and calculate the inverse transform. Easy Peasy, right? Let me know when you’ve solved the first one. Try an easy one first just to get the hang of it.

Combining the Fourier transform function and inverse function with the circuit or system transfer function shows what happens to the input waveform when it reaches the output. A theoretically ideal amplifier would only change its amplitude. However, for various reasons it is important to be able to make desired changes. This is true over a vast range of applications. The combination also is a good way to explain the phenomena of resonance and filtering. Audiophiles for the most part simply don’t understand that they cannot perceive sound above 20 khz if that. They protest that they do not hear sine waves. As I posted previously in the last round about this WE DO NOT LISTEN TO SINE WAVES. However, Fourier analysis helps us understand what we do hear and why reproducing sounds above the audible range of frequencies is worthless and may create other problems.

One thing we hear is the fact that as sounds die out as reflections the Fourier transform shows that high frequencies die out faster than mid and low frequencies. This was a lucky guess that I incorporated into EEAS that was later confirmed. For example, in a typical concert hall the RT is about 2.0 seconds at 1 khz, 1.2 seconds at 8 khz. Each reflection has relatively fewer high frequency harmonics as the sound dies out. This was incorporated into the cheapie patented design version of EEAS by including an adjustable low pass filter in the recursive time delay circuit along with amplitude decrease to control RT. In the full blown development there would be no recursive circuit and each delay would have its own output from series shift registers, its own D/A and its own equalizer for each direction of arrival.

Intervals might be every 50 microseconds or 5 (not sure which.) So as sound dies out it becomes mellower while the first arriving sound has the sharpest transient attack meaning the most high frequency components. This is necessary to reproduce if you are going to have any chance at all of reproducing the tone of acoustical musical instruments as they are heard live in the audience at a concert. As can be seen therefore tone is a dynamic event not captured or reproduced by the current technology. It is important because tonality is one of the four basic elements of music. This is one of many innovations in EEAS technology that makes it far superior to other technologies, it can accurately reproduce tone of live music from recordings. By contrast, whatever frequency response you pick using the steady state analysis such as is measured with an ILG fan at a venue is going to be wrong. Stated simply, except for binaural recordings made in the audience, the current technology cannot accurately reproduce tone in the “you are there” approach to high fidelity sound recording and reproduction. (binaural recording has a fatal flaw, it records and plays two scalar fields, not a vector field but that’s another subject.)

Can Fourier transforms be used in the digital domain? Yes. Generate a series of pulses (ones) without any connected wires to a load. Look at the output in the frequency domain. It will show you the amplitude of the highest frequency. It will also show you the frequencies and amplitude of the jitter. Now connect a wire and a load and look at it again at the input to the load. The highest frequency amplitude will have diminished and the jitter amplitude will have increased. There may be more jitter frequencies added. Now put it through a Schmidt trigger and a shift register operated by a precision clock. The output should be as good or better than the original signal if you have designed your circuit well. And it’s cheap. A new idea? No hardly, at least 50 years old. This is why audiophile voodoo about wires in the digital domain is such a joke. The only problem comes about when the jitter frequency exceeds the shift register time window. This can be corrected by intermediate repeaters that reclock and reform the signal at intervals usually in land line communications networks.

How do you use Fourier transforms Mr. Murison and how many have you calculated?

Fourier transform functions are powerful analytical tools but they are not the final frontier if such a thing even exists. One of the virtually universal gaps in knowledge about sound is the concept of vectors versus scalars. The difference is that scalars have only magnitude. Vectors have magnitude and direction. Electrical signals have only amplitude as a function of time…or frequency. Sound has directions in three planes as well. This makes them entirely different. When a microphone turns sound into electricity, it changes a vector form of one kind of energy into a scalar form of another. Speakers turn them back into vectors and of course they have virtually nothing in common with the vectors that reach your ears from live music. Headphones create vectors too but because they are fixed to your ears and turn when your head turns they are the equivalent of scalars. In short they have no directional properties. This is why binaural sound doesn’t work.

So if you don’t take the vector nature of sound into account in your design, you don’t understand sound and you can’t understand how to recreate it. That is where this industry is right now and that is where it will stay until people with far more knowledge than these people have tackle the problem. Judging from product designs that’s not about to happen anytime soon.

Cat got your tongue Richard?