A fascinating and most gratifying thing happened recently. I wrote two articles--highly
technical articles that discussed the proper meaning behind the words "time" and "frequency" as
we use those words in audio. I wrote those articles with the intention of stretching
the mind, my own as well as the reader's. This mind stretch seems to have aroused
favorable interest, and a great many readers took the time to write and encourage
us to continue with more such articles. Very well, I will try.
The Dilemma of Perception
So there is no misunderstanding, let me state what it is I am doing and how it differs from tradition. I am conducting an exercise in thought by taking a wholly different approach to our present technology. I choose to form no a priori conclusion about the truth or falsity of the mathematical relationships which we now use. Instead, I am attempting to come up to those relationships from a different level of conceptualization. I feel that we should try to find out "why" as well as "how" nature appears to do the things it does.
If you wish to call that a philosophical base, then I do not object.
The reason I search for "why" is simply that right now we have an intense dilemma confronting us whenever we attempt to correlate our physical measurements with human subjective perception. It is no laughing matter that what we measure does not always correlate with what we "hear." It is a disgrace which no amount of finger pointing will eliminate.
I start from the conviction that nature does not solve equations. If we are going to use math, then I want to know why that math should work. It is necessary to start from some primal base. The primal base I present in these articles is that through the study of relationships we can infer which exist within any structure of allowable description. The fancy name is abstract geometry.
It obviously derives from my interest in the subjective/objective problem, since one structure of description can be that of perceptual concepts, while another structure can be that of objective measures of the ingredients of that perception. Can we link the structures? That is my interest.
Alternatives
A few years ago I published a theoretical concept which I call the Principle of Alternatives (1.2.3). It introduces alternatives as a geometric concept. Alternatives are defined as those equally valid descriptions expressed in different frames of reference and corresponding to alternate spaces of representation. Geometrically, alternatives are different ways of looking at the same thing, and the set of all alternatives forms a universe of allowable descriptions.
What this principle asserts is that there is no preferred way of describing anything, either from the standpoint of dimensionality or units of expression.
And therefore, there are an infinity of alternatives for any description we might make.
Abstract? You bet, but very powerful, because we can consider all known classes of description, including those in which all we can determine is the probability with which assignment of properties can be made within a particular frame of reference ("fuzzy" alternatives).
In the previous Audio articles I used this principle to give a new interpretation to the concepts of time and frequency and to the Fourier transform which defines them.
Namely: the "time response" is one of the alternatives to the "frequency response" and the Fourier transform is that map which converts from one alternative to another.
In this discussion, I would like to carry this mind-stretching exercise a little further and ask you to think about the inner meaning which this brings to some relationships common to audio as well as other branches of science. Again I state my caveat: Do not blindly accept everything I state, think about them, mull these things over in your own mind. I believe it is far more important to convince oneself of the reasonableness of such things than to accept as dogma that which might later be overthrown.
Besides, once one becomes accustomed to thinking about such things in the abstract, it is going to be much easier to consider the deeper waters I would eventually like to discuss on the implications this brings to reconciliation of the "meter" and the "ear."
Euclid, Hilbert, and Audio In the previous article I pointed out that "waveness" and "placeness" are possible alternatives of each other. If, for example, we have set up a description in terms of any frame of reference, then it is possible to recast that description into another alternative frame of reference in which each place in the first becomes a wave extending over the whole of the second. One of the infinite number of ways of performing this conversion is that map which we call the Fourier transform.
Very well, let us turn this around.
When two descriptions are related to each other through the Fourier transform, then these descriptions are alternatives of each other.
What special things might we infer about these particular alternatives? One of the first things we can infer is that the number of dimensions will be the same for each of them. There is no way that a three-dimensional description can be related by Fourier transformation to a two-dimensional alternative, for example. There are maps which connect three-dimensional alternatives with two-dimensional alternatives, but the Fourier transform is not one of these.
Let us now consider that special type of geometric framework in which everything obeys the laws of Euclidean geometry. All the postulates of Euclid, including the parallel line postulate, hold. A very little thought about the geometric basis for the Fourier transform, which I gave in the previous article, will reveal that each of the alternatives joined by Fourier transformation are Euclidean in nature. So a Euclidean space is transformed into a Euclidean space.
Let me pause right here and reveal a bit of where we can use this in much later analysis. I contend that the thing we call distortion in audio, both objective and subjective, can be regarded as a warping of the geometry within a given frame of reference. The effect of distortion is to convert a Euclidean representation into a non-Euclidean representation, for example. This warping may possibly be handled as curvature tensors at some later time.
But right now I want to point out that this talk of Euclidean spaces is very important to audio, and it is not part of a "snow job." If what we are describing has a limit to its total energy, as all practical audio measurements do, then we can state that the proper sum of all energy components is finite. Many of the things we measure are such that their squared value is proportional to energy. Sound pressure is such a parameter; so are air particle velocity, voltage, and current. Not obvious now, unless you are into math, but the appropriate sum of the magnitude of such parameters squared is known as the Lebesgue square measure, denoted by the symbol L'. In striving to find some possible deep-seated meaning to properties, whether of perception or physical observation, we are led to search for the most general possible statements about those properties. Statements which are not dependent upon special objects of description, but determined by abstract relations. If we are really successful, our reward is the discovery that we have no words with which to adequately convey those abstract impressions. So we must often double up on the use of certain descriptive terminology which can invoke some appropriate mental analogies. The term geometry, as I use it, in these discussions, is one such word.
Another such word is "space." In the abstract, the word space refers to a set of defined elements together with some agreed upon rules for combining those elements into the analog of a structural configuration. A multidimensional Euclidean space is a readily identified example. In this case, "space" means what we normally mean by the word space.
But there are other ways of defining elements and putting them together to form other "spaces." Another way of saying this is change the frame of reference. An example is what mathematicians call the Hilbert space L2, the infinite-dimensional analog of Euclidean space.
Thus, one alternative for expressing finite energy signals is a space we can identify as a finite-dimensional Euclidean framework. Volts as a function of time is an example which uses the one-dimensional coordinate measured in units of time with the amount of volts at each moment of time being the number representing the signal at that particular coordinate location. Another alternative is the infinite-dimensional Hilbert space L2 in which each possible form of signal which has finite energy is one of the coordinates, and "how much" of that signal is the position along that coordinate. Sure, it is abstract, but that is what Shannon brought into engineering and was the start of that very practical endeavor which we now call Information Theory.
As a technical point, we can thus observe that alternatives can be infinite-dimensional as well as finite dimensional. As a mind-stretch, we should prepare ourselves to grasp the conception of infinite-dimensional spaces. The reason is that in the early parts of this century, mathematicians really began to develop tools for infinite-dimensional representations under the general name of Functional Analysis. There is a great wealth of knowledge to tap here, as Shannon did.
To give you some idea of how we might use it in audio consider this question: What is melody, or even a melodic contour? Stretch the mind a bit. If each allowable tone is assigned as a dimension, then certain groups of tones, bearing particular relations to each other, define subspaces of finite dimensionality. These subspaces may be combinable in a different manner so as to form characteristic patterns which have extremum metric properties relative to subspaces formed from random combinations of tones. That is, the preferred subspaces are more densely packed with less distance separating members of the subspace. I do not know how that would work out on a number cruncher, or whether it may prove to be a silly idea. But the conceptual "distance" between certain notes, and I do not mean where they are on the musical scale but whether they seem to "fit" together, seems to form an attractive way of discussing chords and how they might fit together in the various combinations we might think of as melodies.
And tweak your imagination with this: Might it be possible that such a primal framework, which we could call a gestalt base of analysis, is also tied to other perceptual-observational disciplines, such as psycho-linguistics? Is there an analogy with Noam Chomsky's Theory of Transformational Grammar such that the perception of sound has a deep structure as well as a surface structure? These are indeed important considerations, but discussion of these things lies well ahead of us. And we must get back to the fundamentals I wish to present in this brief article. With the definition of terms cited above and the appreciation for the geometric role that is involved, we can see that the Fourier transform defines a way of changing one representation into another in a special-form-preserving manner. In contemporary mathematical language, the Fourier transform defines an isomorphism of the Hilbert space L2 onto itself.
The consideration I want to place before you is that whenever we run across two types of description, both of which define a Hilbert space and are linked by Fourier transform, then these are alternate descriptions. They both describe the same thing. The coordinates of these two alternatives are versions of each other and can never be considered completely independent.
Observer-Observed
We may have thought, with deepest conviction, that we were assembling a description of an event (or process, or thing) in which there were two types of parameters, both of which were required for a complete characterization. But, if along the way, we discover that these parameters are linked by Fourier transformation, then nature is telling us that they are alternatives.
Should we persist and try to combine both parameters in a common description, we will discover that there is no way we can codetermine an infinitely accurate "place" on both of them. After all, a "place" on one of them is a "wave" on the other; that is what the Fourier transform means.
Those who believe in the adverse perversity of fate--the butter-side down philosophy-might point out that somehow our attempt to measure one of them causes us to lose clarity in the other. Whenever we set up an experiment to determine one of them, our apparatus depends upon the other one to such an extent as to blur complete knowledge of both of them.
The effect is stated correctly: We cannot measure one without calling the other into play, that is because they are different versions of the same thing. But we should never confuse effect for cause.
In audio we want very much to say that frequency and time are both needed to specify a tone. If we try to measure a complex tonal structure with a narrow bandwidth filter, we find that as the bandwidth gets narrower and narrower, the time response of the filter smears out to such an extent that we can no longer say when that frequency component occurs. To a pessimist it might seem that our very attempt at gaining precision in frequency was befouled by nature so as to lose precision in time. The instrument with which the observation was made seems to react with the signal in such a way as to disturb what we are observing.
In other words, if one were not aware of alternatives, it would be very easy to presume an observer-observed limitation to our knowledge. Any attempt at disproving such an interpretation would be doomed to failure on any grounds that attempted to show there was, even conceptually, the possibility of a true infinite accuracy of codetermination of the parameters joined by Fourier transformation. After all, we got into the trouble by the definition which we originally gave these terms plus the assumption we made that they were wholly independent. Therefore, every possible counter experiment we might propose that provides indefinitely accurate joint parameter codetermination will get destroyed when properly analyzed.
What happened? What is wrong? Is nature mad at us because we tried to mix time and frequency? No, nature does not give a darn what frame of reference we choose to use. Nature does its thing whether we are looking or not. Based on this principle of alternatives, I offer the following suggestion: In our subjective evaluation we do indeed perceive properties that are frequency-like and time-like, and they do coexist. But the dimensionality of this alternative is higher than that of the alternative we use to model some of our simpler objective evaluations.
This does not in any way mean that time and frequency form subspaces in a higher-dimensional perceptual space. What I mean is that it is possible to map a one-dimensional space upward to a four-dimensional space if we so wish. Nothing appears in one space that does not also appear in some fashion in the other space. They are alternatives of each other.
A discussion of the mathematical relations for changing from one frame of reference to another when they have a different number of dimensions lies far beyond the points I wish to raise in this discussion. We will eventually get to that problem. But right now I offer this as a suggestion of a way out of the observer-observed dilemma when the properties we think should be independent are actually related by the Fourier transform. There may not be anything whatsoever wrong with the frame of reference, except that the properties are not what we think they are, belonging as they do to a lower dimensional alternative.
Fifty Years of Uncertainty
I leave you with this important fact to ponder. Exactly 50 years ago, Werner Heisenberg made use of the Fourier transform relationship between descriptions in momentum and descriptions in position, the Dirac Jordan transformation theory, and he discovered a most puzzling fact. The narrower one made the region of confinement of a description in position, the broader became the region of confinement of a description in momentum. The complete derivation can be found at the bottom of page 180 in his now-famous paper.(4)
This particular relationship, which we now call the uncertainty principle, is, of course, absolutely correct for the reasons we have discussed. Surprisingly, little recognition seems to have ever been taken of the role played by the Fourier transform or of the implications which this brings to the interpretation of the inner meaning of that relationship.(5. 6, 7, 8)
The inner meaning that the parameters which this relationship ties together are nothing more than different ways of describing the same thing.
REFERENCES
1. R.C. Heyser, "The Delay Plane, Objective Analysis of Subjective Properties, Parts I and IL"). Audio Eng. Soc., Vol. 21, pp. 690-701 (Nov., 1973); PP. 786-791 (Dec., 1973).
2. ,"Geometrical Considerations of Subjective Audio," I. Audio Eng. Soc., Vol. 22, pp. 674-682 (Nov., 1974).
3. , "Perspectives in Audio Analysis: Changing the Frame of Reference, Parts I and II," /. Audio Eng. Soc., Vol. 24, pp. 660-667 (Oct., 1976); pp. 742751 (Nov., 1976).
4. W. Heisenberg, "Uber den Anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik," Zeitschrift fur Physik, Vol. 43, pp. 172198 (1927).
5. M. Jammer, The Philosophy of Quantum Mechanics, John Wiley & Sons, New York, 1974.
6. E. Scheibe, The Logical Analysis of Quantum Mechanics, Pergamon Press, Oxford, 1973.
7. D. Bohm, Quantum Theory, Prentice-Hall Inc., Englewood Cliffs, N.J., 1951.
8. W.C. Price & S.S. Chissick, eds., The Uncertainty Principle and Foundations of Quantum Mechanics, A Fifty Years' Survey, John Wiley & Sons, New York, 1977.
(Source: Audio magazine, Feb. 1978, )
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