
Innovations in Applied Mathematics
relating to Sight and Sound.

Precise Signal Component
Technology
What It Does
The Mathematics behind the Method
Licensing and Related Considerations
Seeking a Partner
Why The Delay for the Demo?
Patent and Related Mathematical Documents
What It Does
Our US patent for Precise Signal Component technology was issued on June 15, 2004. (Applications in other countries are pending at this time.) That is about six months earlier than we had considered the most optimistic issue date. Several other prior commitments and interests must necessarily delay our preparing demonstrations of the technology suitable to put on the web. We are posting the patent itself at this time (download USP 6,751,564 [1.3MB PDF]) and we currently plan to post a few of our original documents that present the material in more detail as we have time to review them  (We have posted several key documents and plan to add more).
We consider Precise Signal Component to be a major breakthrough in signal processing capabilities. We are well aware that "breakthrough" is a widely misused and overused term, but we believe it is completely justified in this case. In our patent we describe how to do what has been widely regarded as an unattainable "holy grail" of signal processing. Precise Signal Component can separate out from a complicated signal, multiple component signals each having its frequency, amplitude, and phase determined to high precision, and it can accurately track components as their characteristics change with time. For example, the individual component signals that make up a note or chord from a piano can be determined as to precise pitch and can be tracked as their volume decays with time. Similarly, the individual component signals that form a violin note or chord can be determined as to pitch and can be tracked as the violinist's expression varies the volume and pitch. Components at closely spaced frequencies can be accurately determined. Musical sounds are just a familiar example – Precise Signal Component has wide application to all timevarying signals which are combinations of multiple individual component signals and such signals occur throughout all areas of nature and technology.
Persons casually familiar with Fourier transform analysis may have the impression that the Fourier transform already can do tracking of precise signal components. Indeed, digital Fourier transform analysis is a very powerful tool that has been widely and very successfully used in many applications. Mathematically speaking, the success of digital Fourier transform analysis stems almost entirely from its being an orthogonal transformation and not from the precision with which it determines frequency components. An intrinsic property of orthogonality is that no information is lost in performing a digital Fourier transform; in physical terms, it just ends up misplaced, blurring the frequency spectrum. Normally digital Fourier transform applications first transform the data into the socalled frequency domain, perform some operations on the result, and then transform the data back into the real timebased world. These two lossless transformations compensate for one another and the final result is both useful and often very close to being what would have been expected if the frequency domain information had been an accurate representation of the true frequency spectrum.
Digital Fourier transform methods are much less satisfactory when the application requires information directly from the frequency domain, such as original signal component frequencies and amplitudes. In the frequency domain, digital Fourier transform methods yield only a gross approximation to the individual component sinusoids that make up the true frequency spectrum. While the method can be fairly accurate in pitch or frequency of the primary components, the amplitude or volume is only a rough approximation and the phase information is so poor an approximation it is normally completely ignored. Just how poor this approximation is may be understood from the fact that using digital Fourier transform methods, the two lines of an instrumental duet cannot reliably be separated into the notes being played. With three or more simultaneous notes being played as part of a musical line – and often with just two notes – the task is impossible.
The Precise Signal Component method provides the tools necessary for tasks as complex as separating the notes of a symphony to approximately the degree a human ear can. Please understand that practical tasks of such complexity are still in the future; the fact we are emphasizing here is that until now they were impossible. Precise Signal Component has made them not only possible, but quite practical to pursue. Precise Signal Component also works well in noisy conditions. It will be very useful in separating human voice components from noisy surroundings and will likely be adapted to cleaning and normalizing voice for speech recognition systems. This is a brand new technology; a long list of possible and likely applications can be drawn up now and that list will grow as the capabilities of the method become better understood by the technical community. Precise Signal Component will open new doors, and could easily prove to be the backbone for entire new area in the electronics industry.
The Mathematics behind the Method
In mathematics, sets of functions are either orthogonal or they are not. Orthogonal sets have a number of unusual, highly desirable qualities that methods like Fourier analysis exploit to great advantage. These helpful qualities of orthogonality are considered to be absent for nonorthogonal function sets. In contrast, in our past engineering mathematical work we have found the concept of nearorthogonality to be extremely useful. Function sets which nearly pass the tests for orthogonality still possess many of the qualities of orthogonal sets to a degree sufficient for engineering purposes. The mathematical methods derived for true orthogonal sets often will not work with nearorthogonal sets and surrogate methods have to be developed to take advantage of the nearorthogonal qualities. In particular, the quality of complete independence of the members of orthogonal sets is compromised. Typically it is necessary to abandon methods which treat members individually and to develop methods which simultaneously take account of subsets of several members. Other techniques improve the nearorthogonality of function sets. Such techniques typically involve the careful selection of members, the number of members, the addition of surrogate members, and the selection of the span over which orthogonality is to be applied. These concepts of nearorthogonlity were key to developing the Precise Signal Component method.
Licensing and Related Considerations
We believe our discovery and development is an important one which may become the cornerstone of a new area of technology. We would like to see the method freely used in the scientific and engineering community. Accordingly, we debated whether it would be best to patent it or to simply publish it. What we saw in either case is that if we are right as to its importance, once our patent is understood the current technologybusiness climate will guarantee a greedy web of patents on frankly obvious extensions of this brand new technology. This can only cause confusion and squelch honest development of the technology.
We will retain control of the patent and its licensing. We intend to have a system of licensing with variable royalty payments. In general, we plan to make the base licensing fees low enough that no one planning to use the technology need fear how royalties will impact the product pricing; there will be no reasonable excuse for not contacting us for a license. Companies and individuals who contact us honestly about licensing before trying to use and sell our technology will get our best low base rates. Companies which we find to be using our technology without first licensing it will be subject to very much higher licensing fees.
We will expect all patents which depend upon ours to be handled by their owners in a similar fashion; that is, to be freely available for licensing and for fees which are in keeping with our intent; that is, the fees should be low enough not to discourage others from building upon them. Our own licensing will be structured to encourage this. What we presently have in mind is to apply extremely high licensing fees of our own in any case which involves the use of a dependent patent that violates this intent. We hope to devise other methods before the time comes that they must be applied, as we would like a way to reasonably distinguish in this between obvious extensions and those which are truly innovative in their own right.
As a token in this direction, after we have a chance to review and reevaluate them, we will be freely publishing some new items on our web site that have been developed since the patent application was written. We believe these items to be patentable ideas in their own right, but we consider them fairly obvious, once the patent technology is known, even though it did take us awhile to think of the "halfshift" as a complete solution to an intrinsic weak point in the original patent method.
Seeking a Partner
We fully expect this patent to go through the same sequence by which other innovative ideas (of both our own and others) have traditionally been received. First, rejection as impossible. Second, reevaluation, usually as the result of seeing some particular demonstration of the impossible. Third, independent investigation of the idea and its principles. Fourth, statements that the idea was "obvious" in the first place. We expect that the first phase may be fairly long, both because it will be awhile before we have good, simple demonstrations prepared and because this is a brand new area of technology. We welcome inquiries from companies that might like to partner with us in fleshing out and implementing the plan we have sketched above. Although the tight formulation of an innovative licensing structure is obviously key, we are more concerned with finding a partner that can handle the litigations that are almost certain to arise in phase four. While we feel certain that litigations against infringers and license violators will be ultimately winnable, in the current court system a large company could easily financially drown us before the case even comes to trial. We seek a partner that can weather such storms and is interested in being part of this newly emerging technology.
Why The Delay for the Demo?
The problem is not in producing a demonstration that the method works; that has been done in the patent. The problem is that there is no alternative way of doing what we are doing and so nothing that can be used for comparison. For developing the method it was necessary to generate artificial signals from known components, with and without added noise, and use the method to extract the components and compare them with the original, as described in the patent. But "real" demonstrations require real world data. When working with real world signals, how can you show whether the method is accurate or not? There is nothing to compare it with. This sort of problem exists at every turn. It is easy to use the method to extract the components of a complex musical line as sinusoids varying with time, but can you report the results in a way that will be meaningful and in a demo and show that they are obviously correct? If you have a single tone, plotting separate time curves for frequency, amplitude, and phase could be meaningful, but even for a single tone with overtones, starting and stopping, this method quickly becomes bewildering to the eye and mind.
We are resolving issues of that nature. We are also taking the opportunity to completely rewrite the core computer codes for the method at the same time in order to take advantage of what we learned while developing the method.
Patent and Related Mathematical Documents
CFS175 Full Frequency Analysis [260KB PDF]
This document has been assembled from the major parts of three documents in which the Precise Signal Component method was originally derived. The original document was assigned the number CFS175 and other parts, including the two assembled here, were assigned CFS175 with some supplementary qualifier. A later, considerably abridged version was prepared for use in the patent application and was assigned the number CFS185. The first part of this document gives the basic considerations for Precise Signal Component and derives the basic form for a single frequency component. The second part, Platform Least Squares, derives a surrogate function to reduce or eliminate confounding between widely spaced frequencies. The third part, Multiple Adjacent Frequencies, derives the means of handling multiple components having frequencies that are closely spaced. This material is identical with the method described in USP 6,751,564, but expands upon it considerably.
CFS245 Integral Least Squares [149KB PDF]
This document predates Precise Signal Component and was not given a CFS number at the time. In reviewing material related to Precise Signal Component I find that it gives a very good development of the method of complex least squares, which is used in Precise Signal Component and rarely (if ever) found in the general literature. It also gets usefully into some areas of orthogonal and nearly orthogonal functions as I was thinking of them at that time. This line of thought led directly to Precise Signal Component. There is a brief excursion into powerrelated frequency functions which I was exploring at the time and which is not relevant to Precise Signal Component, but I have decided to leave the document as it was and assign it a current CFS number.
CFS246 Least Squares Residuals [116KB PDF]
This document derives the details of forming the least squares matrix and extracting the residual sum of squares from the least squares matrix equation. In particular, it includes weighting factors in the development, an element that is usually missing in standard derivations. It was prepared in the process of developing the computer code for Precise Signal Component. It was not given a CFS number at that time, so it has been assigned a current CFS number for release.
CFS224 Displaced Frequency FFTs.pdf [109KB PDF]
The Precise Signal Component method as described in USP 6,751,564 is as powerful a method as that patent implies, but I did find it to have one area of weakness in precision (the only such weakness of which I am aware). This document describes a method for completely eliminating that weakness. I believe this method would have been patentable in its own right, but I offer it here as a public domain addition to the Precise Signal Component method. I intend this to serve as an example of the "obvious" improvements that will arise as Precise Signal Component becomes more familiar and normal science and engineering methods are applied to it. This method depends upon quite venerable technology and certainly will be "obvious" to practitioners in the field once it has been explained. As to whether it would have been obvious given just a statement of the weakness to be addressed, I feel that it should have been, but my own experience was that it was not. It required exploration of a number of blind alleys spread over two years before the concepts aligned in my mind in a way that made it obvious to use an extremely low frequency carrier wave.
Download USP 6,751,564 [1.3MB PDF]
This is the complete patent as released by the USPTO, with a final page of comments and corrections added.
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