Precise Frequency and Amplitude Tracking of Waveforms CFS-175 Web Version August 30 through October 24, 1999

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1 Precise Frequency and Amplitude Tracking of Waveforms CFS-175 We Version August 30 through Octoer 2, 1999 David Dunthorn.c-f-systems.com Note This document has een assemled from the major parts of three documents in hich the Precise Signal Component method as originally derived. The original document as assigned the numer CFS-175 and other parts, including the to assemled here, ere assigned CFS-175 ith some supplementary qualifier. A later, consideraly aridged version as prepared for use in the patent application and as assigned the numer CFS-15. The first part of this document gives the asic considerations for Precise Signal Component and derives the asic form for a single frequency component. The second part, Platform Least Squares, derives a surrogate function to reduce or eliminate confounding eteen idely 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 ith the method descried in USP 6,751,56, ut expands upon it consideraly. This document is vieale only. I elieve this ill e adequate for people ho do not intend to study it. Please contact me through our e site if you need a printale version. I am aare that the no-print can e defeated, ut again I ask that you contact me instead. I really need to kno if and ho people are finding these documents useful, and this seems one of the fe ays I have to encourage feedack.

2 CFS The Fourier Transform and the DFT/FFT The preeminent tool of frequency spectrum analysis is the Fourier transform, typically implemented in the form of the Fast Fourier Transform (FFT) or another variety of Digital Fourier Transform (DFT). The Fourier transform of a signal 1Ð>Ñ is defined as: _ ¹Ð1Ð>Ñß 0Ñ œ ( 1Ð>Ñ/.> _ 130> For those used to thinking of frequency spectrums in terms of sines and cosines, it is 3B helpful to recall that / œ -9=ÐBÑ 3 =3ÐBÑ. Thus the transform can also e expressed in the form: _ ¹ Ð1Ð>Ñß 0Ñ œ ( 1Ð>Ñ-9=Ð 130>Ñ.> 3( 1Ð>Ñ=3Ð 130>Ñ.> _ Hoever, it is often easier to ork ith the complex exponential form. The infinite Fourier transform cannot e used directly in data analysis. We are forced to examine sample signals over a finite time interval. Moreover, the time interval usually must e quite short, as fe signals in the real orld remain of constant character for long periods of time. Thus the finite Fourier transform is evaluated only over a finite sample time 7Þ 3 1> ¹ Ð1Ð>Ñß Ñ œ ( 1Ð>Ñ/ 7.> (2) 7! Note that the frequency 0 has een replaced y Î7. In practice, is taken as an integer 3 and each / 1 > 7 is periodic (see the sine and cosine form) over the sample interval, 7. Over 7, Q evenly spaced samples of 1Ð>Ñ are taken, at > œ!ß 7ÎQß 7ÎQß ÞÞÞß ÐQ Ñ7 ÎQ. The integral is evaluated for frequency numers œ!ß ÞÞÞß Q Þ Note in passing that the standard summation form has the interesting feature that the values 3 start at zero ut stop short of 7Þ Because the functions / 1 > 7 are orthogonal, this approach is roust, and the integral is almost alays approximated directly as the corresponding sum, Q Q 5œ!! 1Ð57Î7Ñ/ 3 15 Q 7, evaluated for each frequency numer, œ!ß ÞÞÞß Q Þ Orthoganality linear independence guarantees that in evaluating this amplitude-measuring integral at one frequency numer there ill e no interference from any of the other frequency components eing measured. (1)

3 CFS Current Approaches to the Base Prolem Fourier analysis is a poerful and easily applied tool. There have een many approaches to modifying the DFT approach in order to ameliorate the prolems descried aove. These have principally een of three varieties. First, indoing, here the data function 1Ð>Ñ is pre-multiplied y another function carefully chosen to reduce the influence of long sample times and changing properties of 1Ð>Ñ ithin the interval. While used for this purpose, indoing has also seen extensive use in minimizing the =3ÐBÑÎB leakage or noise associated ith the Fourier transform, as descried later. The second major category of modifications to the DFT for sounds analysis has een partitioning, here the spectrum is divided into several ranges and each range treated differently using separate DFTs. This method allos coarser DFTs ith shorter sample intervals in the high frequency ranges. The third major category of modifications is the traveling DFT, here the DFT is successively applied to the data, discarding a fe points at the eginning of the sample interval and adding a corresponding numer of ne points to advance the sample interval along the timeline. The result is then analyzed using knoledge gained from experience ith exactly ho the DFT ill change hen passing through various areas of changing signals. While these modifications are useful and may indeed e used in some form to augment the method descried in this document, they do not in themselves provide solid solutions to the ase prolem of the DFT, hich is that the resolution of the frequency and amplitude of component signals to a degree consistent ith human hearing requires sample times that invalidate the constant signal assumption asic to the DFT.

4 CFS-175 The Root Assumption In the folloing, e develop a means of using the tools of standard Fourier analysis herey e are not trapped y the limitations. To do this requires that one simple assumption e made: At any instant, the signal hich e ish to analyze is comprised of a small numer of specific frequency components. By small numer e mean less than infinite and in general, less than the numer of standard Fourier coefficients that ill e computed as part of the method. By specific frequency components, e mean that the frequencies of the actual signals ill not normally correspond exactly to the numered frequencies of any Fourier analysis or DFT that is eing used. By at any instant e mean that the frequency and amplitude of the specific frequency components may change ith time. Such changes ill usually e slo ith respect to the specific frequencies themselves, ut may also involve sudden transients. Development of the Method Returning to quation (2) from aove: 3 > ¹ Ð1Ð>Ñß Ñ œ.> 7 ( 1Ð>Ñ/ 2! Suppose 1Ð>Ñ is a sinusoid of frequency that does not match the integer spacings 1Î7ß 3 ut is rather Ð $17 Ñ Î. Then 1Ð>Ñ œ / 1 ( + $ Ñ> 7, here is a complex constant and 7 ¹ Ð1Ð>Ñß Ñ œ.> 7 ( / 3 1 ( + $ Ñ> / 3 1 > 7 7! [] œ.> 7 ( / 3 $1> 7 ()! Which evaluates to: 3 ˆ 31$ / ¹ Ð1Ð>Ñß Ñ œ Ð9Ñ 1$ The magnitude of this evaluates to: k k=3ð1$ Ñ É ¹ Ð1Ð>Ñß Ñ¹ Ð1Ð>Ñß Ñ œ º º Ð10Ñ 1$ Where the overar indicates complex conjugate. This is the =3ÐBÑ B sideand so

5 CFS commonly associated ith the use of the FFT. Notice that the value is independent of and relates only to $, the displacement from the frequency of interest,. Common practice is to regard these sideands as an unanted form of leakage or noise associated ith the method and to attack the prolem of the sideands y carefully choosing a indoing function, hich hen multiplied into 1Ð>Ñ prior to the transform ill cause the sideands to e as small as possile. In actuality as e shall see elo, the information in these sideands is a key element of full frequency analysis. Windoing typically oscures or destroys this necessary information. quation (10) shos that a reason for sideand leakage is that the signal eing measured does not coincide precisely ith any of the DFT reference frequencies. Let us examine quation (9) more carefully. Note that $ can e any value. It is useful at this point to define $ œ5%, here 5is a principle integer value of $ and % is the difference remaining, typically ut not necessarily a fraction. Thus e have: ¹ Ð1Ð>Ñß Ñ œ 3 ˆ 3 / 1 a5 % 3 ˆ 3 / / 1% œ 1 a5 % 1 a5 % 3 15 ut 5 is an integer value and therefore / œ, so 31% ¹ Ð1Ð>Ñß Ñ œ 3 a / 1 5 % Ð 11 a Ñ In practice e ill e analyzing a set of values of ¹ÐÑ for a sequential range of. {Note: 7/200: The folloing is more tricky than it looks. We are defining a function hich exists only for integer 5 ut succeed in using it as a continuous function. This is nearly as fortuitous for this application as the FFT is for standard Fourier series analysis.} For each of the successive increasing values of, $ ill e one less, since the (fixed) frequency of our 1Ð>Ñ is represented $. Since $ œ 5 %, 5 ill also increase y one for each successive and % ill remain constant. Thus, from a sequence of data C, here e use the shortened notation C ¹ Ð1Ð>ÑßÑ, e ill e trying to determine % from the values of C at different values of 5. The actual function e ill e trying to fit to the data ill e 31% Cœ 3/ a 1 a5 % 3/ a For this analysis, e can recognize that œ 1 is a single complex constant for the purposes of the fit and % is a single real constant. The value of can e found once and % have een determined. This leaves us ith 31% (12)

6 CFS Cœ a5% (13) =3ÐBÑ B quation (13) makes the remarkale statement that the sideand leakage due to actual signal frequencies not matching DFT reference frequencies produces coefficients hich correlate in a very simple ay a linearly iased inverse ith the degree to hich the actual frequency mismatches the nearest DFT frequencies. It is very desirale to put fitting equations into linear form (linear ith respect to the fitting constants) if possile, as it greatly simplifies the computations required to fit the equations. 5œ % C (1) quation (1) is one such form. quation (1) is unusual for to reasons. First, a least squares fit to this form ill e a fit to 5, normally thought of as the independent variale in terms of C, normally thought of as the dependent variale. Second, 5 and % are definitely real, hile and C are complex. Special consideration ill e given oth these unusual features. For the least squares fit to quation (1), normally the sum Š 5 5s Š 5 5s s ould e minimized, here 5 is the value calculated from the fit that corresponds to 5. Here the overar indicates complex conjugate, and e must sum the magnitude differences squared instead of Š 5 5s ecause 5s must e regarded as complex, containing the term. Because 5 is real, the 5 s resulting from the fit should e very C nearly real. Hoever, ordinarily e ould e minimizing!ac CsaC Cs, the sum of the magnitude differences squared for the independent variale in quation (13). We can approximate this y including a eighting factor in the original sum, such that: A AŠ 5 5s Š 5 5s ÐC CÑÐC s CÑ s To do this in this case it is useful to think of this in terms of small differences, herey e can approximate: A k? 5 k k? C k

7 CFS C % k? Ck.C kck A º º œ k? 5 k.5 kk ith.5 resulting from differentiating quation (13) [or quation (1)]. Since k k is a constant, it ill have no effect on the minimization, so the primary eighting factor is: A œ kc k % Note that this eighting factor merely restores the fit so that it more closely matches a least squares fit to quation (13). Other eighting factors may e found to have more desirale characteristics in practice. So, e ish to minimize: A Š 5 5s Š 5 5s A Œ 5 % Œ 5 % C C A Œ 5 % 5 % C C ith respect to the values of the constants real % and complex. Here it is useful to rite œ 3 V M A Œ 5 V M V M % 3 Œ 5 % 3 C C C C To find the minimum, e set the derivatives of the sum equal to zero, for each of %, V, and. M ` V M V M A Œ 5 % 3 Œ 5 % 3 œ! `% C C C C A Œ 5 V M V M % 3 5 % 3 C C C C œ!

8 CFS A Œ 5 % V Œ M3 Œ œ! C C C C C C C C A Œ 5 % V Œ M3 Œ œ! CC CC V/ÐCÑ M7ÐCÑ A 5 % V M œ! kc k kc k here kc k denotes the squared magnitude of C. ` V M V M A œ! ` Œ % Œ % C C C C V 5 5 V V A Œ % % œ! C C C C C C C C V A Œ 5 Œ % Œ œ! C C C C C C C C C C V A Œ 5 Œ % Œ œ! CC CC CC V/ÐCÑ V/ÐCÑ V A 5 % œ! kc k kc k kc k ` V M V M A œ! ` Œ % Œ % C C C C M

9 CFS M A Œ 35 Œ 3% Œ œ! C C C C C C C C C C M A Œ 35 Œ 3% Œ œ! CC CC CC The three equations are: M7ÐCÑ M7ÐCÑ M A 5 % œ! kc k kc k kc k V/ÐCÑ M7ÐCÑ A 5 % V M œ! kc k kc k V/ÐCÑ V/ÐCÑ V A 5 % œ! kc k kc k kc k Partitioning out the sums: In matrix form: M7ÐCÑ M7ÐCÑ M A 5 % œ! kc k kc k kc k A A V/ÐC Ñ M7ÐC Ñ % V A œ A 5 M kc k kc k V/ÐCÑ V/ÐCÑ % A A œ A 5 V kc k kc k kc k M7ÐCÑ M7ÐCÑ % A A œ A 5 M kc k kc k kc k

10 CFS Î!! V/ÐCÑ! M7ÐCÑ A A A Ñ Î! A5 Ñ kc k kck!! Î % Ñ A A! œ V/ÐCÑ! kck kck V/ÐC A5 Ñ V kc k (15) Ð ÓÏ M Ò Ð! A!! A Ï M7ÐCÑ Ó C! A5 k k kck M7ÐC Ñ Ò Ï kc k Ò And from this the three coefficients %, V, and M can e found, giving % and. We can also calculate that appears in quation (13) as œ 13 Î / 3 1% a Þ This gives oth the precise frequency of the tone and its precise amplitude. If e use the principle eighting factor A œ kc k % e otain: Î! %!! Ñ Î! % kck kck V/ÐCÑ kck M7ÐCÑ kck 5 Ñ! k k! Î % Ñ C V/ÐCÑ kc k! œ!kck V/ÐCÑ5 V Ð ÓÏ M Ò Ð Ó! kck M7ÐCÑ!! kck!kck M7ÐCÑ5 Ï Ò Ï Ò Both of these matrices are of the form: (16) Î +,. ÑÎ % Ñ Î] Ñ, -! V œ ] Ï.! - ÒÏ M Ò Ï]$ Ò Using Maple V Version 5, it is easy to otain the algeraic inverse of the 3x3 matrix so that: Î % Ñ Î -,. ÑÎ] Ñ Ð +-.,. œ Ó V, ] Ï Ò +-,. - - M Ï,. +-, ÒÏ Ò. ]$ - - Hœ+-,. a % œ -],].]$ H

11 CFS V œ +-.,. - - Š,] ] ]$ H From quation (15), M œ,. +-, - - Š.] ] ]$ H V/ÐCÑ M7ÐCÑ H œ A A A A kc k kc k kc k +œ A,œ A V/ÐC Ñ k C k -œ A k C k.œ A M7ÐC Ñ k C k ]œ A 5 ]œ A 5 V/ÐC Ñ k C k ]$œ A 5 M7ÐC Ñ k C k

12 CFS And for A œ kc k, from quation (16) % % Hœ kck kck kck V/ÐCÑ kck M7ÐCÑ +œkck %,œkck V/ÐCÑ -œkck.œkck M7ÐCÑ ]œkc k 5 % ]œkc k 5 V/ÐC Ñ ]$œkc k 5 M7ÐC Ñ

13 CFS Platform Least Squares The second major development in the method Using the form derived aove: Cœ a5% [13](1) ill produce reasonaly accurate frequencies and amplitudes of generated mixed frequency signals using the method descried in that report. For example, the folloing results ere otained for a randomly generated mixed frequency signal starting ith a Fast Fourier Transform on 20 (real) points. F in F out Amplitude in Amplitude out Tale I Sample Results of Original Least Squares Fit {Note: 7/200: Understand that the comments here ere made in the course of developing the method and do not refer to it's current state of development.} Using the aove method, the ehavior shon here is typical for the majority of components in generated test frequency signals. Hoever, the results for some components can sho much greater errors, particularly in amplitude. A numer of different phenomena contriute to such errors, and these are currently eing studied. The method has not yet een tested using real-orld data; there currently is no means of determining the precise frequencies and amplitudes in such data and thus no ay of checking hether the method is actually orking properly. It should e noted that the first point in achieving accurate results regards the use of. 5 The temptation is to simply use the index of the array element in the FFT array as 5Þ While mathematically sound, this approach is disastrous in practice for numerical

14 CFS reasons. When 5 is a large numer, the solution of the least squares equations leads to the sutraction of nearly equal large numers to otain a difference that is a tiny fraction of the original operands. Thus it is important to reduce the size of 5 y sutracting a ias from it. It is convenient to use a ias so that the range of 5 either starts at or straddles! if 5 is not used as a divisor in the calculation. This ias is added ack into the frequency at the conclusion of the least squares analysis. It is also clear that much can e gained y tuning the parameters of the least squares fit. This includes the numer of points included in the fit, the determination of the approximate position of frequencies to e analyzed ithin the FFT array, the positioning of the range of points aout that approximate frequency, and the eighting of the individual points ithin the least squares fit. The aove results are essentially a first cut at this, ut no real tuning has een performed at this point. As shon aove, quation [13](1) is reasonaly accurate ut does suffer from the cross interference eteen frequency components. What happens in practice is that the full signal also contains the tails of all the other frequencies and these tend to form a platform under the single frequency signal. Thus a first approximation of hat e have is Cœ a5% here G is a complex constant representing the sum of all the tails of frequencies to either side. Of course G is not really constant and each signal that contriutes to it ill taper off from one side to the other, the direction of the taper depending on hether the interfering signal is aove or elo the frequency of the target signal. Since signals aove the one of interest ill generally taper opposite to those elo, the sum ill gradually tist going along the frequency interval eing analyzed. Thus a etter approximation ould e Multiplying out, e get dividing y 5 Cœ a5% G GW5X5 $ C5% Cœ G5W5 X5 + G% W% 5X% 5 () C Cœ GW5X5 G% W% X% 5% (5) The variales are C and 5, so collecting terms according to constants Cœ GW G a % a % awx% 5X5 % C (') 5 5 This is in the linear form (2) (3)

15 CFS C Cœ+! $ + % (() 5 5 here +! ÞÞÞ+ 3 are complex and + %, hich corresponds to %, is real only. When the constants +! ÞÞÞ+ % are otained, then % œ + % () Xœ+ $ (9) Wœ+ X % (10) Gœ+ W! % (11) œ + G% (12) as efore the original œ 13Î ˆ / 3 1% can e otained from the result. To perform a least squares fit on this form, e need to find the minimum of! A ac CsaC Cs= Œ! $ C A C % Œ C +! $ 5 C + % = C C A Œ C +! $ 5 + % Œ C +! $ 5 + % Here +! ß ÞÞÞß + $ ß and C are the complex quantities, the remainder all eing real variales and constants. Because + % must e maintained as a real, to correspond ith %, it is expedient to expand all the coefficients.! C AŠ C !V!M V 5 M 5 V M $V $M % 5 C!V!M V 5 M 5 V M $V $M % 5 Š C

16 CFS We must successively take the derivative of this sum ith respect to +! R ß+!Mß + 1Rß+ 1 Mß + 2Rß+ 2 Mß + 3Rß+ 3 Mß and + %, set each expression to zero, and solve to find the minimum. ` `+!V! A a ÞÞÞ a ÞÞÞ œ! œ! C AŠ C !V!M V 5 M 5 V M $V $M % 5 C!V!M V 5 M 5 V M $V $M % 5 AŠ C Œ!V V V $V C C A C C % Œ œ! 5 5 Œ V!V V V $V CV A C % œ! 5 5 here the notation C V V/+6ÐCÑ and C M M7+13+<CÐCÑ. ` `+!M! A a ÞÞÞ a ÞÞÞ œ! œ! C A Š 3C !V!M V 5 M 5 V M $V $M % 5 C!V!M V 5 M 5 V M $V $M % 5 AŠ 3C a Œ a!m M M $M C C A 3 C C % œ! 5 5 Œ M!M M M $M CM A C % œ! 5 5 ` `+ V! A a ÞÞÞ a ÞÞÞ œ! œ! C C AŠ !V 5!M 5 V 5 M 5 V M $V $M % 5 C C 5!V 5!M 5 V 5 M 5 V M $V $M % 5 Š

17 CFS C C C C A Œ +!V +V + V + $V 5 + % œ! CV CV A +!V + V + V + $V 5 + % œ! ` `+ M! A a ÞÞÞ a ÞÞÞ œ! œ! C C AŠ !V 5!M 5 V 5 M 5 V M $V $M % 5 C C 5!V 5!M 5 V 5 M 5 V M $V $M % 5 Š C C C C A 3Œ +!M + M + M + $M 5 3+ % œ! CM CM A +!M + M + M + $M 5 + % œ! ` `+ V! A a ÞÞÞ a ÞÞÞ œ! œ! Aˆ C C!V!M V M V M $V $ $M $ % ˆ $ $ C C!V!M V M V M $V $M % ˆ $ A ac C ac C œ!!v V V $V % ˆ $ A C C œ! V!V V V $V % V ` `+ M! A a ÞÞÞ a ÞÞÞ œ! œ! ˆ $ $ A 3C C!V!M V M V M $V $M % ˆ $ $ 3C C!V!M V M V M $V $M %

18 CFS ˆ $ A 3 ac C ac C œ!!m M M $M % ˆ $ A C C œ! M!M M M $M % M ` `+ $V! A a ÞÞÞ a ÞÞÞ œ! œ! ˆ $ $ % % A C C5!V!M V M V M $V $M % ˆ $ % % C $ C 5!V!M V M V M $V $M % ˆ $ % A ac C ac C 5 œ!!v V V $V % ˆ $ % A C C 5 œ! V!V V V $V % V ` `+ $M! A a ÞÞÞ a ÞÞÞ œ! œ! ˆ $ $ % % A 3C C 5!V!M V M V M $V $M % ˆ $ $ % % 3C C 5!V!M V M V M $V $M % ˆ $ % A 3aC C ac C 5 œ!!m M M $M % ˆ $ % A C C 5 œ! M!M M M $M % M! a a ` C C `+ A ÞÞÞ ÞÞÞ œ! œ $M 5 5 CC C C C C CC A 5 +!V 5 3+!M 5 + V 5 3+ M 5 + VC 3+ MC + $VC5 3+ $MC5 + % 5 CC C C C C CC Š 5 +!V 5 3+!M 5 + V 3+ M + VC 3+ M C + $VC5 3+ $MC5 + % 5 5 5! Š

19 CFS CC C C C C C C C C AÐ +!V 3+!M + V 3+ M + VaC C CC 3+ MaC C + $VaC C5 3+ $MaC C5 + % Ñœ! 5 CC CV CM CV CM A Ð +!V +!M + V + M + VCV + MCM CC + $VCV5 + $MCM5 + % Ñ œ! 5 So that the collected nine equations in nine unknons are: Œ V!V V V $V CV A C % œ! 5 5 Œ M!M M M $M CM A C % œ! 5 5 CV CV A +!V + V + V + $V 5 + % œ! CM CM A +!M + M + M + $M 5 + % œ! ˆ $ A C C œ! V!V V V $V % V ˆ $ A C C œ! M!M M M $M % M ˆ $ % A C C 5 œ! V!V V V $V % V ˆ $ % A C C 5 œ! M!M M M $M % M

20 CFS CC CV CM CV CM A Ð +!V +!M + V + M + VCV + MCM CC + $VCV5 + $MCM5 + % Ñ œ! 5 In matrix form, here each element represents a sum over : V Ô A! A 5! A5! A5! A 5! A! A 5! A5! A5 CM Ô AC V A 5 Ô +!V AC M CV A! A! A! A 5! A +!M Ö Ù A C V! A C + V 5 M 5! A! A! A 5 A M A C M 5 A5! A! A5! A5! AC V $ + V œ AC V5! A5! A! A5! A5 $ AC + M M AC M5 $ % + $V A5! A5! A5! A5! AC V5 Ö Ù AC V5 + $ % $M Ö! A5! A5! A5! A5 ACM5 ÖAC M5 Ù ÙÕ + Ø % k CV CM CV CM kck Õ A C k Ø ÕA A A A AC V AC M AC V5 AC M5 A 5 Ø C (13) This system is too complicated to satisfactorily solve symolically ut can e solved easily using numeric matrix inversion or another linear system solution method on a case-y-case asis. As efore, the eighting factor is availale to adjust the method for etter performance. This method has een given preliminary tests in hich it appears to perform quite ell in determining precise frequencies and amplitudes for mixed signals.

21 CFS F in F out Amplitude in Amplitude out Tale II Sample Results of Platform Least Squares Fit Tale II shos that the results of this platform method are, in a ord, amazing. At this point, these are typical results in the same sense that the results shon in Tale I are typical for that method. It ill e noticed in the aove that one component, near 0 œ %%), is glaringly in error. At this point, this happens. Here it is due to to random frequencies eing too close together. The second frequency is completely unreported. Sources of such errors and methods for dealing ith them are eing pursued and it is fully expected that satisfactory resolutions ill e found. The aove methods deal ith single frequencies, ut in some cases it is possile to deal ith multiple frequencies still using linear least squares. For example, ith the frequencies knon, a linear least squares to the corresponding amplitudes is quite feasile: C œ! ÞÞÞ 5% 5% 5% here the % 3 are knon is in simple linear least squares form. It is also possile to determine a second frequency given knon neary frequencies. (1) Cœ! (&) a5% 5% 5% here % and % and perhaps more frequencies are knon and included. quation (15) can e treated in the same manner as quation (3) as treated aove. In fact it ould e quite feasile to add further platform correction terms as as done in quation (3) if this proves desirale. Furthermore, it is also quite feasile to perform a nonlinear least squares fit using one of the aove methods to otain most of the coefficients using linear least squares and a nonlinear method to vary the knon frequencies in order to determine an overall minimum sum of squares.

22 CFS Multiple Adjacent Frequencies Here the method is extended to the general case of multiple closely spaced frequencies The form of quation 13 from the first part aove, Cœ a5% [13](1) as extended to the form Cœ a5% GW5X5 [3](2) of quation 3 in the second part (Platform Least Squares) aove. In Platform Least Squares, it as remarked that the situation of to (or more) neary frequencies could cause errors in an analysis that otherise performed remarkaly ell. In general, this prolem ecomes important if the there are to or more precise frequency signals of significant amplitude ithin the range of frequencies represented y the span of coefficients from the FFT or DFT that are used as data for the least squares fit. The root assumption of this method is at any instant, the signal hich e ish to analyze is comprised of a small numer of specific frequency components, hich e elieve to e the case for many everyday signals. Nonetheless, there is nothing to preclude several of this small numer from eing adjacent to one another. For many practical purposes it ill e sufficient to report such groupings as a single frequency. The ear does not separately detect the three frequencies of a note in the upper octaves of a piano, ut ill detect the eat as a slo variation of amplitude of the single frequency, hich this method is also capale of tracking. Nonetheless, it is likely that there ill e cases in hich it is more desirale to separate groupings of a fe neary frequencies. The most ovious ay of dealing ith the situation of several neary frequencies is to choose the span of data so that it avoids multiple frequencies, even if this requires that the sample e less symmetric aout the target frequency. Hoever, it is also possile to fit multiple precise frequencies that occur ithin a span, as ill e shon. These frequencies can even fall in a single gap eteen to adjacent frequencies of the DFT, although it is to e expected that as the signals ecome more similar, separating them ill ecome less accurate. ach ne frequency that is added adds a complex amplitude coefficient, :, and a real frequency, %:, hich means that to maintain the same excess degrees of freedom in the least squares fit, 1 ne points must e attached to the span, idening it and potentially including more frequencies, hich fact must e alanced against the aility to detect an additional frequency. The 1 point requirement comes aout ecause each frequency adds three ne parameters; to from the complex amplitude and one from the real frequency, hile each additional point from the DFT span adds only to values, from the single complex Fourier coefficient.

23 CFS We address the case of neary frequencies using the form Cœ GW5X5 a 5% : : : (3) for the case of a cluster of : neary frequencies. First e ill deal ith the case of just to neary frequencies and then ith three in order to develop a general formula. Cœ Multiplying out, e get GW5X5 () a5% a5% C5 C5( % + % ) % % Cœ % % a 5G5 G5( % + % ) G% % $ % $ W5 W5 (% + % ) W5% % X5 X5 (% + % ) X5 % % (5) dividing y 5 C œ cg W( % 1+ % 2) X%% 1 2da % 2 % G%% C C cg( % 1+ % 2) W% 1% 2 a d cx( % 1+ % 2) Wd5 X5 ( % 1+ % 2) % 1% 2 (6) This is in the linear form C C C œ +! $ 5 + % 5 + & + ' (() here +! ÞÞÞ+ % are complex and + & and + ', hich correspond to (% 1+ % 2) and % 1% 2, are real only. When the constants +! ÞÞÞ+ 6 are otained, then it can e recognized that ab% 1aB% œb ( % 1+ % 2) B% 1% 2 and thus % 1 and % 2 are the roots of the standard quadratic B + B+ œ!. & ' % 1 œ + & È+ & %+ ' () œ + È + %+ & & ' % (9) Xœ+ % (10) Wœ+ $ X (% + % ) (1) 1 2

24 CFS Gœ+! W( % + % ) X% % (12) G( % 1+ % 2) W% 1% 2 a œ œ + G( % + % ) W% % % % G% % œ % œ + % G% % % œ + + % G( % + % )% W% % % G% % a% % œ+ + % G% W% % G% 1 W% 1% % œ (1 $ ) % % 2 G W + + % œ % 2 % % 2 (1%) % % as efore the original œ 13 Î ˆ 1% / : : : 3 can e otained from the result. No e con continue on to the case of three neary frequencies in order to see ho the system ehaves as e add frequencies. Multiplying out, e get Cœ + + GW5X5 (&) a5% a5% a5% $ C5 C5 (% + % % ) C5a% % % % + % % C% % % œ $ $ $ 2 1 $ 5 5( % + % ) % % 5 5( % + % ) % % ( % + % ) % % $ G5 G5 (% + % % ) G5a% % % % + % % G% % %

25 CFS % $ W5 W5 (% + % % ) W5 a% % % % + % % W5% % % & % $ X5 X5 (% + % % ) X5 a% % % % + % % X5 % % % Ð16Ñ dividing y 5 $ CœcGW( % + % % ) Xa% % % % + % % d a %% $ 2 %% 1 $ 3 %% 1 2G%%% $ c (% + %) (% + %) (% + %) Ga%% %% + %% W%%% d 5 $ 2 1 $ ca G( % + % % ) Wa% % % % + % % X% % % cwx( % + % % ) d5x ( % C C C 1+ % 2 % 3) a% 1% 2 % 1% 3+ % 2% 3 % 1% 2% 3 Ð17Ñ $ This is in the linear form d 5 C C C C œ +! $ 5 $ % & ' ( ) $ here +! ÞÞÞ+ & are complex and + ' ÞÞÞ+ ), hich correspond to (% 1+ % 2 % 3) ß a%% 1 2 %% 1 3+ %% 2 3 and %%% 1 2 3, are real only. When the constants +! ÞÞÞ+ ) are otained, then %%% 1 2 3œ + ) (1) %% %% %% œ ( % 1+ % 2 % 3 œ + ' These can e recognized as the standard form coefficients for a cuic equation ith the roots %, %, and % $ B + ' B + ( B+ ) œab% 1aB% ab% $ œ! Ð19Ñ This equation can e solved directly using the cuic solution (Cardan's solution) D œ $'+ +!)+ )+ É + $+ + &%+ + + )+ + + ' ( ) ' $ $ $ ( ( ' ' ( ) ) ) '

26 CFS Aœ + $ + * ( ' D $ D $ + ' $ ' % œ 'A D $ + 3 È$ D $ œ $A 'A $ ' % ' For this to e valid all three D $ + 3 È$ D $ œ $A 'A $ ' % $ ' % p must e real. Xœ+ & (20) Wœ+ % X( % + % % ) (21) Gœ+! W( % + % % ) Xa% % % % + % % (22) %% %% %% G%%% œ + $ 2 1 $ (% + %) (% + %) (% + %) Ga%% %% + %% W%%% œ + $ 2 1 $ a G( % + % % ) Wa% % % % + % % X% % % œ $ $ W X G c a% 2 % œ $ % 2% $ d% 1 % 1 $ % Ð23Ñ a% % a% % $ W X G c a% œ % $ % % $ d% $ % $ % Ð2Ñ a% % a% % $ W X G $ c a% œ % % % d% $ $ % $ $ % $ Ð25Ñ a% % a% % $ $ As in the case ith to frequencies, once 1 is otained, and 3 can e otained through symmetry, and as efore the original œ 13 Î ˆ 1% / : (26) : : 3

27 CFS can e otained from the result. Clearly, more frequencies can e added using a similar technique. The fitting equation ill remain linear ith respect to constants and in the same form and each added frequency ill require the solution of a standard polynomial of one higher order for the frequencies and a set of linear equations of one higher order for the complex amplitudes. This solution need not e done algeraically; it can e done numerically if that is easier. Looking at quation [7] from Supplement 1 and quations (7) and (1) from this report C Cœ+! $ + % [] ( 5 5 C C C œ +! $ 5 + % 5 + & + ' (() C C C C œ +! $ + % 5 + & 5 + ' + ( + ) these can e expressed as an equation of common form: $ $ (1) C C œ +! :$,: Ð(Ñ 5: 5: here : œ ß ß $ for the aove cases and is extended for cases of more frequencies. The coefficients + are complex and, are real. :œ This corresponds to CœGW5X5 ()) a 5 % :œ : : and in general the % : are the solutions of B : : Ð Ñ, B œ! (*) :œ : hich e have shon for the linear, quadratic, and cuic case to e: % œ, Ð$!Ñ È,, %, % 1 œ ( $ )

28 CFS-175 2, È, %, œ ( $ ) % D œ $',,!), ), É, $,, &%,,, ),,, $ $ $ $ $ $ $ Aœ, $, * D $ D $, œ 'A Ð$$Ñ ' $ % D $, 3 È$ D $ œ $A 'A Ð$%Ñ $ ' % D $, 3 È$ D $ œ $A 'A Ð$&Ñ $ ' % $ and from comining other results: Xœ+ ( $' ) Wœ+ X % ( $( ) : :œ Gœ+ W % X % % ( $) )! : < = :œ ;œ <œ; The coefficients for + $ ÞÞÞ+ along ith the aove results ill formulate into a series of linear equations in the unknons, p. Corresponding to the linear, quadratic, and cuic cases for % aove, the solutions are: œ + $ G% (12) G% 1 W% 1% % œ (1 $ ) % % 2 G W + + % œ % 2 % 3% 2 (1%) % %

29 CFS $ W X G c a% 2 % $ % 2% $ d% 1 5 % 1 3% œ Ð23Ñ a% % a% % $ W X G c a% œ % $ % % $ d% $ 5 % % 3% Ð2Ñ a% % a% % $ W X G $ c a% œ % % % d% $ $ 5 % % $ 3% $ Ð25Ñ a% % a% % $ $ C C œ +! :$,: Ð(Ñ 5: 5: :œ To perform a least squares fit on this form, e need to find the minimum of A ac CsaC Cs= C AC +! :$ :, : : 5 5 :œ C C +! :$, : 5 5 :œ : : = C AC +! :$ :, : : 5 5 :œ C C +! :$ :, : : 5 5 :œ Here +! ß ÞÞÞß + $ ß and C are the complex quantities, the remainder all eing real variales and constants., ÞÞÞ, must e maintained as real ecause all the,: are additive and multiplicative cominations of the % ;. Due to the mixture of real and imaginary coefficients, it is expedient to expand all the coefficients as the sums of real and imaginary parts.

30 CFS !! C AC + R! 3+ M! + V 5 3+ M 5 + V 5 3+ M 5 + V:$ : 3+ 5 M:$ :, 5 : : 5 :œ C C + R! 3+ M! + V 5 3+ M 5 + V 5 3+ M 5! + V:$ : 3+ M:$ :, 5 5 : : 5 :œ We must successively take the derivative of this sum ith respect to + V! ß + M! ß ÞÞÞß + V$ ß + M$ and, ÞÞÞ,, set each expression to zero, and solve to find the minimum. Using the notation C V V/+6ÐCÑ and C M M7+13+<CÐCÑ, first ork the complex + 's for ; œ!þþþ hich are the terms outside the inner sum. ` `+ V; ;! A a ÞÞÞ a ÞÞÞ œ! œ! AŠ C ; ; ; ; ; ; R! M! V M ; V M ;: ;: ;:! , C 5 V:$ M:$ : :œ ; ; ; ; ; ; ; R! M! V M V M ;: ;: ;:! V:$ M:$ : :œ AŠC , C 5 œ AŠ ac C ; ; ; ; R! V V ;: ;: + 5, ac C 5 œ! :œ V:$ : AŠ C ; ; ; ; V R! V V + 5, C 5 œ! :œ ;: ;: V:$ : V (2) ` `+ M;! A a ÞÞÞ a ÞÞÞ œ! œ! AŠ3C ; ; ; ; ; ; R! M! V M ; V M ;: ;: ;:! , C 5 :œ V:$ M:$ :

31 CFS ; ; ; ; ; ; ; R! M! V M V M ;: ;: ;:! V:$ M:$ : :œ AŠ3C , C 5 œ AŠ3aC C ; ; ; M! M ; M ;: ;: + 5 3, ac C 5 œ :œ M:$ : AŠ C ; ; ; M M! M ; M + 5, C 5 œ! :œ ;: ;: M:$ : M (29) Next, ork the complex + ;'s for ; œ $ÞÞÞ, hich are the + coefficient terms inside the inner sum. ` `+ V;! A a ÞÞÞ a ÞÞÞ œ! œ! AŠ C ; ; ; $; $; %; %; R! M! V M V M ;: ;: ;: ; ;! , C 5 A ŠC V:$ M:$ : R! :œ ; $; $; %; M! V M V %; ;: ;: ;: M! V:$ M:$ : :œ , C 5 œ AŠ ac C ; ; $; %; R! V V + 5, ac C 5 œ :œ ;: ;: V:$ :

32 CFS AŠ C ; ; $; %; V R! V V + 5, C 5 œ! :œ ;: ;: V:$ : V (30) ` `+ M;! A a ÞÞÞ a ÞÞÞ œ! œ! A Š 3C ; ; ; $; $; %; %; R! M! V M V M ;: ;: ;: ;! , C5 AŠ3C5 3 V:$ M:$ : :œ ; ; $; $; R! M! V M %; %; ;: ;: ;: V M V:$ M:$ : :œ ! , C5 œ A Š 3 ac C ; ; $; %; M! M M + 5 3, ac C 5 œ :œ ;: ;: M:$ : AŠ C ; ; $; %; M M! M M + 5, C 5 œ! :œ ;: ;: M:$ : M (31) Finally e need to ork the real,;'s for ; œ ÞÞÞ. `! ` ; AaÞÞÞaÞÞÞ œ! œ! ; ; ; ; ; ; Š R! M! V M V ; ;: ;: ;: + M C5! + V:$ C5 3+ M:$ C5,CC5 : + :œ ; ; ; ; ; ; Š R! M! V M V ; ;: ;: ;: + M C5! + V:$ C5 3+ M:$ C5,CC5 : œ! :œ A CC5 + C5 3+ C5 + C5 3+ C5 + C5 3 A CC5 + C5 3+ C5 + C5 3+ C5 + C5 3

33 CFS AŠ kck 5 + C 5 + C 5 + C 5 + C 5 + C 5 ; ; ; ; ; ; R! V M! M V V M M V V ; ;: ;: ;: + C 5 + C 5 + C 5, kc k 5 œ! Ð$Ñ M M V:$ V M:$ M : :œ In matrix form, these equations can e expressed as: Q! œ here! is the vector of coefficients and is the vector of dependent variales, here each element represents a sum over Þ Ô AC V + V! Ô AC M + Ö Ù M! AC V5 + V AC M5 + M + AC V5 V + AC M5 M + AC! œ V$ V5 œ + M$ AC M5 ã ã + V AC V5 + M AC M5, Ö Ù AkC k 5 ã Õ, Ø Ö ã Ù Õ A kc k 5 Ø In the matrix Q elo, each element represents a represents a sum over and also carries an implicit A, the eighting factor that is common to all elements. For example, in the representation elo Q œ! A and the last element of the first ro Q œ! A V/+6ÐC Ñ5. ß$ ß

34 CFS Ô! 5! 5! 5! á 5! CV5 á CV5!! 5! 5! 5 á! 5 CM5 á CM5 $ 5! 5! 5!! á 5! CV á CV5 $! 5! 5! 5! á! 5 CM á CM5 $ % 5! 5! 5! 5! á 5! CV5 á CV5 $ %! 5! 5! 5! 5 á! 5 CM5 á CM5 5!! 5! 5! á 5! CV5 á CV5! 5!! 5! 5 á! 5 CM5 á CM5 ã ã ã ã ã ã ã ã á ã ã ã á ã 5! 5! 5! 5! á 5! CV5 á CV5! 5! 5! 5! 5 á!! CM5 á CM5 CV5 CM5 CV CM CV5 CM5 CV5 CM5 á CV5 CM5 C 5 k k á kck 5 Ö ã ã ã ã ã ã ã ã á ã ã ã ã ã Ù Õ C 5 C 5 C 5 C 5 C 5 C 5 C 5 C 5 á C 5 C 5 kck 5 á kck 5 Ø V M V M V M V M V M As efore, the coefficients! can e otained y solving Q! œ, hich can e done y inverting the matrix! œq or y other means. The frequencies and amplitudes are then otained from the linear coefficients as shon aove. Preliminary results This method as programmed and has given highly accurate results using test cases - in line ith those for single frequencies taulated in Supplement 1. In fact, it as possile to separate sets of to and three precise frequency signals hich ere all eteen to successive frequencies of the FFT eing used. Although the treatment of three adjacent frequencies is a convenient and practical place to stop, there is no reason hy even more adjacent signals could not e treated. At this point it does appear important to determine ho many signals are present in an interval and to perform the corresponding analysis in order to otain maximum precision. This may e done efore the fact, y examining the patters present in the Fourier coefficients, or after the fact, y examining the results of the fits after they are performed. The next step in development ill e defining an efficient procedure to determine this.