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61 AMPLITUDE-FREQUENCY RELATIONSHIPS Fourier analysis results have one further surprise in store-the equation of the dotted upper boundary connecting coarse structure peaks m Figure A I-l is: . = - , Where " " is a constant. If we now assume the reality of a line spectrum, and ignore the possible existence of modulating Unes for the sake of simplification, each component of the coarse structure is denoted as follows: Cj- = a,- sin (coft + 1) Differentiating: = atOj cos (oj/ + 0,) And conadering the previously noted relationship between a, and a?,: ,- = cos (w,-f + 0,) This relationship impUes that the maximum time rate of change of each spectral element in the coarse structure is identical to that of every other line in the spectrum-and that this deUcate balance is maintained over many calendar years by a precise and particular relationship between amplitude and frequency! Once more we are forced to question why such an orderly and interesting relationship should exist in the spectra of stock prices. And what about the implication regardmg the nature of human decision processes which are responsible for price change and the resulting ordered spectral signature observed? The above finding regarding equaUty of the maximum time rate of change of prices due to each spectral component is quite useful in apphcations, and is the basis of the simple technique of noting how many components are up or down at a given time in order to resolve chart patterns, etc. The very fact that this technique works constitutes a test of the general hypothesis. A detailed study of Figure IX-4 reveals just how effectively this test is passed. THE USE OF COMB HLTERS The results of Fourier analysis are useful, but quite general in nature. In addition, numerical analysis based on equi-spaced digital data can sometimes lead to erroneous conclusions. The results obtained can be validated and considerably extended through the use of overlapping combs of digital filters. A typical such comb of filters is shown in Figure A 1-2. In this particular case, the response bands of the individual filters were arranged with identically equal frequency spacings, and so constructed that each frequency was viewed by at least four separate filters. In this way, if spectral energy is not separated into discrete lines (as implied by the Fourier analysis), we should expect filter output more or less equally in all regions
IDEALIZED COMB FILTER FIGURE A1-2 I 23456 78 9 10 1112 13141516 17 1819 2021 22 23  OjO IIjO ANGULAR FREQUENCY RAO./YR. I2X> IZA Setting Up Overlapping Bandpass Filters
5-24-40 +2 FC-i -2 +2 FC-« -2 ♦2 FC-S -2 +2 FC-4 0 -2 FC-5 0 -2 +2 FC-e -2 +2 FC-7 -2 PC-e 0 +2 FC-9 0 -2 +4 FCHO 0  COMB OUTPUT EXAMPLE FIGURE AI-3 9-4 -I 0 25 50 75 0 125 150 175 200 225 250 WKS. Typical "Comb" Filter Results
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