A New Method for Spectral Analysis and Its Application System
There are many spectral analysis methods using the orthogonal systems other than trigonometric functions such as Wavelet or Walsh functions, and some of them already have been used widely in practical applications. Here a new orthogonal system is introduced. It is useful specifically for analyzing the signals in which many spikes exist such as heart pulsations.
Firstly we consider the function system {fn(x) (n = 0, ±1, ±2, ...)} on [0, 2p] expressed as,
fn(x) = lim m® ¥ (4mp)-1/2 Sj=-mm exp{i(2j-1)(nx-b)} (b: constant and 0 £ b £ p)
The following figures are the illustrations of that system.
pre-brick functions
The top row of left figures corresponds to f1(x) mentioned above and the second one is for f2(x), and the top row of right figures corresponds to f-1(x) and the second one is for f-2(x) and so on. Each function has only three values: 0 and ±¥. From the shape of these functions, here they are called pre-brick functions. When a function g(x) in C[0, p] is given, and the coefficient aj and the function g#(x) are defined as,
aj = ò02p dx fj(x) g(x)
and
g#(x) = Sj=-¥¥ aj fj(x) ...1)
then g#(x) is not only different from g(x), but also it is discontinuous everywhere. Nevertheless when the constant b appeared in the definition of fn(x) is rational number, it is known that the function system {fn(x) (n = 0, ±1, ±2, ...)} is the complete orthogonal normal system on C[0, 2p] (Þ proof). Namely the following equation is satisfied in general.
ò02p dx |g#(x) - g(x)|2 = 0
Since aj are infinitesimal values (Þ remark), it is not convenient for practical analysis to use aj. Hence we modify the above pre-brick functions as,
bn(x) = 1/p (when x = (b+2kp)/n, k: any integer)
= -1/p (when x = (b+(2k+1)p)/n, k: any integer)
= 0 (other cases)
brick functions
The top row of left figures above corresponds to b1(x) and the second one is for b2(x), and the top row of right figures corresponds to b-1(x) and the second one is for b-2(x) and so on. Each function has only three values: 0 and ±1/p.
Here the functions bn(x) are called brick functions. The brick function system {bn(x) (n = ±1, ±2, ...)} is not normalized system. In return using this system, the former expansion equation (1) is rewritten with finite coefficients cj as,
g#(x) = Sj=-¥¥ cj bj(x) (j ¹ 0) ...1)'
where the coefficients cj are calculated in the range of [0, 2p] as,
It is easily imagined that the brick function system is more adequate than trigonometric function system on the spectral analysis when the input signals contain many spikes because the narrower the width of a spike, the fewer the brick functions which have the expansion coefficient other than null, while the narrower the width of a spike, the more the trigonometric functions having different frequency in Fourier transformation.
Based on the method described above, FBT - fast brick function transformation - Spectral Analyzer has been developed. The following figures are the examples obtained from the analysis on virtual cardiograms using that system.
1) Normal heart action
Input Signals - Normal sinus rhythm
Spectral Analysis by FBT
Spectral Analysis by usual DFT
The upper figure is the simulation of normal heart action. The lower left figure is the result of spectral analysis by FBT Spectral Analyzer. The software used for it has been developed with Borland C++ and in order to analyze the signals successively, a method similar to that of FerFT has been employed. As the reference, the result of spectral analysis on the same signals as the former one by usual discrete Fourier transformation is shown in the lower right. Obviously the spectra shown in the lower right figure disperse more widely than that of left one, or the spectra obtained from FBT. As another noticeable point, the peak of spectra obtained from FBT seems to be a little higher than that obtained from usual DFT. This inclination can be seen generally.
2) Abnormally slow heart action
Input Signals - Sinus bradycardia
Spectral Analysis by FBT
Spectral Analysis by usual DFT
As well as the former example, the dispersion of spectra on FBT is narrower than that of usual DFT. Also by comparing the feature of spectra between these two examples obtained from FBT, it is thought to be possible to distinguish the bradycardia from normal sinus rhythm.
3) Irregular heart rhythm
Input Signals - Sinus arrhythmia
Spectral Analysis by FBT
Spectral Analysis by usual DFT
In both cases of FBT and DFT, the dispersions of spectra are wider than the previous cases and it seems the difference between them is less distinctive.
4) Coronary insufficiency
Input Signals - Myocardial ischemia
Spectral Analysis by FBT
Spectral Analysis by usual DFT
As well as the case (1) and (2), the dispersion of spectra obtained from FBT is narrower than that of usual DFT. However when comparing with the case (1), we notice the spectra obtained from FBT in the case of (4) are almost same as that of case (1), although after increasing the resolution we can get some fine differences between them. On the other hand the spectral analysis by usual DFT shows more distinctive difference between the case (1) and (4) particularly on the first spectrum. Therefore in some cases such as the case (4), FBT requires finer resolution than that of usual DFT to analyze the pattern of input signals.
The software for FBT Spectral Analyzer is available to download by click here (277kb). You can simulate the above results by yourself. The hardware has been developed based on Graham Cattley's "Pocket Sampler" published on Electronics Australia in August 1996 and is completely compatible with DATAKit #80-112 (for the detail of DATAKit #80-112, please visit: Minute Man Electronics - Pocket Sampler/Data Logger for PC
Philmore DATAKit # 80-112.)
