njit-etd2003-081 - New Jersey Institute of Technology

177
different values of A, over the length of f(x), the signals are multiplied, and the area
under the product of the two signals is computed. If the cross-correlation function is
normalized by the product of the square root of the auto cross-correlation of each
function, then the normalized cross correlation will have a range from 0 to 1. For two
signals that are the same, the cross-correlation at zero lag (when A= 0) is equal to 1.
The closer the cross-correlation value is to 1, the more similar the waveforms are to one
another.
It should be pointed out that the power spectra (Fourier and wavelets) in this study
are not continuous, but discrete. Therefore, to compute the discrete cross-correlation, the
following equation was used [26]:
where N is the number of points in the signal. To compute the cross-correlations for this
section, a Matlab program was written and is listed in Appendix B.
Based on the cross-correlation method presented, the cross-correlation coefficient
between the Fourier power spectrum and the wavelet power spectrum was calculated for
each COPD subject. Table 5.1 tabulates the cross-correlation results. It should be noted
that the table displays the maximum absolute value of the cross-correlation function.
From the table and the graph in figure 5.17, the values are from 0.2 to over 0.9 but all
values indicated that the Morlet wavelet is the best representation.