Truncation of the Infinite Fourier Transform - Signal Processing of Random Physiological Signals - page 161

expect about a 9% error at each region of discontinuity in a signal. It should be noted
that discontinuities may occur at the beginning and end points of the analysis window.
Can one correct for the discontinuities and the Gibbs phenomenon? There are
a few approaches that may be used. For example, one could use the low-pass filter the
signal to remove the high-frequency components, thus smoothing the signal, for example,
rounding off the QRS in the electrocardiogram (ECG) signal. Or, one could avoid any
analysis in the region of discontinuity by “Piece-meal” or segment analysis.
14.2 SUMMARY
Keep in mind that all Fourier analysis are approximations due to truncation in the
number of Fourier coefficients; therefore, from “lessons learned,” one should do the best
approximation possible, knowing that 100% accuracy is not attainable. The smoother
the signal, the faster it will converge, thus requiring fewer Fourier coefficient terms.
Discontinuities either at the edges or in the middle of the signal will result in greater
discrepancy (error). Remember that the Gibbs phenomenon indicates that approximately
a 9% error will occur in the region of any discontinuity.