Window Functions and Spectral Leakage - Signal Processing of Random Physiological Signals - page 175

CHAPTER 16

Window Functions and
Spectral Leakage

The purpose of window functions is to minimize a phenomenon called “Spectral
Leakage,” which is the result of selecting a finite time interval and a nonorthogonal
trigonometric basis function over the interval of analysis. Only frequencies that coincide
with the basis function will project on a single basis vector, such as shown in Fig. 16.1.
Signals, which include frequencies not on the basis set, will not be periodic in the observ-
ing interval (often referred to as the “Observation or Observing Window”). The periodic
extension of a signal with a period that is not equal to the fundamental period of the
window (as in Fig. 16.2) will produce discontinuities at the boundaries of the observing
interval. It is these discontinuities that cause the frequency leakage of the signal across
the spectrum.
The sine function in Fig. 16.2 does not fit perfectly in the window, since its value
at the beginning is not the same as the value at the end of the observation interval [4].
Since the window is supposed to contain a periodic function, one should note that in
the next cycle, the value must go from 1 to 0 at the same instant of time. Hence, the
discontinuity at the edge of the window will result in the spectral power of a specific
frequency leaking out of the main lobe or specific spectral bandwidth to adjacent spectral
side lobes. Leakage into the side lobes may occur in either side (direction) of the main
lobe. The decrease of power from the main lobe adds to the power in the sideband
harmonics resulting in an erroneous estimate of power in the sideband harmonics.