Windowing Techniques



The Gibbs effect occurs whenever a discontinuous periodic signal is approximated with a truncated Fourier series. The amplitude of the overshoots at the discontinuities does not decrease as more harmonics are included. However, additional harmonics cause the duration of the overshoots to decrease, consistent with the expected reduction of integral squared-error.

By attenuating the amplitude of the included harmonics, the Gibbs effect can be removed. However, the approximation will no longer be a mimimum integral squared-error approximation. This is called windowing. There are many windowing methods available; two are illustrated here: the F� Window and the Hamming Window.

F� Window

For the F� window, if N harmonics are included in an approximation, the amplitude of the kth harmonic is multiplied by the factor (N-k)/N.
The 12 Term Fourier Series of a Square Wave with a F� Window
N=12
Amplitude Frequency (rad/sec) Phase (degrees)
(12-1)/12 � 1 1 0
(12-3)/12 � 1/3 3 0
(12-5)/12 � 1/5 5 0
(12-7)/12 � 1/7 7 0
(12-9)/12 � 1/9 9 0
(12-11)/12 � 1/11 11 0


The overshoot is decreased. Including more harmonics further decreases the overshoot.


The 24 Term Fourier Series of a Square Wave with a F� Window
N=24
Amplitude Frequency (rad/sec) Phase (degrees)
(24-1)/24 � 1 1 0
(24-3)/24 � 1/3 3 0
: : :
: : :
(24-21)/24 � 1/21 21 0
(24-23)/24 � 1/23 23 0

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