NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW
VERSION: "The Digital Audio
Resampling Home Page", by Julius O. Smith III,
Copyright © 2016-05-17 by Julius O. Smith III -
Center for Computer Research
in Music and Acoustics (CCRMA), Stanford University
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From Theory to Practice
The summation in Eq. (1) cannot
be implemented in practice because the ``ideal lowpass filter'' impulse
response 
actually extends from minus infinity to infinity. It is
necessary in practice to windowthe ideal impulse response so
as to make it finite. This is the basis of the window method
for digital filter
design[Digital Signal
Processing Committee 1979a,Rabiner
and Gold 1975]. While many other filter design techniques
exist, the window method is simple and robust, especially for very
long impulse responses. In the case of the algorithm presented
below, the filter impulse response is very long because it is
heavily oversampled. Another approach is to design optimal
decimated ``sub-phases'' of the filter impulse response, which are
then interpolated to provide the ``continuous'' impulse response
needed for the algorithm [Putnam
and Smith 1997].
Figure 3
shows the frequency
response of the ideal lowpass filter. This is just the Fourier transform of .
Figure 3:Frequency
response of the ideal lowpass filter.
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If we truncate at the fifth zero-crossing to the left and the right of
the origin, we obtain the frequency response shown in
Fig. 4.
Note that the stopband exhibits only slightly more than 20 dB rejection.
Figure 4:Frequency
response of the ideal lowpass filter after rectangularly windowing
the ideal (sinc) impulse response at the fifth zero
crossing to the left and right of the time origin. The vertical
axis is in units of decibels(dB), and the horizontal axis is labeled in
units of spectral samples between plus and minus half the sampling rate.
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If we instead use the Kaiser window [Kaiser 1974,Smith 1992]
to taper to zero by the fifth zero-crossing to the left and the
right of the origin, we obtain the frequency response shown in
Fig. 5.
Note that now the stopband starts out close to
dB. The Kaiser window has a single parameter which can be
used to modify the stop-band attenuation, trading it against the
transition width from pass-band to stop-band.
Figure 5:Frequency
response of the ideal lowpass filter Kaiser windowed at the fifth
zero crossing to the left and right.
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