Frequency Sampling and the Kaiser Window.

Cuthbert Nyack
The frustration that may be encountered in finding transition sample values could lead one to think that it might be easier to use windows to obtain lower stop band attenuation. This is possible but it comes with the cost of having to use longer filters.

The applet below is included for anyone who might be interested in experimenting with this approach.
The samples are divided into 5 groups.
n1(0) samples set scrollbar 0 with amplitude an1(1) set by scrollbar 1.
n2(2) samples set scrollbar 2 with amplitude an2(3) set by scrollbar 3. etc
There are 4 reference lines set by scrollbars 58 to 61 which can be used to see the gain at 4 frequencies.
Using the Kaiser window with the frequency sampling approach may be seen as being more flexible than using it with the Fourier Series because the transition samples may be used as additional parameters to control the frequency response derived by using the window.
On the down side it can be difficult to adjust the pass band edge or to set the attenuation at the pass band edge accurately. This gets worse as the width of the transition band reduces.



A Low Pass filter with a stop band attenuation of 100dB. In this case scrollbars 3, 5 and 7 may be used to adjust transition samples to increase the stop band attenuation or to make small adjustments to the pass band edge. The 1 dB frequency is ~0.145ws and the 100dB frequency is ~0.245ws.
A Low Pass filter with a stop band attenuation of 131dB. The 1 dB frequency is ~0.173ws and the 131dB frequency is ~0.283ws.
A Low Pass filter with a stop band attenuation of 153dB. The 1 dB frequency is ~0.198ws and the 131dB frequency is ~0.273ws.
A High Pass filter with a stop band attenuation of 119dB.
A Band Pass filter with a stop band attenuation of 105dB.
A Band Stop filter with a stop band attenuation of 104dB.

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COPYRIGHT 2008 Cuthbert Nyack.