'Design' of FIR Filters with the Fourier Series and the Kaiser Window.

Cuthbert Nyack
Kaiser window w(t) = Io [q(1 - (2t/t)2)n] /Io(q) for |t| <=t/2.
The power n is normally taken as 0.5 but in the applet here n is treated as a variable.
q is a variable parameter for the Kaiser window and increasing q increases the stop band attenuation but with some broadening of the transition region.

The standard window parameters have been chosen to give acceptable filters over a wide range of cases. With the interactive approach used here, it is easy to adjust the parameter n to give a more suitable filter for any particular case.

Empirical Relations have been worked out for N and q for a given application. The last line in the applet below shows N and q for different dp/ds, ds and DF.



It is possible to 'design' FIR LP, HP, BP and BS filters using the applet below. Design is put in inverted commas because the window approach does not allow independent specification of all relevant filter parameters.

Scrollbar 0 sets the type of filter, values from 0 to 3 sets LP, HP, BP and BS filters. 4 to 7 allows LP, HP, BP and BS filters with slope to be examined. Allowing a finite slope instead of a sharp discontinuity at the cut off frequency has a similar effect to windowing and allows a smaller q for the same stop band attenuation but does not produce significantly better filters. The slope is set by scrollbar 10.

For all filters, the filter length is set by scrollbar 31, the Kaiser parameter q is set by scrollbars 1(coarse adjustment) and 2(fine adjustment). n is set by scrollbar 3. The frequencies of the band edges are shown on the applet. eg for LP and HP the cut off frequency is set by scrollbar 4(coarse adjustment) and 5(fine adjustment) etc.

There are 4 frequency reference lines White(set by scrollbar 26), Orange(set by scrollbar 27), Blue(set by scrollbar 28) and Pink(set by scrollbar 29). Gn(WL) etc show the gain at the frequency of the white line etc.
There are 2 dB reference lines Pink Magenta adjusted by scrollbar 24 and blue magenta adjusted by scrollbar 25.

The final result is the impulse response which is shown as I(S30)(n) and all elements of the impulse response can be seen by varying scrollbar 30.

To derive a filter first set the frequency reference lines to the band edges and the dB reference line to the desired stop band attenuation. N, q, n and wc are then adjusted iteratively until the specifications are met.



A Low Pass filter is shown below.
The attenuation is < 0.1dB for w < 0.16ws and > 80dB for w > 0.22ws.

Interpretation of the first line of white text:-
The white/orange/blue/pink frequency reference lines are moved by scrollbars 26/27/28/29 and are located at 0.160/0.177/0.216/0.220 ws respectively.
The pink magenta/blue magenta dB reference lines are moved by scrollbars 24/25 and are located at -87/-81dB.

Interpretation of the second line of white text:-
The gains of the green curve at the white/orange/blue/pink lines are -0.1/-3.057/-93.49/-85.81dB.
The impulse response shown in cyan can be obtained numerically by changing scrollbar 30. I(2) = -1.2449438E-4.

Interpretation of the third line of white text:-
Fn is set by scrollbar 0 and is 0 which illustrates a Low Pass filter.
The length of the filter is set by scrollbar 31 and is 79.
The Kaiser parameter q is set by scrollbars 1(coarse adjustment) and 2(fine adjustnment) and is 6.949600. Kaiser parameter n is set by scrollbar 3 and is 0.531.
Cut off frequency is set by scrollbar 4(coarse adjustment) and 5(fine adjustment) and is 0.183060ws.

A High Pass filter is shown below.
The attenuation is < 0.1dB for w > 0.32ws and > 100dB for w < 0.25ws.

A Band Pass filter is shown below.
The attenuation is < 0.1dB for w > 0.12ws and w < 0.32ws and > 100dB for w < 0.053ws and w > 0.386ws .

A Band Stop filter is shown below.
The attenuation is < 0.1dB for w < 0.18ws and w > 0.37ws and > 60dB for w > 0.23ws and w < 0.32ws .

A Low Pass filter with slope is shown below.
The attenuation is < 0.1dB for w < 0.14ws and > 100dB for w > 0.21ws.

A High Pass filter with slope is shown below.
The attenuation is < 0.1dB for w > 0.31ws and > 100dB for w < 0.24ws.

A Band Pass filter with slope is shown below.
The attenuation is < 0.1dB for w > 0.15ws and w < 0.28ws and > 100dB for w < 0.085ws and w > 0.345ws .

A Band Stop filter with slope is shown below.
The attenuation is < 0.1dB for w < 0.1ws and w > 0.33ws and > 100dB for w > 0.154ws and w < 0.275ws .

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COPYRIGHT 2008 Cuthbert Nyack.