# Low Pass FIR Filter with the Fourier Series and the Kaiser Window.

Cuthbert Nyack
This applet enables the study of low pass filters with the Kaiser window. The effect of N, q and n on the filter characteristics can be examined.

eg parameters (31, 0.15, 2.0, 0.55, NA, -65, -80, 69, 171, 2) has a 0.1dB to 80dB bandwidth of ~0.228 - 0.092 = ~0.136ws. Changing N to 91 reduces the bandwidth to ~0.048ws.

With N = 91, q = 1p, n = 0.5 the minimum stop band attenuation is ~39dB. Increasing q to 3.5p increases the minimum stop band attenuation to 109dB. dB/q ~ 8.9 and dBp/q ~28. These numbers are only a rough approximation.

The image below show a low pass filter with a minimum stop band attenuation of 100dB. The filter length is 91 and the 0.1dB to 100dB transition bandwidth is ~0.061ws. If n is kept at 0.5, then the minimum stop band attenuation is only 85dB. Changing n to 0.547 increases the stop band attenuation and can produce a more equiripple stop band characteristic at the cost of an increase in bandwidth. The pass band has a varying ripple which can be measured by moving one of the frequency reference lines through the passband. The interpretation of the white text is:-
First line:-
The white frequency reference line is at 0.198ws. The gain of the filter with the standard Kaiser window at 0.198ws is -87.17dB, that of the variable Kaiser window is 100.7dB and that of the rectangular window is 55.59dB.
Second line:-
The frequency of the Orange frequency reference line is 0.137ws. Gain of the filter with the standard Kaiser, variable Kaiser and rectangular window is -0.07248dB, -0.1076dB and -0.3797dB. The phase with the variable Kaiser window is -61.83°.

Third line:-
When k = 2 the impulse responses I(k) for the standard Kaiser, variable Kaiser and Rectangular window are -2.4274211E-5, -1.9789742E-5 and -5.0673979E-3.

Changing fc/fs to 0.161 and q/ p to 3.21 gives similar charactreistics for the standard Kaiser window but the stop band response is not as flat. 