Various FIR Filters from the Fourier Series.

Cuthbert Nyack

Frequency and impulse response of various FIR filters derived from a Fourier Series of the frequency response are shown by the applet below.
N is the length of the impulse response which can be changed by the last scrollbar to the right.
Frequency response is in red(linear) and green(log) The green plot is effectively a logarithmic plot of the Gibbs' phenomenon.
Impulse response is in pink.
Changing the Scrollbar labelled 0 on the left changes the type of filter. Possible filter types are Low Pass, High Pass, Band Pass, Band Stop, 2 Band Pass, Triangle and Band Pass with a sloping 'top'.

As N is varied for the Low Pass case, the attenuation of the first peak in the stop band varies between 20 and 22dB. The minimum attenuation occurs when the impulse response is truncated near one of its peaks and a maximum attenuation occurs when the truncation occurs near one of the zeros of the impulse response.
hanging the Scrollbar labelled 1 changes the cut off frequency of the filter.
The parameters (N = 51, wc = 0.15ws, W Line(S28) = 0.15ws) show that the attenuation at the cutoff frequency is -6.368dB. To get an attenuation of -3dB at w = 0.15ws then the cut off frequency must be increased to ~-0.155ws.
The frequency period of the ripples is ~ ws/ (N - 1)/2.

The case of a band pass filter is shown below.
Interpretation of white text:-
The white line is a frequency marker and its frequency is shown on the first line as 0.178ws
The gain of the frequency response shown in green is -0.835013dB at 0.178ws.
The second term in the impulse is I(S0)(2) = 0.011692665. Other values of the impulse response can be seen by changing scrollbar 30.

The second line shows that the length of the filter is 51 and is set by scrollbar 31. The lower cut off frequency is 0.170ws and is set by scrollbar 3.
The upper cut off frequency is 0.350ws and is set by scrollbar 4.

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