void FIdKaiserLPF (double h, int N, double Fc, double alpha)
Generate a Kaiser windowed lowpass filter
This routine calculates the impulse response of a symmetric lowpass filter
which is obtained by applying a Kaiser window to a sin(x)/x function.
Consider the continuous-time version of this response
sin(2 pi Fc t)
h(t) = Kw(t/T, a) -------------- ,
where Kw(t/T,a) is a Kaiser window with parameter a, spanning -T <= t <= T.
The cutoff of the lowpass filter is Fc. The gain of the filter is set to
approximate unity in the passband. If it is desired to have the peak
value of the time response equal to unity, h(t) must be divided by 2*Fc.
The discrete-time impulse response of length N is obtained by setting the
sampling rate to one; t = n - T and T = (N-1)/2. The cutoff frequency Fc
is then a normalized frequency.
The parameter a (alpha) determines the shape of the window, with increasing
a giving a wider transition region but better stopband attenuation. The
tradeoff is shown in the following table.
stopband a transition passband
attenuation width D ripple
30 dB 2.210 1.536 +/- 0.270 dB
40 dB 3.384 2.228 +/- 0.0864 dB
50 dB 4.538 2.926 +/- 0.0274 dB
60 dB 5.658 3.621 +/- 0.00868 dB
70 dB 6.764 4.317 +/- 0.00275 dB
80 dB 7.865 5.015 +/- 0.00089 dB
90 dB 8.960 5.712 +/- 0.00027 dB
100 dB 10.056 6.408 +/- 0.00009 dB
The transition width parameter D = (N-1) dF, where dF is the normalized
Consider a lowpass filter with a 90 dB stopband attenuation to be used for
interpolation by a factor of 8. The cutoff frequency is then 0.5/8 = 0.0625.
The transition bandwidth should be less than 10 % of the cutoff frequency,
giving dF = 0.00625. The 90 dB attenuation is reached for a = 8.960.
This gives D = 5.712. Then N >= D/dF + 1, giving N = 915. This is a long
filter because of the narrow transition region and high attenuation
specification, but in fact, for interpolation only 1/8 of these samples are
used at any one time.
J. F. Kaiser, "Nonrecursive digital filter design using the I0-sinh window
function", Proc. 1974 IEEE Int. Symp. on Circuits and Syst., pp. 20-23,
A. Antoniou, Digital Filters: Analysis, Design and Applications, 2nd ed.,
<- double h
Array containing the impulse response samples. The reference point in
the array is at offset (N-1)/2.
-> int N
Number of impulse response samples
-> double Fc
Lowpass filter cutoff (normalized frequency, 0 to 1/2)
-> double alpha
Kaiser window parameter
Author / revision
/ Revision 1.1 2005/02/01
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