## FIdKaiserLPF

#### Routine

void FIdKaiserLPF (double h[], int N, double Fc, double alpha)

#### Purpose

Generate a Kaiser windowed lowpass filter

#### Description

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)  -------------- ,
                              pi t

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 transition width.

Example:

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.

References:

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, April 1974.

A. Antoniou, Digital Filters: Analysis, Design and Applications, 2nd ed., McGraw-Hill, 1993.

#### Parameters

<- 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

P. Kabal / Revision 1.1 2005/02/01

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