Note 36. Designing FIR Filters: Kaiser Window Method
The Kaiser window is the window most often used in the windowing technique for designing linear-phase FIR filters that is described in Note 35. In order to exploit the adjustability of the Kaiser window most effectively, the filter design techniques presented in Note 35 must be modified, as shown for the lowpass case in Recipe 36.1. Comparing Recipe 35.1 and Recipe 36.1 reveals that Recipe 36.1 can be made generic with respect to band configuration if changes are made to steps 2, 8, and 9 to obtain the result shown in Recipe 36.2.
Recipe 36.1. Designing Lowpass FIR Filters Using the Kaiser Window
Perform the following steps to approximate the ideal lowpass filter response.
- Based on the intended application, determine the following:
• the normalized passband edge frequency, λP
• the normalized stopband edge frequency, λS
• the maximum tolerable passband ripple, δP
• the maximum tolerable stopband ripple, δS
- Determine the transition width, Δλ, as
Δλ = λs – λp
- Set δ equal to the smaller of δP or δS, and compute A as
A = –20log10δ
- Use the value of A computed in step 3 to determine the value for β* as
- Determine the required number of taps, N, as
- Compute the window coefficients as
- For
, compute h[n] and h[N – 1 – n] as
where m = n –(N – 1)/2.
- If N is odd, compute h[(N – 1)/2] as
- Check the filter response against the conditions specified in step 1. If the filter meets the specified conditions, the design is complete. If the filter response does not meet the specifications, increase N by 1, and return to step 6.
* This formula for β was determined empirically by Kaiser and reported in [3].
Recipe 36.2. Designing FIR Filters Using the Kaiser Window
Perform the following steps to approximate the ideal filter response.
- Based on the intended application, determine the following specifications:
a. The maximum tolerable passband ripple, δP, and maximum tolerable stopband ripple, δS. Bandpass filters have two stopbands. If the tolerable ripple is different for these two stopbands, set δS to the to the more stringent of the two. Bandstop filters have two passbands; if the tolerable ripple is different for these two passbands, set δP to the more stringent of the two.
b. Critical frequencies, as depicted in Figure 35.1
• the normalized passband edge frequencies: λP for LP, HP; λPL and λPU for BP, BS
• the normalized passband edge frequencies: λS for LP, HP; λSL and λSU for BP, BS
- Determine the transition width, Δλ, as
- Set δ equal to the smaller of δP or δS, and compute A as
A = –20log10 δ
- Use the value of A computed in step 3 to determine the value for β as
- Determine the required number of taps, N, as
- Compute the window coefficients as
- For n = 0, 1, . . .,, compute h[n] and h[N – 1 – n] as
- If N is odd, compute h[(N – 1)/2] as
- Check the filter response against the conditions specified in step 1. If the filter meets the specified conditions, the design is complete. If the filter response does not meet the specifications, increase N by 1, and return to step 6.
References
1. D. Slepian and H. Pollak, “Prolate-Spheroidal Wave Functions, Fourier Analysis and Uncertainty-I,” Bell Syst. Tech. J., vol. 40, January 1961, pp. 43–64.
2. J. F. Kaiser, “Digital Filters,” Chapter 7 in System Analysis by Digital Computer, F. F. Kuo and J. F. Kaiser eds., John Wiley & Sons, 1966.
3. J. F. Kaiser, “Nonrecursive Digital Filter Design Using the I0-sinh Window Function,” Proc. 1974 IEEE Int. Symp. on Circuits and Syst., April 22–25, 1974, pp. 20–23.
4. A. Antoniou, Digital Filters: Analysis and Design, McGraw-Hill, 1979.
5. f. j. harris, “On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform,” Proc. IEEE, vol. 66, no. 1, January 1978, pp. 51–83.