Computing digital filter order

Digital filters generally have frequency domain specifications like that drawn in Fig. 5.10.1a (cf. Fig. 4.2.2) where

fp = maximum passband frequency (Hz) for Mp (dB) maximum ripple
fs = minimum stopband frequency (Hz) for Mr (dB) minimum rejection
fs = sampling frequency (Hz) = 1/T seconds


fpT = 360 fp/fs = normalized passband frequency in degrees
frT = 360 fr/fs = normalized stopband frequency in degrees
Od = wsT/wpT = passband/stopband frequency ratio of digital filter

The normalized stopband frequency Od of the digital filter equals Od = wsT/wpT = fs/fp. For such (Mp, Mr, Od) specifications, the digital filter order must be determined. To do this, the classsical, optimum, and adaptive filter transfer functions of Chap. 4 must be combined with the digital transformations of Chap. 5 which have just been discussed. This can be easily done using magnitude response nomographs.[4]

The nomographs for analog filters are well-established and have the form shown in Fig. 5.10.2. The frequency domain gain requirements for the analog filter (Mp, Mr, Oa) shown in Fig. 5.10.1b are transferred onto the nomograph. The filter order n is determined by reading the order of the first curve falling above the data point (v,Oa).

The analog filter nomograph can also be used for digital filters by relating the digital filter frequency Od and analog filter frequency Oa. This can be easily done as will be shown in a moment. Then by plotting this relation on a graph directly below the nomograph as shown in Fig. 5.10.2, the digital filter frequency Od is graphically converted into the equivalent analog filter frequency Oa. Then the analog filter nomograph is used as usual to obtain the required filter order n. Now let us investigate the relation between Od and Oa.

Fig. 5.10.1

Consider the bilinear transform H011 given by Eq. 5.9.4a where

p = (2/p) (z-1)/(z+1)
Setting the analog frequency variable p = u+jv and the digital frequency variable z = exp(jwT), then
jv = j tan(wT/2)/(T/2) = jw tan(wT/2)/(wT/2)
using the reciprocal of Eq. 5.5.3. This relates the digital filter frequency w (or normalized frequency wT) to the analog filter frequency v (or normalized frequency vT). Since the digital filter specification requires a normalized passband frequency of wpT and a normalized stopband frequency of wrT, then the equivalent analog filter frequencies are
2 pi (fp'/fs) = 2 tan(pi fp/fs) rad or fp' = fp tan(pi fp/fs)/(pi fp/fs) Hz
2 pi (fr'/fs) = 2 tan(pi fr/fs) rad or fr' = fr tan(pi fr/fs)/(pi fr/fs) Hz
in Hz. fi is the digital filter frequency and fi' is the equivalent analog filter frequency. The analog frequency ratio Oa equals
Oa = fr'/fp' = tan(180 fr/fs)/tan(180 fp/fs) = tan(wrT/2)/tan(wpT/2)

Fig. 5.10.2

© C.S. Lindquist, Adaptive and Digital Signal Processing with Digital Filtering Applications, vol. 2, pp. 390-392, Steward & Sons, 1989.