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FIR Filters - Bessel Filters

These coefficients are scaled by 1000 to get integer values. Thus coefficient value becomes: -27, -13, 4, 12, 12, 4, -13 and -27. Distributed arithmetic look up table values (total 256 values) are generated by the executable program ‘dagen.exe’ provided with [2]. | Chapter 6 Bessel Filters Bessel filters are designed to have maximally flat group-delay characteristics. As a consequence there is no ringing in the impulse and step responses. 6.1 Transfer Function The general expression for the transfer function of an nth-order Bessel lowpass filter is given by H s A- 6.1 Qn s where qn s X bksk k 1 2n k k 2n k kl n ky. The following recursion can be used to determine qn s from Jn ifs and Q -2 s q 2n-l qn_1 s2 7 _2 Table 6.1 lists qn s for n 2 through n 8. These values were generated by the C function besselCoefficients provided in Listini 6.1. This function is used by other Bessel filter routines presented later in this section. Unlike the transfer function for Butterworth and Chebyshev filters Eq. 6.1 does not provide an explicit expression for the poles of the Bessel filter. The numerator of 6.1 will be a polynomial in s upon which numerical root-finding methods such as Algorithm 2.1 must be used to determine the pole locations for H s . Table 6.2 lists approximate pole locations for n 2 through n 8. 109 110 Chapter Six table 6.1 Denominator Polynomials for Transfer Functions of Bessel Filters Normalized to Have Unit Delay at ai 0 n q s 2 s2 3s 3 3 s3 6s2 15s 15 4 s4 10s3 45s2 105s 105 5 s6 15s4 105s3 420s2 945s 945 6 s6 21s6 210s4 1260s3 4725s2 10 395s 10 395 7 s7 28s6 378s6 3150s4 17 325s3 62 370s2 135 135s 135 135 8 s8 36s7 630s6 6930s6 9450s4 270 270s3 945 945s2 2 027 025s 2 027 025 TABLE 6.2 Poles of Bessel Filter Normalized to Have Unit Delay at co 0 n Pole values 2 1.5 0.8660 3 -2.3222 -1.8390 1.7543 4 -2.1039 2.6575 -2.8961 0.8672 5 -3.6467 -2.3247 3.5710 -3.3520 1.7427 6 -2.5158 4.4927 -3.7357 2.6263 -4.2484 0.8675 7 -4.9716 -2.6857 5.4206 -4.0701 3.5173 -4.7584 1.7393 8 -5.2049 2.6162 -4.3683 4.4146 -2.8388 6.3540 -5.5878 0.8676 The transfer functions given by 6.1 are for Bessel filters normalized to have unit delay at co 0. The poles pk and denominator coefficients bk can be renormalized for a 3-dB frequency of co 1