EXAMPLE 12.2.1 The transfer function of a 4th-order elliptic lowpass filter with 0.28 dB ripple and a 1 KHz bandwidth equals

H(z) =
0.01201 (z+1)^2 (z^2 - 0.7981z + 1)/
(z^2 - 1.360z + 0.5133)(z^2 - 1.427z + 0.8160)

This H(z) was determined in Example 11.2.1 using a bilinear transform. Implement H(z) using a 1F multiple feedback structure.

SOLUTION This filter has the structure shown in Fig. 12.2.1. The ak and bk coefficients result when H(z) of Eq. 12.2.8 is written in a summation form as

H(Z) =
(0.01201 + 0.01443Z + 0.004850Z^2 + 0.01443Z^3 + 0.01201Z^4)/
1 - 2.787Z + 3.269Z^2 - 1.842Z^3 + 0.4189Z^4

where Z = z^(-1). Therefore matching the coefficients of Eq. 12.2.9 to those in Eq. 12.1.1 gives

a0=0.01201, a1=0.01443, a2=0.004850, a3=0.01443, a4=0.01201
b0=1, b1=-2.787, b2=3.269, b3=- 1.842, b4=0.4189

The 1F canonical realization of H(z) is shown in Fig. 12.2.5. Since N=4, we see that this filter form requires N=4 delay elements, 2N+1=9 multipliers, and 2 summers as listed in Table 12.10.1.

© C.S. Lindquist, Adaptive and Digital Signal Processing with Digital Filtering Applications, vol. 2, pp. 783-784, Steward & Sons, 1989.