Filter Realizations - Introduction to Digital Signal Processing and Filter Design - page 345

Σ Σ

− 0.5089

0.4482

− 0.4482

0.5089

z − 1

Σ

z − 1

Σ

0.5

X (z)

Y (z)

Σ

Σ Σ Σ

− 0.3385

0.8717

0.3667 0.3385

− 0.3667 − 0.8717

z − 1 z − 1 z − 1

Σ Σ Σ

Figure 6.27 Two lattice-ladder allpass structures connected in parallel to realize a fifth-
order lowpass elliptic filter.

minor operations on binary numbers. The direct-form IIR filter for a fifth-order
filter would also require more than five multipliers, whereas the filter shown in
Figure 6.27, which has lattice-ladder, coupled allpass filters, requires 10 mul-
tipliers. Therefore we conclude that the parallel connection of allpass filters as
shown in Figure 6.25 requires a minimum number of delay elements and thus
offers an advantage over the other structures.
The realization of IIR filters as a parallel connection of allpass filters has
another advantage, as explained below. It was pointed out that the magnitude
response of allpass filters does not change when the multiplier constants are
quantized to finite wordlength. The other advantage is that there are many struc-
tures for realizing allpass filters that contain a minimum number of multipliers
(and delay elements). In the method of realizing the lowpass filter by a connec-
tion of two allpass filters in parallel, we used Property (6.4) in Equation (6.42)
which is reproduced below:
G(e jω ) =

2 e jθ 1 (ω)

e jθ 2 (ω) =

2 1 +

e j(θ 1 (ω) − θ 2 (ω)) ≤ 1

1 1

+

This shows that the lowpass filter containing the two allpass structures in parallel
has a magnitude response equal to or less than unity. The magnitude response
in Figure 6.28 illustrates this property in the passband and attains the maximum
value at three frequencies in the passband, which are marked by arrows. As