Filter Realizations - Introduction to Digital Signal Processing and Filter Design - page 341

X( z) 1

−

X( z) Y (z)

z − 1 ∑ ∑

z − 1 ∑ ∑

− 1

d 1

d 1

z − 1

Y(z)

∑

( a ) ( b )

X(z) Y(z)

z − 1

∑ ∑

z − 1

∑

z − 1

− 1

d 1

d 1

− 1

Y(z)

∑ ∑

( c ) ( d )

Figure 6.23 First-order allpass structures.

parallel form, or lattice-ladder form. But the class of allpass functions of first
and second orders can be realized by many structures that employ the fewest
multipliers [1]. A few examples of first-order and second-order allpass filters
are shown in Figures 6.23 and 6.24, respectively. Their transfer functions are
respectively given by

z − 1

d 1 +

A I (z) =

1 + d 1 z − 1

d 1 z − 1 z − 2 d 1 d 2 + +

A II (z) =

d 1 d 2 z − 2
We choose the simpler structure of second-order allpass filter from Figure 6.24a
for A 1 and A 2 (z) , which requires fewer delay elements than do the remaining four
second-order structures. When these two allpass filters are connected in parallel
(as shown in Fig. 6.16), we get the structure shown in Figure 6.25 for the transfer
function G(z) of the fifth-order elliptic lowpass filter chosen in this example:

1 +

d 1 z − 1 +

z 2 − 1 . 329 z + 1 . 965

A 1 (z) = (6.69)

1 . 965 z 2 − 1 . 329 z + 1

1 − 1 . 329 z − 1 + 1 . 965 z − 2

=

1 . 965 − 1 . 329 z − 1 + z − 2

0 . 6763 z − 1 z − 2 0 . 5089 − +

=

(6.70)

1 − 0 . 6763 z − 1 + 0 . 5089 z − 2