Filter Realizations - Introduction to Digital Signal Processing and Filter Design - page 326

having these zeros, we get the numerator of A 2 (z) , which has the zeros at z − 1
k

.

We identify the zeros of P(z) + Q(z) that are outside the unit circle as the

(N + 2 ,...,N) of A 1 (z) . By reversing the order of

the coefficients of the numerator polynomial having these zeros, we obtain the
denominator polynomial of A 1 (z) .Ithas (N

− r) zeros z j (j = r + 1 ,r

r) zeros at z − j as shown in (6.54).
This completes the design procedure used to obtain A 1 (z) and A 2 (z) from G(z) .
An example is worked out in Section 6.5.

−

6.4.2 Lattice-Ladder Realization

Another well-known realization of an IIR transfer function of the form (6.1) is
shown in Figure 6.19a, and is known as the lattice-ladder realization .Itisalso
called the autoregressive moving-average (ARMA) model in the literature on
speech processing, adaptive filters, and signal processing in general. When the
numerator of (6.1) is a constant, we have an all-pole model also known as the
autoregressive (AR) model , and the structure to realize a third-order AR model
is shown in Figure 6.19b, whereas when the denominator is a constant, we get
an FIR model called the moving-average (MA) model. The structure shown in
Figure 6.17a is the model for a third-order FIR filter function or the MA model,
and the structure in Figure 6.17b is its transpose. We do not present the theoreti-
cal analysis of lattice structures for these models as it is beyond the scope of this
book but explain the use of a MATLAB function tf2latc in the next section to
derive the structures. This function implements the theoretical procedure, and
for the ARMA model, it gives the N lattice parameters k i , i = 1 , 2 ,...,N

and also the values of the N + 1 ladder coefficients v i , i = 0 , 1 ,...,N .But

X(z) Y(z)

Σ Σ Σ

K 2

K 3
K 3

K 1
K 1

K 2

z − 1 z − 1 z − 1

Σ Σ

Σ V(z)

( a )

X(z)

Y(z)

Σ Σ Σ

K 2 K 1

K 1

K 3
K 3 K 2

U(z)

z − 1 z − 1 Σ Σ

z − 1

Σ

( b )

Figure 6.17 (a) Lattice structure for an FIR filter; (b) transpose of the lattice structure
for the FIR filter in (a).