Filter Realizations - Introduction to Digital Signal Processing and Filter Design - page 327

in Figure 6.19a, the value of the ladder coefficient v 5 happens to be zero for the
numerical example, and therefore the multiplier v 5 is zero. The lattice parameters
are also known as the reflection coefficients , and it has been shown that the poles
of the IIR filter function are inside the unit circle of the z plane if

| k i | ≤ 1. So

this method is used to test whether an IIR filter is stable.

6.5 REALIZATION OF FIR AND IIR FILTERS USING MATLAB

Many of the computations involved in the realization of FIR and IIR filters as
presented in this chapter can be carried out by MATLAB functions. For example,
an FIR filter realization in the cascaded structure can be obtained by finding the
roots of the transfer function and then finding the second-order polynomials with
complex conjugate pair of the roots or a pair of two real zeros.
To find the roots of a polynomial H(z) = n = 0 b(n)z − n ,weusetheMATLAB

function R = roots(b) where the vector b = [b(0), b(1), b(2), ···
b(N)] and R is the vector of the N roots. Choosing a pair of complex conju-
gate roots or a pair of real roots, we construct the second-order polynomials
using the MATLAB function P k =poly(R k ) ,where R k is the list of two roots and
P k is the vector of the coefficients of the second-order polynomial. Of course, if
H(z) is an odd-order polynomial, one first-order polynomial with a single real
root will be left as a term in the decomposition of H(z) .

Example 6.11

Using the MATLAB commands

b = [1.965 -3.202 4.435 -3.14 1.591 -0.3667];
R= roots(b)

we get the roots

0.2682 + 0.8986i
0.2682 - 0.8986i
0.3383 + 0.6284i
0.3383 - 0.6284i
0.4166

Then we continue

R1=[0.2682+0.8986*i 0.2682-0.8986*i];
P1=poly(R1)
P1=
1.0000 -0.5364 0.8794
R2=[0.3383+0.6284*i 0.3383-0.6284*i];
P2=poly(R2)