Filter Realizations - Introduction to Digital Signal Processing and Filter Design - page 334

The lattice structure for this filter is shown in Figure 6.19b, where H(z − 1 ) =

Y(z)/X(z) .
Suppose that we select an allpass transfer function H ap (z − 1 )

0 . 0154 − 0 . 0504 z − 1 − 0 . 0205 z − 2 z − 3 +

H ap (z − 1 ) = (6.67)

1 − 0 . 2051 z − 1 − 0 . 0504 z − 2 + 0 . 0154 z − 3

and use the function [k,v]= tf2latc(num,den) we get the following outputs
for the vectors k and v :

k =− 0 . 2145

− 0 . 0473

0 . 0154

v = 0 . 0000

0 . 0000
0 . 0000
1 . 0000

Although this allpass transfer function has a numerator and a denominator and
hence is not an AR model, the lattice structure for realizing it is the same as the
lattice structure for an AR model in Figure 6.19b, but the output is V(z) and not
Y(z) . Hence the allpass transfer function realized is H ap (z − 1 ) V(z)/X(z) .

When we compare Figure 6.17 for the lattice structure for the third-order
FIR (MA) filter and Figure 6.19b for the lattice structure for the third-order
IIR all-pole (AR) or the allpass (AP) filter, carefully note the direction of the
multipliers and their signs, which are different. Also note that the output terminals
are different for the all-pole filter and allpass filters in Figure 6.19b.

=

6.5.1 MATLAB Program Used to Find Allpass Filters in Parallel

A MATLAB program lp2apx.m , developed by the author, to obtain the decom-
position of a lowpass filter as the sum of two allpass filters is given below.
The input data consist of the cutoff frequency W p of the passband, the stopband
frequency W s , the maximum attenuation A p in the passband, and the minimum
attenuation A s in the stopband, which are entered according to the choice of
the Butterworth, Chebyshev I, Chebyshev II, and elliptic lowpass filters. If the
order of the filter derived from the functions buttord, cheb1ord, cheb2ord
or ellipord is found to be an even integer, it is automatically increased by
one to make it an odd-order integer. The program immediately makes avail-
able the magnitude response of the specified lowpass filter G(z) , the response
of the lowpass filter computed from the two allpass filters A 1 (z) and A 2 (z) ,and
also the highpass filter H(z) . The coefficients of the numerator and denomina-
tor polynomials of A 1 (z) and A 2 (z) are obtained by typing A1N , A1D , A2N ,and