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Parallel Form

In the canonical parallel form, the transfer function H(z) is expanded into partial fractions. H(z) is then realized as a sum of a constant, first-order, and second-order transfer functions, as shown:

Hi(z)=u(z)e(z)=K+H1(z)+H2(z)++Hp(z).

This expansion, where K is a constant and the Hi(z) are the first- and second-order transfer functions, follows.

As in the series canonical form, there is no unique description for the first-order and second-order transfer function. Because of the nature of the Sum block, the ordering of the individual filters doesn't matter. However, because of the constant K, you can choose the first-order and second-order transfer functions such that their forms are simpler than those for the series cascade form described in the preceding section. This is done by expanding H(z) as

H(z)=K+i=1jHi(z)+i=j+1pHi(z)=K+i=1jbi1+aiz1+i=j+1pei+fiz11+ciz1+diz2.

The first-order diagram for H(z) follows.

The second-order diagram for H(z) follows.

The parallel form example transfer function is given by

Hex(z)=5.55563.46391+0.1z1+1.0916+3.0086z110.6z1+0.9z2.

The realization of Hex(z) using fixed-point Simulink® blocks is shown in the following figure. You can display this model by typing

fxpdemo_parallel_form

at the MATLAB® command line.