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:

$H_{i} (z) = \frac{u (z)}{e (z)} = K + H_{1} (z) + H_{2} (z) + \dots + H_{p} (z) .$

This expansion, where K is a constant and the H_i(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

$\begin{matrix} H (z) = K + \sum_{i = 1}^{j} H_{i} (z) + \sum_{i = j + 1}^{p} H_{i} (z) \\ = K + \sum_{i = 1}^{j} \frac{b_{i}}{1 + a_{i} z^{- 1}} + \sum_{i = j + 1}^{p} \frac{e_{i} + f_{i} z^{- 1}}{1 + c_{i} z^{- 1} + d_{i} z^{- 2}} . \end{matrix}$

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

$H_{e x} (z) = 5.5556 - \frac{3.4639}{1 + 0.1 z^{- 1}} + \frac{- 1.0916 + 3.0086 z^{- 1}}{1 - 0.6 z^{- 1} + 0.9 z^{- 2}} .$

The realization of H_ex(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.

Documentation

Parallel Form

Fixed-Point Designer Documentation

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