Series Cascade Form

In the canonical series cascade form, the transfer function H(z) is written as a product of first-order and second-order transfer functions.

This equation yields the canonical series cascade form shown below.

Factoring H(z) into H_i(z) where i = 1,2,3,...,p can be done in a number of ways. Using the poles and zeros of H(z), you can obtain H_i(z) by grouping pairs of conjugate complex poles and pairs of conjugate complex zeros to produce second-order transfer functions, or by grouping real poles and real zeros to produce either first-order or second-order transfer functions. You could also group two real zeros with a pair of conjugate complex poles or vice versa. Since there are many ways to obtain H_i(z), it is desirable to compare the various groupings to see which produces the best results for the transfer function under consideration.

For example, one factorization of H(z) might be

You must also take into consideration that the ordering of the individual H_i(z)'s will lead to systems with different numerical characteristics. You may want to try various orderings for a given set of H_i(z)'s to determine which gives the best numerical characteristics.

The first order diagram for H(z) is given below.

The second order diagram for H(z) is given below.

The series cascade form example transfer function is given by

The realization of H_ex(z) using the Fixed-Point Blockset is shown below. You can display this model by typing

fxpdemo_series_cascade_form

at the MATLAB command line.