Z-transform partial-fraction expansion

`residuez`

converts a discrete time system, expressed as the ratio of two
polynomials, to partial fraction expansion, or residue, form. It also converts the partial
fraction expansion back to the original polynomial coefficients.

Numerically, the partial fraction expansion of a ratio of polynomials is an ill-posed problem. If the denominator polynomial is near a polynomial with multiple roots, then small changes in the data, including round-off errors, can cause arbitrarily large changes in the resulting poles and residues. You should use state-space or pole-zero representations instead.

`residuez`

applies standard MATLAB^{®} functions and partial fraction techniques to find `r`

,
`p`

, and `k`

from `b`

and
`a`

. It finds

The direct terms

`a`

using`deconv`

(polynomial long division) when`length(b)`

>`length(a)-1`

.The poles using

`p`

=`roots`

`(a)`

.Any repeated poles, reordering the poles according to their multiplicities.

The residue for each nonrepeating pole

*p*by multiplying_{j}*b*(*z*)/*a*(*z*) by 1/(1 -*p*_{j}*z*^{−1}) and evaluating the resulting rational function at*z*=*p*._{j}The residues for the repeated poles by solving

S2*r2 = h - S1*r1

for

`r2`

using`\`

.`h`

is the impulse response of the reduced*b*(*z*)/*a*(*z*),`S1`

is a matrix whose columns are impulse responses of the first-order systems made up of the nonrepeating roots, and`r1`

is a column containing the residues for the nonrepeating roots. Each column of matrix`S2`

is an impulse response. For each root*p*of multiplicity_{j}*s*,_{j}`S2`

contains*s*columns representing the impulse responses of each of the following systems._{j}$$\frac{1}{1-{p}_{j}{z}^{-1}},\frac{1}{{(1-{p}_{j}{z}^{-1})}^{2}},\cdots ,\frac{1}{{(1-{p}_{j}{z}^{-1})}^{{s}_{j}}}$$

The vector

`h`

and matrices`S1`

and`S2`

have`n`

`+`

`xtra`

rows, where`n`

is the total number of roots and the internal parameter`xtra`

, set to 1 by default, determines the degree of over-determination of the system of equations.

[1] Oppenheim, Alan V., Ronald W.
Schafer, and John R. Buck. *Discrete-Time Signal Processing*. 2nd Ed.
Upper Saddle River, NJ: Prentice Hall, 1999.