residuez

Signal Processing Toolbox

z-transform partial-fraction expansion.

[r,p,k] = residuez(b,a)
[b,a] = residuez(r,p,k)

Description

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.

[r,p,k] = residuez(b,a)

finds the residues, poles, and direct terms of a partial fraction expansion of the ratio of two polynomials, b(z) and a(z). Vectors b and a specify the coefficients of the polynomials of the discrete-time system b(z)/a(z) in descending powers of z:

If there are no multiple roots and a > n-1,

The returned column vector r contains the residues, column vector p contains the pole locations, and row vector k contains the direct terms. The number of poles is

n = length(a)-1 = length(r) = length(p)

The direct term coefficient vector k is empty if length(b) < length(a); otherwise

length(k) = length(b) - length(a) + 1

If p(j) = ... = p(j+s-1) is a pole of multiplicity s, then the expansion includes terms of the form

[b,a] = residuez(r,p,k)

with three input arguments and two output arguments, converts the partial fraction expansion back to polynomials with coefficients in row vectors b and a.

The residue function in the MATLAB environment is very similar to residuez. It computes the partial fraction expansion of continuous-time systems in the Laplace domain (see reference [1]), rather than discrete-time systems in the z-domain as does residuez.

Algorithm

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.
It finds the poles using p = roots(a). mpoles finds repeated poles and reorders the poles according to their multiplicities.
It finds the residue for each nonrepeating pole p_i by multiplying b(z)/a(z) by
1/(1-p_iz^-1) and evaluating the resulting rational function at z = p_i.
It finds 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_j of multiplicity s_j, S2 contains s_j columns representing the impulse responses of each of the following systems:

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 overdetermination of the system of equations.

Diagnostics

If a(1) == 0 while computing the partial fraction decomposition using
[r,p,k] = residuez(b,a), residuez gives the following error message:

First coefficient in A vector must be nonzero.

If the number of residues r and poles p is not the same, residuez gives the following error message:

R and P vectors must be the same size.

`convmtx`	Convolution matrix.
`deconv`	Deconvolution and polynomial division (see MATLAB Function Reference).
`poly`	Polynomial with specified roots (see MATLAB Function Reference).
`prony`	Prony's method for time domain IIR filter design.
`residue`	Partial fraction expansion (see MATLAB Function Reference).
`roots`	Polynomial roots (see MATLAB Function Reference).
`ss2tf`	State-space to zero-pole-gain conversion.
`tf2ss`	Transfer function to state-space conversion.
`tf2zp`	Transfer function to zero-pole-gain conversion.
`zp2ss`	Zero-pole-gain to state-space conversion.