Signal Processing Toolbox

residuez

z-transform partial-fraction expansion.

Syntax

[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) is less than 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 standard MATLAB language 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:

1. The direct terms a using deconv (polynomial long division) when length(b) > length(a)-1.
2. The poles using p = roots(a).
3. Any repeated poles, reordering the poles according to their multiplicities.
4. 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.
5. 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.

See Also

convmtx	Convolution matrix.
deconv	Deconvolution and polynomial division (see the MATLAB documentation).
poly	Polynomial with specified roots (see the MATLAB documentation).
prony	Prony's method for time domain IIR filter design.
residue	Partial fraction expansion (see the MATLAB documentation).
roots	Polynomial roots (see the MATLAB documentation).
ss2tf	Convert state-space filter parameters to zero-pole-gain form.
tf2ss	Convert transfer function filter parameters to state-space form.
tf2zp	Convert transfer function filter parameters to zero-pole-gain form.
zp2ss	Convert zero-pole-gain filter parameters to state-space form.

References

[1] Oppenheim, A.V., and R.W. Schafer, Digital Signal Processing, Prentice-Hall, Englewood Cliffs, NJ, 1975, pp. 166-170.

resample

rlevinson