MATLAB Function Reference

residue

Convert between partial fraction expansion and polynomial coefficients

Syntax

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

Description

The residue function converts a quotient of polynomials to pole-residue representation, and back again.

[r,p,k] = residue(b,a) finds the residues, poles, and direct term of a partial fraction expansion of the ratio of two polynomials, b(s) and a(s), of the form:

[b,a] = residue(r,p,k) converts the partial fraction expansion back to the polynomials with coefficients in b and a.

Definition

If there are no multiple roots, then:

The number of poles n is

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

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

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

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

Arguments

b,a	Vectors that specify the coefficients of the polynomials in descending powers of s
r	Column vector of residues
p	Column vector of poles
k	Row vector of direct terms

Algorithm

The residue function is an M-file. It first obtains the poles with roots. Next, if the fraction is nonproper, the direct term k is found using deconv, which performs polynomial long division. Finally, the residues are determined by evaluating the polynomial with individual roots removed. For repeated roots, the M-file resi2 computes the residues at the repeated root locations.

Limitations

Numerically, the partial fraction expansion of a ratio of polynomials represents an ill-posed problem. If the denominator polynomial, a(s), is near a polynomial with multiple roots, then small changes in the data, including roundoff errors, can make arbitrarily large changes in the resulting poles and residues. Problem formulations making use of state-space or zero-pole representations are preferable.

See Also

deconv, poly, roots

References

[1] Oppenheim, A.V. and R.W. Schafer, Digital Signal Processing, Prentice-Hall, 1975, p. 56.

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