# RESIDUE

## Purpose:

Finds the partial fraction expansion of a rational polynomial.

## Syntax:

RESIDUE(b, a)

(r, p, k) = RESIDUE(b, a)

 b - A series. The numerator (i.e. zero) coefficients in descending powers. a - A series. The denominator (i.e. pole) coefficients in descending powers

## Alternate Syntax:

RESIDUE(r, p, k)

(b, a) = RESIDUE(r, p, k)

 r - A series. The residues representing the numerator terms of the partial fraction expansion. p - A series. The poles of the partial fraction expansion. k - A series. The numerator coefficients for the direct terms of the partial fraction expansion.

## Returns:

(r, p, k) = RESIDUE(b, a) returns the partial fraction expansion of the rational polynomial.

R, p and k are series where r represents the residues of the partial fraction expansion, p are the pole locations and k represents the direct terms (if any).

(b, a) = RESIDUE(r, p, k) returns the inverse partial fraction expansion, converting the partial fraction expansion back into b(s) / a(s) form. Series b and a are the numerator and denominator terms of the rational polynomial with the partial fraction expansion represented by r, p, and k. Series k represents the remainder terms (if any).

RESIDUE(r, p, k) with one or zero output arguments returns b and a in one series of two columns where b == col(1) and a == col(2).

RESIDUE(f) or (b, a) = RESIDUE(f) assumes f is a three column series with r, p and k as each of the columns. Thus:

residue(residue(b, a)) == {{b/a, a/a}}.

## Example: (r, p, k) = residue({1}, {1, 4, 3})

r == {-0.5, 0.5}

p == {-3, -1}

k == {}

representing the partial fraction expansion: The impulse response for t >= 0 can be found by inspection: Now, performing the inverse transform:

(b, a, c) = residue(r, p, k)

b == {0, 1}

a == {1, 4, 3}

c == {}

The series b and a represent the numerator and denominator terms of the original rational polynomial.

## Example: (r, p, k) = residue({1, 1, 1}, {1, -5, 8, -4})

r == {3, -2, 7}

p == {1, 2, 2}

k == {}

Since the polynomial contains two repeated poles, the result represents the partial fraction expansion: ## Remarks:

Given the rational polynomial H(s) = b(s) / a(s) where: s = jω complex frequency N = number of numerator terms M = number of denominator terms

If a ≠ 1, the numerator and denominator terms are normalized by dividing each coefficient by a.

If there are no repeated roots, the partial fraction expansion of the rational polynomial is of the form: If there are K repeated roots (closer than 1.0e-3), then the partial fraction expansion includes terms such as: See RESIDUEZ to find the partial fraction expansion of a Z-transform.

FREQS

IMPS

INVFREQS

INVFREQZ

POLY

REPROOT

RESIDUEZ

ROOTS

SFREQ

SPLANE