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[1], a/a[1]}}.

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

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.