Finds the partial fraction expansion of a rational polynomial.
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 |
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. |
(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
RESIDUE(r, p, k) with one or
zero output arguments returns b and a in one series of two columns where b ==
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]}}.
(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.
(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:
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
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