Convert between partial fraction expansion and polynomial coefficients
Syntax
Description
The residue
function converts a quotient of polynomials to pole-residue representation, and back again.
finds the residues, poles, and direct term of a partial fraction expansion of the ratio of two polynomials, and , of the form[r,p,k] = residue(b,a)
where and are the j
th elements of the input vectors b
and a
.
converts the partial fraction expansion back to the polynomials with coefficients in [b,a] = residue(r,p,k)
b
and a
.
Definition
If there are no multiple roots, then
The direct term coefficient vector is empty if length(b)
<
length(a)
; otherwise
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 |
r | Column vector of residues |
p | Column vector of poles |
k | Row vector of direct terms |
Algorithm
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, 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, , 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.
Examples
If the ratio of two polynomials is expressed as
and you can calculate the partial fraction expansion as
Now, convert the partial fraction expansion back to polynomial coefficients.
The result can be expressed as
Note that the result is normalized for the leading coefficient in the denominator.
See Also
References
[1] Oppenheim, A.V. and R.W. Schafer, Digital Signal Processing, Prentice-Hall, 1975, p. 56.
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