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Convert between partial fraction expansion and polynomial coefficients

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

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

where *b _{j}* and

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

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

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 |

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.

If the ratio of two polynomials is expressed as

then

b = [ 5 3 -2 7] a = [-4 0 8 3]

and you can calculate the partial fraction expansion as

[r, p, k] = residue(b,a) r = -1.4167 -0.6653 1.3320 p = 1.5737 -1.1644 -0.4093 k = -1.2500

Now, convert the partial fraction expansion back to polynomial coefficients.

[b,a] = residue(r,p,k) b = -1.2500 -0.7500 0.5000 -1.7500 a = 1.0000 -0.0000 -2.0000 -0.7500

The result can be expressed as

Note that the result is normalized for the leading coefficient in the denominator.

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

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