MATLAB Function Reference |

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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