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Partial fraction expansion (partial fraction decomposition)

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

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

`[`

finds
the residues, poles, and direct term of a Partial Fraction Expansion of the ratio of two polynomials,
where the expansion is of the form`r`

,`p`

,`k`

]
= residue(`b`

,`a`

)

$$\frac{b(s)}{a(s)}=\frac{{b}_{m}{s}^{m}+{b}_{m-1}{s}^{m-1}+\dots +{b}_{1}s+{b}_{0}}{{a}_{n}{s}^{n}+{a}_{n-1}{s}^{n-1}+\dots +{a}_{1}s+{a}_{0}}=\frac{{r}_{n}}{s-{p}_{n}}+\mathrm{...}+\frac{{r}_{2}}{s-{p}_{2}}+\frac{{r}_{1}}{s-{p}_{1}}+k\left(s\right).$$

The inputs to `residue`

are vectors of coefficients
of the polynomials `b = [bm ... b1 b0]`

and ```
a
= [an ... a1 a0]
```

. The outputs are the residues ```
r
= [rn ... r2 r1]
```

, the poles `p = [pn ... p2 p1]`

,
and the polynomial `k`

. For most textbook problems, `k`

is `0`

or
a constant.

`residue`

first obtains the poles using `roots`

.
Next, if the fraction is nonproper, the direct term `k`

is
found using `deconv`

, which performs polynomial long
division. Finally, `residue`

determines the residues
by evaluating the polynomial with individual roots removed. For repeated
roots, `resi2`

computes the residues at the repeated
root locations.

Numerically, the partial fraction expansion of a ratio of polynomials
represents an ill-posed problem. If the denominator polynomial, * a*(

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

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