Note: This page has been translated by MathWorks. Please click here

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

Partial fraction expansion (partial fraction decomposition)

`[`

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.

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

Was this topic helpful?