Z-transform partial-fraction expansion
[r,p,k] = residuez(b,a)
[b,a] = residuez(r,p,k)
residuez
converts a discrete time system,
expressed as the ratio of two polynomials, to partial fraction expansion, or residue, form. It also
converts the partial fraction expansion back to the original polynomial
coefficients.
Note: Numerically, the partial fraction expansion of a ratio of polynomials is an ill-posed problem. If the denominator polynomial is near a polynomial with multiple roots, then small changes in the data, including round-off errors, can cause arbitrarily large changes in the resulting poles and residues. You should use state-space or pole-zero representations instead. |
[r,p,k] = residuez(b,a)
finds
the residues, poles, and direct terms of a partial fraction expansion
of the ratio of two polynomials, b(z)
and a(z). Vectors b
and a
specify
the coefficients of the polynomials of the discrete-time system b(z)/a(z)
in descending powers of z.
$$\begin{array}{c}B(z)={b}_{0}+{b}_{1}{z}^{-1}+{b}_{2}{z}^{-2}+\cdots +{b}_{m}{z}^{-m}\\ A(z)={a}_{0}+{a}_{1}{z}^{-1}+{a}_{2}{z}^{-2}+\cdots +{a}_{n}{z}^{-n}\end{array}$$
If there are no multiple roots and a
> n-1
,
$$\frac{B(z)}{A(z)}=\frac{r(1)}{1-p(1){z}^{-1}}+\cdots +\frac{r(n)}{1-p(n){z}^{-1}}+k(1)+k(2){z}^{-1}+\cdots +k(m-n+1){z}^{-(m-n)}$$
The returned column vector r
contains the
residues, column vector p
contains the pole locations,
and row vector k
contains the direct terms. The
number of poles is
n = length(a)-1 = length(r) = length(p)
The direct term coefficient vector k
is empty
if length(b)
is less than length(a)
;
otherwise:
length(k) = length(b) - length(a) + 1
If p(j) = ... = p(j+s-1)
is a pole of multiplicity s
,
then the expansion includes terms of the form
$$\frac{r(j)}{1-p(j){z}^{-1}}+\frac{r(j+1)}{{(1-p(j){z}^{-1})}^{2}}+\cdots +\frac{r(j+{s}_{r}-1)}{{(1-p(j){z}^{-1})}^{s}}$$
with three input
arguments and two output arguments, converts the partial fraction
expansion back to polynomials with coefficients in row vectors [b,a] = residuez(r,p,k)
b
and a
.
The residue
function
in the standard MATLAB^{®} language is very similar to residuez
.
It computes the partial fraction expansion of continuous-time systems
in the Laplace domain (see reference [1]),
rather than discrete-time systems in the z-domain
as does residuez
.
[1] Oppenheim, Alan V., Ronald W. Schafer, and John R. Buck. Discrete-Time Signal Processing. 2nd Ed. Upper Saddle River, NJ: Prentice Hall, 1999.