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residuez() - Signal Processing

```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
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.B(z)=b0+b1z−1+b2z−2+⋯+bmz−mA(z)=a0+a1z−1+a2z−2+⋯+anz−nIf there are no multiple roots and a > n-1,B(z)A(z)=r(1)1−p(1)z−1+⋯+r(n)1−p(n)z−1+k(1)+k(2)z−1+⋯+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 isn = 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 formr(j)1−p(j)z−1+r(j+1)(1−p(j)z−1)2+⋯+r(j+sr−1)(1−p(j)z−1)s[b,a] = residuez(r,p,k)   with three input
arguments and two output arguments, converts the partial fraction
expansion back to polynomials with coefficients in row vectors 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 ),
rather than discrete-time systems in the z-domain
as does residuez.```

Syntax

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

Example

`n = length(a)-1 = length(r) = length(p)`