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# residuez() - Signal Processing

### Syntax

### Example

### Output / Return Value

### Limitations

### Alternatives / See Also

### Reference

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.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 [1]), rather than discrete-time systems in the z-domain as does residuez.

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

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