scipy.signal.invres(r, p, k, tol=0.001, rtype='avg')[source]#

Compute b(s) and a(s) from partial fraction expansion.

If M is the degree of numerator b and N the degree of denominator a:

        b(s)     b[0] s**(M) + b[1] s**(M-1) + ... + b[M]
H(s) = ------ = ------------------------------------------
        a(s)     a[0] s**(N) + a[1] s**(N-1) + ... + a[N]

then the partial-fraction expansion H(s) is defined as:

    r[0]       r[1]             r[-1]
= -------- + -------- + ... + --------- + k(s)
  (s-p[0])   (s-p[1])         (s-p[-1])

If there are any repeated roots (closer together than tol), then H(s) has terms like:

  r[i]      r[i+1]              r[i+n-1]
-------- + ----------- + ... + -----------
(s-p[i])  (s-p[i])**2          (s-p[i])**n

This function is used for polynomials in positive powers of s or z, such as analog filters or digital filters in controls engineering. For negative powers of z (typical for digital filters in DSP), use invresz.


Residues corresponding to the poles. For repeated poles, the residues must be ordered to correspond to ascending by power fractions.


Poles. Equal poles must be adjacent.


Coefficients of the direct polynomial term.

tolfloat, optional

The tolerance for two roots to be considered equal in terms of the distance between them. Default is 1e-3. See unique_roots for further details.

rtype{‘avg’, ‘min’, ‘max’}, optional

Method for computing a root to represent a group of identical roots. Default is ‘avg’. See unique_roots for further details.


Numerator polynomial coefficients.


Denominator polynomial coefficients.