scipy.signal.

residue#

scipy.signal.residue(b, a, tol=0.001, rtype='avg')[source]#

Compute partial-fraction expansion of b(s) / a(s).

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 residuez.

See Notes for details about the algorithm.

Parameters:

barray_like: Numerator polynomial coefficients.
aarray_like: Denominator polynomial coefficients.
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.

Returns:

rndarray: Residues corresponding to the poles. For repeated poles, the residues are ordered to correspond to ascending by power fractions.
pndarray: Poles ordered by magnitude in ascending order.
kndarray: Coefficients of the direct polynomial term.