scipy.signal.

invresz#

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

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

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

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

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

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

If there are any repeated roots (closer than tol), then the partial fraction expansion has terms like:

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

This function is used for polynomials in negative powers of z, such as digital filters in DSP. For positive powers, use invres.

Parameters:

rarray_like: Residues corresponding to the poles. For repeated poles, the residues must be ordered to correspond to ascending by power fractions.
parray_like: Poles. Equal poles must be adjacent.
karray_like: 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.

Returns:

bndarray: Numerator polynomial coefficients.
andarray: Denominator polynomial coefficients.

See also

residuez, unique_roots, invres

Examples

The following example builds b(z) / a(z) back from its partial-fraction expansion:

>>> import numpy as np
>>> from scipy import signal
>>> r = np.array([-2.2, 3.2])
>>> p = np.array([0.25, 0.5])
>>> k = np.array([])
>>> b, a = signal.invresz(r, p, k)
>>> b
array([1. , 0.3])
>>> a
array([ 1.   , -0.75 ,  0.125])

Use residuez to get back the expansion:

>>> signal.residuez(b, a)
(array([-2.2,  3.2]), array([0.25, 0.5 ]), array([], dtype=float64))

A non-empty k adds the direct term:

>>> r = np.array([0.4])
>>> p = np.array([0.5])
>>> k = np.array([1.6, -0.2])
>>> b, a = signal.invresz(r, p, k)
>>> b, a
(array([ 2. , -1. ,  0.1]), array([ 1. , -0.5]))