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.


r : array_like


p : array_like


k : array_like

Coefficients of the direct polynomial term.

tol : float, optional

The tolerance for two roots to be considered equal. Default is 1e-3.

rtype : {‘max’, ‘min, ‘avg’}, optional

How to determine the returned root if multiple roots are within tol of each other.

  • ‘max’: pick the maximum of those roots.
  • ‘min’: pick the minimum of those roots.
  • ‘avg’: take the average of those roots.

b : ndarray

Numerator polynomial coefficients.

a : ndarray

Denominator polynomial coefficients.

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