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# scipy.signal.invresz

`scipy.signal.``invresz`(r, p, k, tol=0.001, rtype='avg')

  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: | r : array_like Residues. p : array_like Poles. 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.
  Returns: | b : ndarray Numerator polynomial coefficients. a : ndarray Denominator polynomial coefficients.

  See also

  `residuez`, `unique_roots`, `invres`