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

    p array_like

      Poles. Equal poles must be adjacent.

    k array_like

      Coefficients of the direct polynomial term.

    tol float, 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:
    b ndarray

      Numerator polynomial coefficients.

    a ndarray

      Denominator polynomial coefficients.

  See also

  `residuez`, `unique_roots`, `invres`