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Module « scipy.signal »

Fonction residue - module scipy.signal

Signature de la fonction residue

def residue(b, a, tol=0.001, rtype='avg') 

Description

help(scipy.signal.residue)

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
----------
b : array_like
    Numerator polynomial coefficients.
a : array_like
    Denominator polynomial coefficients.
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
-------
r : ndarray
    Residues corresponding to the poles. For repeated poles, the residues
    are ordered to correspond to ascending by power fractions.
p : ndarray
    Poles ordered by magnitude in ascending order.
k : ndarray
    Coefficients of the direct polynomial term.

See Also
--------
invres, residuez, numpy.poly, unique_roots

Notes
-----
The "deflation through subtraction" algorithm is used for
computations --- method 6 in [1]_.

The form of partial fraction expansion depends on poles multiplicity in
the exact mathematical sense. However there is no way to exactly
determine multiplicity of roots of a polynomial in numerical computing.
Thus you should think of the result of `residue` with given `tol` as
partial fraction expansion computed for the denominator composed of the
computed poles with empirically determined multiplicity. The choice of
`tol` can drastically change the result if there are close poles.

References
----------
.. [1] J. F. Mahoney, B. D. Sivazlian, "Partial fractions expansion: a
       review of computational methodology and efficiency", Journal of
       Computational and Applied Mathematics, Vol. 9, 1983.

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