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def zpk2sos(z, p, k, pairing=None, *, analog=False)
Return second-order sections from zeros, poles, and gain of a system
Parameters
----------
z : array_like
Zeros of the transfer function.
p : array_like
Poles of the transfer function.
k : float
System gain.
pairing : {None, 'nearest', 'keep_odd', 'minimal'}, optional
The method to use to combine pairs of poles and zeros into sections.
If analog is False and pairing is None, pairing is set to 'nearest';
if analog is True, pairing must be 'minimal', and is set to that if
it is None.
analog : bool, optional
If True, system is analog, otherwise discrete.
.. versionadded:: 1.8.0
Returns
-------
sos : ndarray
Array of second-order filter coefficients, with shape
``(n_sections, 6)``. See `sosfilt` for the SOS filter format
specification.
See Also
--------
sosfilt
Notes
-----
The algorithm used to convert ZPK to SOS format is designed to
minimize errors due to numerical precision issues. The pairing
algorithm attempts to minimize the peak gain of each biquadratic
section. This is done by pairing poles with the nearest zeros, starting
with the poles closest to the unit circle for discrete-time systems, and
poles closest to the imaginary axis for continuous-time systems.
``pairing='minimal'`` outputs may not be suitable for `sosfilt`,
and ``analog=True`` outputs will never be suitable for `sosfilt`.
*Algorithms*
The steps in the ``pairing='nearest'``, ``pairing='keep_odd'``,
and ``pairing='minimal'`` algorithms are mostly shared. The
``'nearest'`` algorithm attempts to minimize the peak gain, while
``'keep_odd'`` minimizes peak gain under the constraint that
odd-order systems should retain one section as first order.
``'minimal'`` is similar to ``'keep_odd'``, but no additional
poles or zeros are introduced
The algorithm steps are as follows:
As a pre-processing step for ``pairing='nearest'``,
``pairing='keep_odd'``, add poles or zeros to the origin as
necessary to obtain the same number of poles and zeros for
pairing. If ``pairing == 'nearest'`` and there are an odd number
of poles, add an additional pole and a zero at the origin.
The following steps are then iterated over until no more poles or
zeros remain:
1. Take the (next remaining) pole (complex or real) closest to the
unit circle (or imaginary axis, for ``analog=True``) to
begin a new filter section.
2. If the pole is real and there are no other remaining real poles [#]_,
add the closest real zero to the section and leave it as a first
order section. Note that after this step we are guaranteed to be
left with an even number of real poles, complex poles, real zeros,
and complex zeros for subsequent pairing iterations.
3. Else:
1. If the pole is complex and the zero is the only remaining real
zero*, then pair the pole with the *next* closest zero
(guaranteed to be complex). This is necessary to ensure that
there will be a real zero remaining to eventually create a
first-order section (thus keeping the odd order).
2. Else pair the pole with the closest remaining zero (complex or
real).
3. Proceed to complete the second-order section by adding another
pole and zero to the current pole and zero in the section:
1. If the current pole and zero are both complex, add their
conjugates.
2. Else if the pole is complex and the zero is real, add the
conjugate pole and the next closest real zero.
3. Else if the pole is real and the zero is complex, add the
conjugate zero and the real pole closest to those zeros.
4. Else (we must have a real pole and real zero) add the next
real pole closest to the unit circle, and then add the real
zero closest to that pole.
.. [#] This conditional can only be met for specific odd-order inputs
with the ``pairing = 'keep_odd'`` or ``'minimal'`` methods.
.. versionadded:: 0.16.0
Examples
--------
Design a 6th order low-pass elliptic digital filter for a system with a
sampling rate of 8000 Hz that has a pass-band corner frequency of
1000 Hz. The ripple in the pass-band should not exceed 0.087 dB, and
the attenuation in the stop-band should be at least 90 dB.
In the following call to `ellip`, we could use ``output='sos'``,
but for this example, we'll use ``output='zpk'``, and then convert
to SOS format with `zpk2sos`:
>>> from scipy import signal
>>> import numpy as np
>>> z, p, k = signal.ellip(6, 0.087, 90, 1000/(0.5*8000), output='zpk')
Now convert to SOS format.
>>> sos = signal.zpk2sos(z, p, k)
The coefficients of the numerators of the sections:
>>> sos[:, :3]
array([[0.0014152 , 0.00248677, 0.0014152 ],
[1. , 0.72976874, 1. ],
[1. , 0.17607852, 1. ]])
The symmetry in the coefficients occurs because all the zeros are on the
unit circle.
The coefficients of the denominators of the sections:
>>> sos[:, 3:]
array([[ 1. , -1.32544025, 0.46989976],
[ 1. , -1.26118294, 0.62625924],
[ 1. , -1.2570723 , 0.8619958 ]])
The next example shows the effect of the `pairing` option. We have a
system with three poles and three zeros, so the SOS array will have
shape (2, 6). The means there is, in effect, an extra pole and an extra
zero at the origin in the SOS representation.
>>> z1 = np.array([-1, -0.5-0.5j, -0.5+0.5j])
>>> p1 = np.array([0.75, 0.8+0.1j, 0.8-0.1j])
With ``pairing='nearest'`` (the default), we obtain
>>> signal.zpk2sos(z1, p1, 1)
array([[ 1. , 1. , 0.5 , 1. , -0.75, 0. ],
[ 1. , 1. , 0. , 1. , -1.6 , 0.65]])
The first section has the zeros {-0.5-0.05j, -0.5+0.5j} and the poles
{0, 0.75}, and the second section has the zeros {-1, 0} and poles
{0.8+0.1j, 0.8-0.1j}. Note that the extra pole and zero at the origin
have been assigned to different sections.
With ``pairing='keep_odd'``, we obtain:
>>> signal.zpk2sos(z1, p1, 1, pairing='keep_odd')
array([[ 1. , 1. , 0. , 1. , -0.75, 0. ],
[ 1. , 1. , 0.5 , 1. , -1.6 , 0.65]])
The extra pole and zero at the origin are in the same section.
The first section is, in effect, a first-order section.
With ``pairing='minimal'``, the first-order section doesn't have
the extra pole and zero at the origin:
>>> signal.zpk2sos(z1, p1, 1, pairing='minimal')
array([[ 0. , 1. , 1. , 0. , 1. , -0.75],
[ 1. , 1. , 0.5 , 1. , -1.6 , 0.65]])
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