 ``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]])

N