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 : 'nearest', 'keep_odd', optional The method to use to combine pairs of poles and zeros into sections. See Notes below.
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.
*Algorithms*
The current algorithms are designed specifically for use with digital filters. (The output coefficients are not correct for analog filters.)
The steps in the ``pairing='nearest'`` and ``pairing='keep_odd'`` 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. The algorithm steps and are as follows:
As a pre-processing step, 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 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'`` method.
.. 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 `signal.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 >>> 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([ 0.0014154 , 0.00248707, 0.0014154 ],
[ 1. , 0.72965193, 1. ],
[ 1. , 0.17594966, 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. , -1.32543251, 0.46989499],
[ 1. , -1.26117915, 0.6262586 ],
[ 1. , -1.25707217, 0.86199667])
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
, 0.75
, and the second section has the zeros
1, 0
}
and poles
.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.