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speech2derive.old/lib/python3.8/site-packages/scipy/signal/filter_design.py

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"""Filter design."""
import math
import operator
import warnings
import numpy
import numpy as np
from numpy import (atleast_1d, poly, polyval, roots, real, asarray,
resize, pi, absolute, logspace, r_, sqrt, tan, log10,
arctan, arcsinh, sin, exp, cosh, arccosh, ceil, conjugate,
zeros, sinh, append, concatenate, prod, ones, full, array,
mintypecode)
from numpy.polynomial.polynomial import polyval as npp_polyval
from numpy.polynomial.polynomial import polyvalfromroots
from scipy import special, optimize, fft as sp_fft
from scipy.special import comb
from scipy._lib._util import float_factorial
__all__ = ['findfreqs', 'freqs', 'freqz', 'tf2zpk', 'zpk2tf', 'normalize',
'lp2lp', 'lp2hp', 'lp2bp', 'lp2bs', 'bilinear', 'iirdesign',
'iirfilter', 'butter', 'cheby1', 'cheby2', 'ellip', 'bessel',
'band_stop_obj', 'buttord', 'cheb1ord', 'cheb2ord', 'ellipord',
'buttap', 'cheb1ap', 'cheb2ap', 'ellipap', 'besselap',
'BadCoefficients', 'freqs_zpk', 'freqz_zpk',
'tf2sos', 'sos2tf', 'zpk2sos', 'sos2zpk', 'group_delay',
'sosfreqz', 'iirnotch', 'iirpeak', 'bilinear_zpk',
'lp2lp_zpk', 'lp2hp_zpk', 'lp2bp_zpk', 'lp2bs_zpk',
'gammatone', 'iircomb']
class BadCoefficients(UserWarning):
"""Warning about badly conditioned filter coefficients"""
pass
abs = absolute
def _is_int_type(x):
"""
Check if input is of a scalar integer type (so ``5`` and ``array(5)`` will
pass, while ``5.0`` and ``array([5])`` will fail.
"""
if np.ndim(x) != 0:
# Older versions of NumPy did not raise for np.array([1]).__index__()
# This is safe to remove when support for those versions is dropped
return False
try:
operator.index(x)
except TypeError:
return False
else:
return True
def findfreqs(num, den, N, kind='ba'):
"""
Find array of frequencies for computing the response of an analog filter.
Parameters
----------
num, den : array_like, 1-D
The polynomial coefficients of the numerator and denominator of the
transfer function of the filter or LTI system, where the coefficients
are ordered from highest to lowest degree. Or, the roots of the
transfer function numerator and denominator (i.e., zeroes and poles).
N : int
The length of the array to be computed.
kind : str {'ba', 'zp'}, optional
Specifies whether the numerator and denominator are specified by their
polynomial coefficients ('ba'), or their roots ('zp').
Returns
-------
w : (N,) ndarray
A 1-D array of frequencies, logarithmically spaced.
Examples
--------
Find a set of nine frequencies that span the "interesting part" of the
frequency response for the filter with the transfer function
H(s) = s / (s^2 + 8s + 25)
>>> from scipy import signal
>>> signal.findfreqs([1, 0], [1, 8, 25], N=9)
array([ 1.00000000e-02, 3.16227766e-02, 1.00000000e-01,
3.16227766e-01, 1.00000000e+00, 3.16227766e+00,
1.00000000e+01, 3.16227766e+01, 1.00000000e+02])
"""
if kind == 'ba':
ep = atleast_1d(roots(den)) + 0j
tz = atleast_1d(roots(num)) + 0j
elif kind == 'zp':
ep = atleast_1d(den) + 0j
tz = atleast_1d(num) + 0j
else:
raise ValueError("input must be one of {'ba', 'zp'}")
if len(ep) == 0:
ep = atleast_1d(-1000) + 0j
ez = r_['-1',
numpy.compress(ep.imag >= 0, ep, axis=-1),
numpy.compress((abs(tz) < 1e5) & (tz.imag >= 0), tz, axis=-1)]
integ = abs(ez) < 1e-10
hfreq = numpy.around(numpy.log10(numpy.max(3 * abs(ez.real + integ) +
1.5 * ez.imag)) + 0.5)
lfreq = numpy.around(numpy.log10(0.1 * numpy.min(abs(real(ez + integ)) +
2 * ez.imag)) - 0.5)
w = logspace(lfreq, hfreq, N)
return w
def freqs(b, a, worN=200, plot=None):
"""
Compute frequency response of analog filter.
Given the M-order numerator `b` and N-order denominator `a` of an analog
filter, compute its frequency response::
b[0]*(jw)**M + b[1]*(jw)**(M-1) + ... + b[M]
H(w) = ----------------------------------------------
a[0]*(jw)**N + a[1]*(jw)**(N-1) + ... + a[N]
Parameters
----------
b : array_like
Numerator of a linear filter.
a : array_like
Denominator of a linear filter.
worN : {None, int, array_like}, optional
If None, then compute at 200 frequencies around the interesting parts
of the response curve (determined by pole-zero locations). If a single
integer, then compute at that many frequencies. Otherwise, compute the
response at the angular frequencies (e.g., rad/s) given in `worN`.
plot : callable, optional
A callable that takes two arguments. If given, the return parameters
`w` and `h` are passed to plot. Useful for plotting the frequency
response inside `freqs`.
Returns
-------
w : ndarray
The angular frequencies at which `h` was computed.
h : ndarray
The frequency response.
See Also
--------
freqz : Compute the frequency response of a digital filter.
Notes
-----
Using Matplotlib's "plot" function as the callable for `plot` produces
unexpected results, this plots the real part of the complex transfer
function, not the magnitude. Try ``lambda w, h: plot(w, abs(h))``.
Examples
--------
>>> from scipy.signal import freqs, iirfilter
>>> b, a = iirfilter(4, [1, 10], 1, 60, analog=True, ftype='cheby1')
>>> w, h = freqs(b, a, worN=np.logspace(-1, 2, 1000))
>>> import matplotlib.pyplot as plt
>>> plt.semilogx(w, 20 * np.log10(abs(h)))
>>> plt.xlabel('Frequency')
>>> plt.ylabel('Amplitude response [dB]')
>>> plt.grid()
>>> plt.show()
"""
if worN is None:
# For backwards compatibility
w = findfreqs(b, a, 200)
elif _is_int_type(worN):
w = findfreqs(b, a, worN)
else:
w = atleast_1d(worN)
s = 1j * w
h = polyval(b, s) / polyval(a, s)
if plot is not None:
plot(w, h)
return w, h
def freqs_zpk(z, p, k, worN=200):
"""
Compute frequency response of analog filter.
Given the zeros `z`, poles `p`, and gain `k` of a filter, compute its
frequency response::
(jw-z[0]) * (jw-z[1]) * ... * (jw-z[-1])
H(w) = k * ----------------------------------------
(jw-p[0]) * (jw-p[1]) * ... * (jw-p[-1])
Parameters
----------
z : array_like
Zeroes of a linear filter
p : array_like
Poles of a linear filter
k : scalar
Gain of a linear filter
worN : {None, int, array_like}, optional
If None, then compute at 200 frequencies around the interesting parts
of the response curve (determined by pole-zero locations). If a single
integer, then compute at that many frequencies. Otherwise, compute the
response at the angular frequencies (e.g., rad/s) given in `worN`.
Returns
-------
w : ndarray
The angular frequencies at which `h` was computed.
h : ndarray
The frequency response.
See Also
--------
freqs : Compute the frequency response of an analog filter in TF form
freqz : Compute the frequency response of a digital filter in TF form
freqz_zpk : Compute the frequency response of a digital filter in ZPK form
Notes
-----
.. versionadded:: 0.19.0
Examples
--------
>>> from scipy.signal import freqs_zpk, iirfilter
>>> z, p, k = iirfilter(4, [1, 10], 1, 60, analog=True, ftype='cheby1',
... output='zpk')
>>> w, h = freqs_zpk(z, p, k, worN=np.logspace(-1, 2, 1000))
>>> import matplotlib.pyplot as plt
>>> plt.semilogx(w, 20 * np.log10(abs(h)))
>>> plt.xlabel('Frequency')
>>> plt.ylabel('Amplitude response [dB]')
>>> plt.grid()
>>> plt.show()
"""
k = np.asarray(k)
if k.size > 1:
raise ValueError('k must be a single scalar gain')
if worN is None:
# For backwards compatibility
w = findfreqs(z, p, 200, kind='zp')
elif _is_int_type(worN):
w = findfreqs(z, p, worN, kind='zp')
else:
w = worN
w = atleast_1d(w)
s = 1j * w
num = polyvalfromroots(s, z)
den = polyvalfromroots(s, p)
h = k * num/den
return w, h
def freqz(b, a=1, worN=512, whole=False, plot=None, fs=2*pi, include_nyquist=False):
"""
Compute the frequency response of a digital filter.
Given the M-order numerator `b` and N-order denominator `a` of a digital
filter, compute its frequency response::
jw -jw -jwM
jw B(e ) b[0] + b[1]e + ... + b[M]e
H(e ) = ------ = -----------------------------------
jw -jw -jwN
A(e ) a[0] + a[1]e + ... + a[N]e
Parameters
----------
b : array_like
Numerator of a linear filter. If `b` has dimension greater than 1,
it is assumed that the coefficients are stored in the first dimension,
and ``b.shape[1:]``, ``a.shape[1:]``, and the shape of the frequencies
array must be compatible for broadcasting.
a : array_like
Denominator of a linear filter. If `b` has dimension greater than 1,
it is assumed that the coefficients are stored in the first dimension,
and ``b.shape[1:]``, ``a.shape[1:]``, and the shape of the frequencies
array must be compatible for broadcasting.
worN : {None, int, array_like}, optional
If a single integer, then compute at that many frequencies (default is
N=512). This is a convenient alternative to::
np.linspace(0, fs if whole else fs/2, N, endpoint=include_nyquist)
Using a number that is fast for FFT computations can result in
faster computations (see Notes).
If an array_like, compute the response at the frequencies given.
These are in the same units as `fs`.
whole : bool, optional
Normally, frequencies are computed from 0 to the Nyquist frequency,
fs/2 (upper-half of unit-circle). If `whole` is True, compute
frequencies from 0 to fs. Ignored if worN is array_like.
plot : callable
A callable that takes two arguments. If given, the return parameters
`w` and `h` are passed to plot. Useful for plotting the frequency
response inside `freqz`.
fs : float, optional
The sampling frequency of the digital system. Defaults to 2*pi
radians/sample (so w is from 0 to pi).
.. versionadded:: 1.2.0
include_nyquist : bool, optional
If `whole` is False and `worN` is an integer, setting `include_nyquist` to True
will include the last frequency (Nyquist frequency) and is otherwise ignored.
.. versionadded:: 1.5.0
Returns
-------
w : ndarray
The frequencies at which `h` was computed, in the same units as `fs`.
By default, `w` is normalized to the range [0, pi) (radians/sample).
h : ndarray
The frequency response, as complex numbers.
See Also
--------
freqz_zpk
sosfreqz
Notes
-----
Using Matplotlib's :func:`matplotlib.pyplot.plot` function as the callable
for `plot` produces unexpected results, as this plots the real part of the
complex transfer function, not the magnitude.
Try ``lambda w, h: plot(w, np.abs(h))``.
A direct computation via (R)FFT is used to compute the frequency response
when the following conditions are met:
1. An integer value is given for `worN`.
2. `worN` is fast to compute via FFT (i.e.,
`next_fast_len(worN) <scipy.fft.next_fast_len>` equals `worN`).
3. The denominator coefficients are a single value (``a.shape[0] == 1``).
4. `worN` is at least as long as the numerator coefficients
(``worN >= b.shape[0]``).
5. If ``b.ndim > 1``, then ``b.shape[-1] == 1``.
For long FIR filters, the FFT approach can have lower error and be much
faster than the equivalent direct polynomial calculation.
Examples
--------
>>> from scipy import signal
>>> b = signal.firwin(80, 0.5, window=('kaiser', 8))
>>> w, h = signal.freqz(b)
>>> import matplotlib.pyplot as plt
>>> fig, ax1 = plt.subplots()
>>> ax1.set_title('Digital filter frequency response')
>>> ax1.plot(w, 20 * np.log10(abs(h)), 'b')
>>> ax1.set_ylabel('Amplitude [dB]', color='b')
>>> ax1.set_xlabel('Frequency [rad/sample]')
>>> ax2 = ax1.twinx()
>>> angles = np.unwrap(np.angle(h))
>>> ax2.plot(w, angles, 'g')
>>> ax2.set_ylabel('Angle (radians)', color='g')
>>> ax2.grid()
>>> ax2.axis('tight')
>>> plt.show()
Broadcasting Examples
Suppose we have two FIR filters whose coefficients are stored in the
rows of an array with shape (2, 25). For this demonstration, we'll
use random data:
>>> np.random.seed(42)
>>> b = np.random.rand(2, 25)
To compute the frequency response for these two filters with one call
to `freqz`, we must pass in ``b.T``, because `freqz` expects the first
axis to hold the coefficients. We must then extend the shape with a
trivial dimension of length 1 to allow broadcasting with the array
of frequencies. That is, we pass in ``b.T[..., np.newaxis]``, which has
shape (25, 2, 1):
>>> w, h = signal.freqz(b.T[..., np.newaxis], worN=1024)
>>> w.shape
(1024,)
>>> h.shape
(2, 1024)
Now, suppose we have two transfer functions, with the same numerator
coefficients ``b = [0.5, 0.5]``. The coefficients for the two denominators
are stored in the first dimension of the 2-D array `a`::
a = [ 1 1 ]
[ -0.25, -0.5 ]
>>> b = np.array([0.5, 0.5])
>>> a = np.array([[1, 1], [-0.25, -0.5]])
Only `a` is more than 1-D. To make it compatible for
broadcasting with the frequencies, we extend it with a trivial dimension
in the call to `freqz`:
>>> w, h = signal.freqz(b, a[..., np.newaxis], worN=1024)
>>> w.shape
(1024,)
>>> h.shape
(2, 1024)
"""
b = atleast_1d(b)
a = atleast_1d(a)
if worN is None:
# For backwards compatibility
worN = 512
h = None
if _is_int_type(worN):
N = operator.index(worN)
del worN
if N < 0:
raise ValueError('worN must be nonnegative, got %s' % (N,))
lastpoint = 2 * pi if whole else pi
# if include_nyquist is true and whole is false, w should include end point
w = np.linspace(0, lastpoint, N, endpoint=include_nyquist and not whole)
if (a.size == 1 and N >= b.shape[0] and
sp_fft.next_fast_len(N) == N and
(b.ndim == 1 or (b.shape[-1] == 1))):
# if N is fast, 2 * N will be fast, too, so no need to check
n_fft = N if whole else N * 2
if np.isrealobj(b) and np.isrealobj(a):
fft_func = sp_fft.rfft
else:
fft_func = sp_fft.fft
h = fft_func(b, n=n_fft, axis=0)[:N]
h /= a
if fft_func is sp_fft.rfft and whole:
# exclude DC and maybe Nyquist (no need to use axis_reverse
# here because we can build reversal with the truncation)
stop = -1 if n_fft % 2 == 1 else -2
h_flip = slice(stop, 0, -1)
h = np.concatenate((h, h[h_flip].conj()))
if b.ndim > 1:
# Last axis of h has length 1, so drop it.
h = h[..., 0]
# Rotate the first axis of h to the end.
h = np.rollaxis(h, 0, h.ndim)
else:
w = atleast_1d(worN)
del worN
w = 2*pi*w/fs
if h is None: # still need to compute using freqs w
zm1 = exp(-1j * w)
h = (npp_polyval(zm1, b, tensor=False) /
npp_polyval(zm1, a, tensor=False))
w = w*fs/(2*pi)
if plot is not None:
plot(w, h)
return w, h
def freqz_zpk(z, p, k, worN=512, whole=False, fs=2*pi):
r"""
Compute the frequency response of a digital filter in ZPK form.
Given the Zeros, Poles and Gain of a digital filter, compute its frequency
response:
:math:`H(z)=k \prod_i (z - Z[i]) / \prod_j (z - P[j])`
where :math:`k` is the `gain`, :math:`Z` are the `zeros` and :math:`P` are
the `poles`.
Parameters
----------
z : array_like
Zeroes of a linear filter
p : array_like
Poles of a linear filter
k : scalar
Gain of a linear filter
worN : {None, int, array_like}, optional
If a single integer, then compute at that many frequencies (default is
N=512).
If an array_like, compute the response at the frequencies given.
These are in the same units as `fs`.
whole : bool, optional
Normally, frequencies are computed from 0 to the Nyquist frequency,
fs/2 (upper-half of unit-circle). If `whole` is True, compute
frequencies from 0 to fs. Ignored if w is array_like.
fs : float, optional
The sampling frequency of the digital system. Defaults to 2*pi
radians/sample (so w is from 0 to pi).
.. versionadded:: 1.2.0
Returns
-------
w : ndarray
The frequencies at which `h` was computed, in the same units as `fs`.
By default, `w` is normalized to the range [0, pi) (radians/sample).
h : ndarray
The frequency response, as complex numbers.
See Also
--------
freqs : Compute the frequency response of an analog filter in TF form
freqs_zpk : Compute the frequency response of an analog filter in ZPK form
freqz : Compute the frequency response of a digital filter in TF form
Notes
-----
.. versionadded:: 0.19.0
Examples
--------
Design a 4th-order digital Butterworth filter with cut-off of 100 Hz in a
system with sample rate of 1000 Hz, and plot the frequency response:
>>> from scipy import signal
>>> z, p, k = signal.butter(4, 100, output='zpk', fs=1000)
>>> w, h = signal.freqz_zpk(z, p, k, fs=1000)
>>> import matplotlib.pyplot as plt
>>> fig = plt.figure()
>>> ax1 = fig.add_subplot(1, 1, 1)
>>> ax1.set_title('Digital filter frequency response')
>>> ax1.plot(w, 20 * np.log10(abs(h)), 'b')
>>> ax1.set_ylabel('Amplitude [dB]', color='b')
>>> ax1.set_xlabel('Frequency [Hz]')
>>> ax1.grid()
>>> ax2 = ax1.twinx()
>>> angles = np.unwrap(np.angle(h))
>>> ax2.plot(w, angles, 'g')
>>> ax2.set_ylabel('Angle [radians]', color='g')
>>> plt.axis('tight')
>>> plt.show()
"""
z, p = map(atleast_1d, (z, p))
if whole:
lastpoint = 2 * pi
else:
lastpoint = pi
if worN is None:
# For backwards compatibility
w = numpy.linspace(0, lastpoint, 512, endpoint=False)
elif _is_int_type(worN):
w = numpy.linspace(0, lastpoint, worN, endpoint=False)
else:
w = atleast_1d(worN)
w = 2*pi*w/fs
zm1 = exp(1j * w)
h = k * polyvalfromroots(zm1, z) / polyvalfromroots(zm1, p)
w = w*fs/(2*pi)
return w, h
def group_delay(system, w=512, whole=False, fs=2*pi):
r"""Compute the group delay of a digital filter.
The group delay measures by how many samples amplitude envelopes of
various spectral components of a signal are delayed by a filter.
It is formally defined as the derivative of continuous (unwrapped) phase::
d jw
D(w) = - -- arg H(e)
dw
Parameters
----------
system : tuple of array_like (b, a)
Numerator and denominator coefficients of a filter transfer function.
w : {None, int, array_like}, optional
If a single integer, then compute at that many frequencies (default is
N=512).
If an array_like, compute the delay at the frequencies given. These
are in the same units as `fs`.
whole : bool, optional
Normally, frequencies are computed from 0 to the Nyquist frequency,
fs/2 (upper-half of unit-circle). If `whole` is True, compute
frequencies from 0 to fs. Ignored if w is array_like.
fs : float, optional
The sampling frequency of the digital system. Defaults to 2*pi
radians/sample (so w is from 0 to pi).
.. versionadded:: 1.2.0
Returns
-------
w : ndarray
The frequencies at which group delay was computed, in the same units
as `fs`. By default, `w` is normalized to the range [0, pi)
(radians/sample).
gd : ndarray
The group delay.
Notes
-----
The similar function in MATLAB is called `grpdelay`.
If the transfer function :math:`H(z)` has zeros or poles on the unit
circle, the group delay at corresponding frequencies is undefined.
When such a case arises the warning is raised and the group delay
is set to 0 at those frequencies.
For the details of numerical computation of the group delay refer to [1]_.
.. versionadded:: 0.16.0
See Also
--------
freqz : Frequency response of a digital filter
References
----------
.. [1] Richard G. Lyons, "Understanding Digital Signal Processing,
3rd edition", p. 830.
Examples
--------
>>> from scipy import signal
>>> b, a = signal.iirdesign(0.1, 0.3, 5, 50, ftype='cheby1')
>>> w, gd = signal.group_delay((b, a))
>>> import matplotlib.pyplot as plt
>>> plt.title('Digital filter group delay')
>>> plt.plot(w, gd)
>>> plt.ylabel('Group delay [samples]')
>>> plt.xlabel('Frequency [rad/sample]')
>>> plt.show()
"""
if w is None:
# For backwards compatibility
w = 512
if _is_int_type(w):
if whole:
w = np.linspace(0, 2 * pi, w, endpoint=False)
else:
w = np.linspace(0, pi, w, endpoint=False)
else:
w = np.atleast_1d(w)
w = 2*pi*w/fs
b, a = map(np.atleast_1d, system)
c = np.convolve(b, a[::-1])
cr = c * np.arange(c.size)
z = np.exp(-1j * w)
num = np.polyval(cr[::-1], z)
den = np.polyval(c[::-1], z)
singular = np.absolute(den) < 10 * EPSILON
if np.any(singular):
warnings.warn(
"The group delay is singular at frequencies [{0}], setting to 0".
format(", ".join("{0:.3f}".format(ws) for ws in w[singular]))
)
gd = np.zeros_like(w)
gd[~singular] = np.real(num[~singular] / den[~singular]) - a.size + 1
w = w*fs/(2*pi)
return w, gd
def _validate_sos(sos):
"""Helper to validate a SOS input"""
sos = np.atleast_2d(sos)
if sos.ndim != 2:
raise ValueError('sos array must be 2D')
n_sections, m = sos.shape
if m != 6:
raise ValueError('sos array must be shape (n_sections, 6)')
if not (sos[:, 3] == 1).all():
raise ValueError('sos[:, 3] should be all ones')
return sos, n_sections
def sosfreqz(sos, worN=512, whole=False, fs=2*pi):
r"""
Compute the frequency response of a digital filter in SOS format.
Given `sos`, an array with shape (n, 6) of second order sections of
a digital filter, compute the frequency response of the system function::
B0(z) B1(z) B{n-1}(z)
H(z) = ----- * ----- * ... * ---------
A0(z) A1(z) A{n-1}(z)
for z = exp(omega*1j), where B{k}(z) and A{k}(z) are numerator and
denominator of the transfer function of the k-th second order section.
Parameters
----------
sos : array_like
Array of second-order filter coefficients, must have shape
``(n_sections, 6)``. Each row corresponds to a second-order
section, with the first three columns providing the numerator
coefficients and the last three providing the denominator
coefficients.
worN : {None, int, array_like}, optional
If a single integer, then compute at that many frequencies (default is
N=512). Using a number that is fast for FFT computations can result
in faster computations (see Notes of `freqz`).
If an array_like, compute the response at the frequencies given (must
be 1-D). These are in the same units as `fs`.
whole : bool, optional
Normally, frequencies are computed from 0 to the Nyquist frequency,
fs/2 (upper-half of unit-circle). If `whole` is True, compute
frequencies from 0 to fs.
fs : float, optional
The sampling frequency of the digital system. Defaults to 2*pi
radians/sample (so w is from 0 to pi).
.. versionadded:: 1.2.0
Returns
-------
w : ndarray
The frequencies at which `h` was computed, in the same units as `fs`.
By default, `w` is normalized to the range [0, pi) (radians/sample).
h : ndarray
The frequency response, as complex numbers.
See Also
--------
freqz, sosfilt
Notes
-----
.. versionadded:: 0.19.0
Examples
--------
Design a 15th-order bandpass filter in SOS format.
>>> from scipy import signal
>>> sos = signal.ellip(15, 0.5, 60, (0.2, 0.4), btype='bandpass',
... output='sos')
Compute the frequency response at 1500 points from DC to Nyquist.
>>> w, h = signal.sosfreqz(sos, worN=1500)
Plot the response.
>>> import matplotlib.pyplot as plt
>>> plt.subplot(2, 1, 1)
>>> db = 20*np.log10(np.maximum(np.abs(h), 1e-5))
>>> plt.plot(w/np.pi, db)
>>> plt.ylim(-75, 5)
>>> plt.grid(True)
>>> plt.yticks([0, -20, -40, -60])
>>> plt.ylabel('Gain [dB]')
>>> plt.title('Frequency Response')
>>> plt.subplot(2, 1, 2)
>>> plt.plot(w/np.pi, np.angle(h))
>>> plt.grid(True)
>>> plt.yticks([-np.pi, -0.5*np.pi, 0, 0.5*np.pi, np.pi],
... [r'$-\pi$', r'$-\pi/2$', '0', r'$\pi/2$', r'$\pi$'])
>>> plt.ylabel('Phase [rad]')
>>> plt.xlabel('Normalized frequency (1.0 = Nyquist)')
>>> plt.show()
If the same filter is implemented as a single transfer function,
numerical error corrupts the frequency response:
>>> b, a = signal.ellip(15, 0.5, 60, (0.2, 0.4), btype='bandpass',
... output='ba')
>>> w, h = signal.freqz(b, a, worN=1500)
>>> plt.subplot(2, 1, 1)
>>> db = 20*np.log10(np.maximum(np.abs(h), 1e-5))
>>> plt.plot(w/np.pi, db)
>>> plt.ylim(-75, 5)
>>> plt.grid(True)
>>> plt.yticks([0, -20, -40, -60])
>>> plt.ylabel('Gain [dB]')
>>> plt.title('Frequency Response')
>>> plt.subplot(2, 1, 2)
>>> plt.plot(w/np.pi, np.angle(h))
>>> plt.grid(True)
>>> plt.yticks([-np.pi, -0.5*np.pi, 0, 0.5*np.pi, np.pi],
... [r'$-\pi$', r'$-\pi/2$', '0', r'$\pi/2$', r'$\pi$'])
>>> plt.ylabel('Phase [rad]')
>>> plt.xlabel('Normalized frequency (1.0 = Nyquist)')
>>> plt.show()
"""
sos, n_sections = _validate_sos(sos)
if n_sections == 0:
raise ValueError('Cannot compute frequencies with no sections')
h = 1.
for row in sos:
w, rowh = freqz(row[:3], row[3:], worN=worN, whole=whole, fs=fs)
h *= rowh
return w, h
def _cplxreal(z, tol=None):
"""
Split into complex and real parts, combining conjugate pairs.
The 1-D input vector `z` is split up into its complex (`zc`) and real (`zr`)
elements. Every complex element must be part of a complex-conjugate pair,
which are combined into a single number (with positive imaginary part) in
the output. Two complex numbers are considered a conjugate pair if their
real and imaginary parts differ in magnitude by less than ``tol * abs(z)``.
Parameters
----------
z : array_like
Vector of complex numbers to be sorted and split
tol : float, optional
Relative tolerance for testing realness and conjugate equality.
Default is ``100 * spacing(1)`` of `z`'s data type (i.e., 2e-14 for
float64)
Returns
-------
zc : ndarray
Complex elements of `z`, with each pair represented by a single value
having positive imaginary part, sorted first by real part, and then
by magnitude of imaginary part. The pairs are averaged when combined
to reduce error.
zr : ndarray
Real elements of `z` (those having imaginary part less than
`tol` times their magnitude), sorted by value.
Raises
------
ValueError
If there are any complex numbers in `z` for which a conjugate
cannot be found.
See Also
--------
_cplxpair
Examples
--------
>>> a = [4, 3, 1, 2-2j, 2+2j, 2-1j, 2+1j, 2-1j, 2+1j, 1+1j, 1-1j]
>>> zc, zr = _cplxreal(a)
>>> print(zc)
[ 1.+1.j 2.+1.j 2.+1.j 2.+2.j]
>>> print(zr)
[ 1. 3. 4.]
"""
z = atleast_1d(z)
if z.size == 0:
return z, z
elif z.ndim != 1:
raise ValueError('_cplxreal only accepts 1-D input')
if tol is None:
# Get tolerance from dtype of input
tol = 100 * np.finfo((1.0 * z).dtype).eps
# Sort by real part, magnitude of imaginary part (speed up further sorting)
z = z[np.lexsort((abs(z.imag), z.real))]
# Split reals from conjugate pairs
real_indices = abs(z.imag) <= tol * abs(z)
zr = z[real_indices].real
if len(zr) == len(z):
# Input is entirely real
return array([]), zr
# Split positive and negative halves of conjugates
z = z[~real_indices]
zp = z[z.imag > 0]
zn = z[z.imag < 0]
if len(zp) != len(zn):
raise ValueError('Array contains complex value with no matching '
'conjugate.')
# Find runs of (approximately) the same real part
same_real = np.diff(zp.real) <= tol * abs(zp[:-1])
diffs = numpy.diff(concatenate(([0], same_real, [0])))
run_starts = numpy.nonzero(diffs > 0)[0]
run_stops = numpy.nonzero(diffs < 0)[0]
# Sort each run by their imaginary parts
for i in range(len(run_starts)):
start = run_starts[i]
stop = run_stops[i] + 1
for chunk in (zp[start:stop], zn[start:stop]):
chunk[...] = chunk[np.lexsort([abs(chunk.imag)])]
# Check that negatives match positives
if any(abs(zp - zn.conj()) > tol * abs(zn)):
raise ValueError('Array contains complex value with no matching '
'conjugate.')
# Average out numerical inaccuracy in real vs imag parts of pairs
zc = (zp + zn.conj()) / 2
return zc, zr
def _cplxpair(z, tol=None):
"""
Sort into pairs of complex conjugates.
Complex conjugates in `z` are sorted by increasing real part. In each
pair, the number with negative imaginary part appears first.
If pairs have identical real parts, they are sorted by increasing
imaginary magnitude.
Two complex numbers are considered a conjugate pair if their real and
imaginary parts differ in magnitude by less than ``tol * abs(z)``. The
pairs are forced to be exact complex conjugates by averaging the positive
and negative values.
Purely real numbers are also sorted, but placed after the complex
conjugate pairs. A number is considered real if its imaginary part is
smaller than `tol` times the magnitude of the number.
Parameters
----------
z : array_like
1-D input array to be sorted.
tol : float, optional
Relative tolerance for testing realness and conjugate equality.
Default is ``100 * spacing(1)`` of `z`'s data type (i.e., 2e-14 for
float64)
Returns
-------
y : ndarray
Complex conjugate pairs followed by real numbers.
Raises
------
ValueError
If there are any complex numbers in `z` for which a conjugate
cannot be found.
See Also
--------
_cplxreal
Examples
--------
>>> a = [4, 3, 1, 2-2j, 2+2j, 2-1j, 2+1j, 2-1j, 2+1j, 1+1j, 1-1j]
>>> z = _cplxpair(a)
>>> print(z)
[ 1.-1.j 1.+1.j 2.-1.j 2.+1.j 2.-1.j 2.+1.j 2.-2.j 2.+2.j 1.+0.j
3.+0.j 4.+0.j]
"""
z = atleast_1d(z)
if z.size == 0 or np.isrealobj(z):
return np.sort(z)
if z.ndim != 1:
raise ValueError('z must be 1-D')
zc, zr = _cplxreal(z, tol)
# Interleave complex values and their conjugates, with negative imaginary
# parts first in each pair
zc = np.dstack((zc.conj(), zc)).flatten()
z = np.append(zc, zr)
return z
def tf2zpk(b, a):
r"""Return zero, pole, gain (z, p, k) representation from a numerator,
denominator representation of a linear filter.
Parameters
----------
b : array_like
Numerator polynomial coefficients.
a : array_like
Denominator polynomial coefficients.
Returns
-------
z : ndarray
Zeros of the transfer function.
p : ndarray
Poles of the transfer function.
k : float
System gain.
Notes
-----
If some values of `b` are too close to 0, they are removed. In that case,
a BadCoefficients warning is emitted.
The `b` and `a` arrays are interpreted as coefficients for positive,
descending powers of the transfer function variable. So the inputs
:math:`b = [b_0, b_1, ..., b_M]` and :math:`a =[a_0, a_1, ..., a_N]`
can represent an analog filter of the form:
.. math::
H(s) = \frac
{b_0 s^M + b_1 s^{(M-1)} + \cdots + b_M}
{a_0 s^N + a_1 s^{(N-1)} + \cdots + a_N}
or a discrete-time filter of the form:
.. math::
H(z) = \frac
{b_0 z^M + b_1 z^{(M-1)} + \cdots + b_M}
{a_0 z^N + a_1 z^{(N-1)} + \cdots + a_N}
This "positive powers" form is found more commonly in controls
engineering. If `M` and `N` are equal (which is true for all filters
generated by the bilinear transform), then this happens to be equivalent
to the "negative powers" discrete-time form preferred in DSP:
.. math::
H(z) = \frac
{b_0 + b_1 z^{-1} + \cdots + b_M z^{-M}}
{a_0 + a_1 z^{-1} + \cdots + a_N z^{-N}}
Although this is true for common filters, remember that this is not true
in the general case. If `M` and `N` are not equal, the discrete-time
transfer function coefficients must first be converted to the "positive
powers" form before finding the poles and zeros.
"""
b, a = normalize(b, a)
b = (b + 0.0) / a[0]
a = (a + 0.0) / a[0]
k = b[0]
b /= b[0]
z = roots(b)
p = roots(a)
return z, p, k
def zpk2tf(z, p, k):
"""
Return polynomial transfer function representation from zeros and poles
Parameters
----------
z : array_like
Zeros of the transfer function.
p : array_like
Poles of the transfer function.
k : float
System gain.
Returns
-------
b : ndarray
Numerator polynomial coefficients.
a : ndarray
Denominator polynomial coefficients.
"""
z = atleast_1d(z)
k = atleast_1d(k)
if len(z.shape) > 1:
temp = poly(z[0])
b = np.empty((z.shape[0], z.shape[1] + 1), temp.dtype.char)
if len(k) == 1:
k = [k[0]] * z.shape[0]
for i in range(z.shape[0]):
b[i] = k[i] * poly(z[i])
else:
b = k * poly(z)
a = atleast_1d(poly(p))
# Use real output if possible. Copied from numpy.poly, since
# we can't depend on a specific version of numpy.
if issubclass(b.dtype.type, numpy.complexfloating):
# if complex roots are all complex conjugates, the roots are real.
roots = numpy.asarray(z, complex)
pos_roots = numpy.compress(roots.imag > 0, roots)
neg_roots = numpy.conjugate(numpy.compress(roots.imag < 0, roots))
if len(pos_roots) == len(neg_roots):
if numpy.all(numpy.sort_complex(neg_roots) ==
numpy.sort_complex(pos_roots)):
b = b.real.copy()
if issubclass(a.dtype.type, numpy.complexfloating):
# if complex roots are all complex conjugates, the roots are real.
roots = numpy.asarray(p, complex)
pos_roots = numpy.compress(roots.imag > 0, roots)
neg_roots = numpy.conjugate(numpy.compress(roots.imag < 0, roots))
if len(pos_roots) == len(neg_roots):
if numpy.all(numpy.sort_complex(neg_roots) ==
numpy.sort_complex(pos_roots)):
a = a.real.copy()
return b, a
def tf2sos(b, a, pairing='nearest'):
"""
Return second-order sections from transfer function representation
Parameters
----------
b : array_like
Numerator polynomial coefficients.
a : array_like
Denominator polynomial coefficients.
pairing : {'nearest', 'keep_odd'}, optional
The method to use to combine pairs of poles and zeros into sections.
See `zpk2sos`.
Returns
-------
sos : ndarray
Array of second-order filter coefficients, with shape
``(n_sections, 6)``. See `sosfilt` for the SOS filter format
specification.
See Also
--------
zpk2sos, sosfilt
Notes
-----
It is generally discouraged to convert from TF to SOS format, since doing
so usually will not improve numerical precision errors. Instead, consider
designing filters in ZPK format and converting directly to SOS. TF is
converted to SOS by first converting to ZPK format, then converting
ZPK to SOS.
.. versionadded:: 0.16.0
"""
return zpk2sos(*tf2zpk(b, a), pairing=pairing)
def sos2tf(sos):
"""
Return a single transfer function from a series of second-order sections
Parameters
----------
sos : array_like
Array of second-order filter coefficients, must have shape
``(n_sections, 6)``. See `sosfilt` for the SOS filter format
specification.
Returns
-------
b : ndarray
Numerator polynomial coefficients.
a : ndarray
Denominator polynomial coefficients.
Notes
-----
.. versionadded:: 0.16.0
"""
sos = np.asarray(sos)
result_type = sos.dtype
if result_type.kind in 'bui':
result_type = np.float64
b = np.array([1], dtype=result_type)
a = np.array([1], dtype=result_type)
n_sections = sos.shape[0]
for section in range(n_sections):
b = np.polymul(b, sos[section, :3])
a = np.polymul(a, sos[section, 3:])
return b, a
def sos2zpk(sos):
"""
Return zeros, poles, and gain of a series of second-order sections
Parameters
----------
sos : array_like
Array of second-order filter coefficients, must have shape
``(n_sections, 6)``. See `sosfilt` for the SOS filter format
specification.
Returns
-------
z : ndarray
Zeros of the transfer function.
p : ndarray
Poles of the transfer function.
k : float
System gain.
Notes
-----
The number of zeros and poles returned will be ``n_sections * 2``
even if some of these are (effectively) zero.
.. versionadded:: 0.16.0
"""
sos = np.asarray(sos)
n_sections = sos.shape[0]
z = np.zeros(n_sections*2, np.complex128)
p = np.zeros(n_sections*2, np.complex128)
k = 1.
for section in range(n_sections):
zpk = tf2zpk(sos[section, :3], sos[section, 3:])
z[2*section:2*section+len(zpk[0])] = zpk[0]
p[2*section:2*section+len(zpk[1])] = zpk[1]
k *= zpk[2]
return z, p, k
def _nearest_real_complex_idx(fro, to, which):
"""Get the next closest real or complex element based on distance"""
assert which in ('real', 'complex')
order = np.argsort(np.abs(fro - to))
mask = np.isreal(fro[order])
if which == 'complex':
mask = ~mask
return order[np.nonzero(mask)[0][0]]
def zpk2sos(z, p, k, pairing='nearest'):
"""
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, 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.
"""
# TODO in the near future:
# 1. Add SOS capability to `filtfilt`, `freqz`, etc. somehow (#3259).
# 2. Make `decimate` use `sosfilt` instead of `lfilter`.
# 3. Make sosfilt automatically simplify sections to first order
# when possible. Note this might make `sosfiltfilt` a bit harder (ICs).
# 4. Further optimizations of the section ordering / pole-zero pairing.
# See the wiki for other potential issues.
valid_pairings = ['nearest', 'keep_odd']
if pairing not in valid_pairings:
raise ValueError('pairing must be one of %s, not %s'
% (valid_pairings, pairing))
if len(z) == len(p) == 0:
return array([[k, 0., 0., 1., 0., 0.]])
# ensure we have the same number of poles and zeros, and make copies
p = np.concatenate((p, np.zeros(max(len(z) - len(p), 0))))
z = np.concatenate((z, np.zeros(max(len(p) - len(z), 0))))
n_sections = (max(len(p), len(z)) + 1) // 2
sos = zeros((n_sections, 6))
if len(p) % 2 == 1 and pairing == 'nearest':
p = np.concatenate((p, [0.]))
z = np.concatenate((z, [0.]))
assert len(p) == len(z)
# Ensure we have complex conjugate pairs
# (note that _cplxreal only gives us one element of each complex pair):
z = np.concatenate(_cplxreal(z))
p = np.concatenate(_cplxreal(p))
p_sos = np.zeros((n_sections, 2), np.complex128)
z_sos = np.zeros_like(p_sos)
for si in range(n_sections):
# Select the next "worst" pole
p1_idx = np.argmin(np.abs(1 - np.abs(p)))
p1 = p[p1_idx]
p = np.delete(p, p1_idx)
# Pair that pole with a zero
if np.isreal(p1) and np.isreal(p).sum() == 0:
# Special case to set a first-order section
z1_idx = _nearest_real_complex_idx(z, p1, 'real')
z1 = z[z1_idx]
z = np.delete(z, z1_idx)
p2 = z2 = 0
else:
if not np.isreal(p1) and np.isreal(z).sum() == 1:
# Special case to ensure we choose a complex zero to pair
# with so later (setting up a first-order section)
z1_idx = _nearest_real_complex_idx(z, p1, 'complex')
assert not np.isreal(z[z1_idx])
else:
# Pair the pole with the closest zero (real or complex)
z1_idx = np.argmin(np.abs(p1 - z))
z1 = z[z1_idx]
z = np.delete(z, z1_idx)
# Now that we have p1 and z1, figure out what p2 and z2 need to be
if not np.isreal(p1):
if not np.isreal(z1): # complex pole, complex zero
p2 = p1.conj()
z2 = z1.conj()
else: # complex pole, real zero
p2 = p1.conj()
z2_idx = _nearest_real_complex_idx(z, p1, 'real')
z2 = z[z2_idx]
assert np.isreal(z2)
z = np.delete(z, z2_idx)
else:
if not np.isreal(z1): # real pole, complex zero
z2 = z1.conj()
p2_idx = _nearest_real_complex_idx(p, z1, 'real')
p2 = p[p2_idx]
assert np.isreal(p2)
else: # real pole, real zero
# pick the next "worst" pole to use
idx = np.nonzero(np.isreal(p))[0]
assert len(idx) > 0
p2_idx = idx[np.argmin(np.abs(np.abs(p[idx]) - 1))]
p2 = p[p2_idx]
# find a real zero to match the added pole
assert np.isreal(p2)
z2_idx = _nearest_real_complex_idx(z, p2, 'real')
z2 = z[z2_idx]
assert np.isreal(z2)
z = np.delete(z, z2_idx)
p = np.delete(p, p2_idx)
p_sos[si] = [p1, p2]
z_sos[si] = [z1, z2]
assert len(p) == len(z) == 0 # we've consumed all poles and zeros
del p, z
# Construct the system, reversing order so the "worst" are last
p_sos = np.reshape(p_sos[::-1], (n_sections, 2))
z_sos = np.reshape(z_sos[::-1], (n_sections, 2))
gains = np.ones(n_sections, np.array(k).dtype)
gains[0] = k
for si in range(n_sections):
x = zpk2tf(z_sos[si], p_sos[si], gains[si])
sos[si] = np.concatenate(x)
return sos
def _align_nums(nums):
"""Aligns the shapes of multiple numerators.
Given an array of numerator coefficient arrays [[a_1, a_2,...,
a_n],..., [b_1, b_2,..., b_m]], this function pads shorter numerator
arrays with zero's so that all numerators have the same length. Such
alignment is necessary for functions like 'tf2ss', which needs the
alignment when dealing with SIMO transfer functions.
Parameters
----------
nums: array_like
Numerator or list of numerators. Not necessarily with same length.
Returns
-------
nums: array
The numerator. If `nums` input was a list of numerators then a 2-D
array with padded zeros for shorter numerators is returned. Otherwise
returns ``np.asarray(nums)``.
"""
try:
# The statement can throw a ValueError if one
# of the numerators is a single digit and another
# is array-like e.g. if nums = [5, [1, 2, 3]]
nums = asarray(nums)
if not np.issubdtype(nums.dtype, np.number):
raise ValueError("dtype of numerator is non-numeric")
return nums
except ValueError:
nums = [np.atleast_1d(num) for num in nums]
max_width = max(num.size for num in nums)
# pre-allocate
aligned_nums = np.zeros((len(nums), max_width))
# Create numerators with padded zeros
for index, num in enumerate(nums):
aligned_nums[index, -num.size:] = num
return aligned_nums
def normalize(b, a):
"""Normalize numerator/denominator of a continuous-time transfer function.
If values of `b` are too close to 0, they are removed. In that case, a
BadCoefficients warning is emitted.
Parameters
----------
b: array_like
Numerator of the transfer function. Can be a 2-D array to normalize
multiple transfer functions.
a: array_like
Denominator of the transfer function. At most 1-D.
Returns
-------
num: array
The numerator of the normalized transfer function. At least a 1-D
array. A 2-D array if the input `num` is a 2-D array.
den: 1-D array
The denominator of the normalized transfer function.
Notes
-----
Coefficients for both the numerator and denominator should be specified in
descending exponent order (e.g., ``s^2 + 3s + 5`` would be represented as
``[1, 3, 5]``).
"""
num, den = b, a
den = np.atleast_1d(den)
num = np.atleast_2d(_align_nums(num))
if den.ndim != 1:
raise ValueError("Denominator polynomial must be rank-1 array.")
if num.ndim > 2:
raise ValueError("Numerator polynomial must be rank-1 or"
" rank-2 array.")
if np.all(den == 0):
raise ValueError("Denominator must have at least on nonzero element.")
# Trim leading zeros in denominator, leave at least one.
den = np.trim_zeros(den, 'f')
# Normalize transfer function
num, den = num / den[0], den / den[0]
# Count numerator columns that are all zero
leading_zeros = 0
for col in num.T:
if np.allclose(col, 0, atol=1e-14):
leading_zeros += 1
else:
break
# Trim leading zeros of numerator
if leading_zeros > 0:
warnings.warn("Badly conditioned filter coefficients (numerator): the "
"results may be meaningless", BadCoefficients)
# Make sure at least one column remains
if leading_zeros == num.shape[1]:
leading_zeros -= 1
num = num[:, leading_zeros:]
# Squeeze first dimension if singular
if num.shape[0] == 1:
num = num[0, :]
return num, den
def lp2lp(b, a, wo=1.0):
r"""
Transform a lowpass filter prototype to a different frequency.
Return an analog low-pass filter with cutoff frequency `wo`
from an analog low-pass filter prototype with unity cutoff frequency, in
transfer function ('ba') representation.
Parameters
----------
b : array_like
Numerator polynomial coefficients.
a : array_like
Denominator polynomial coefficients.
wo : float
Desired cutoff, as angular frequency (e.g. rad/s).
Defaults to no change.
Returns
-------
b : array_like
Numerator polynomial coefficients of the transformed low-pass filter.
a : array_like
Denominator polynomial coefficients of the transformed low-pass filter.
See Also
--------
lp2hp, lp2bp, lp2bs, bilinear
lp2lp_zpk
Notes
-----
This is derived from the s-plane substitution
.. math:: s \rightarrow \frac{s}{\omega_0}
Examples
--------
>>> from scipy import signal
>>> import matplotlib.pyplot as plt
>>> lp = signal.lti([1.0], [1.0, 1.0])
>>> lp2 = signal.lti(*signal.lp2lp(lp.num, lp.den, 2))
>>> w, mag_lp, p_lp = lp.bode()
>>> w, mag_lp2, p_lp2 = lp2.bode(w)
>>> plt.plot(w, mag_lp, label='Lowpass')
>>> plt.plot(w, mag_lp2, label='Transformed Lowpass')
>>> plt.semilogx()
>>> plt.grid()
>>> plt.xlabel('Frequency [rad/s]')
>>> plt.ylabel('Magnitude [dB]')
>>> plt.legend()
"""
a, b = map(atleast_1d, (a, b))
try:
wo = float(wo)
except TypeError:
wo = float(wo[0])
d = len(a)
n = len(b)
M = max((d, n))
pwo = pow(wo, numpy.arange(M - 1, -1, -1))
start1 = max((n - d, 0))
start2 = max((d - n, 0))
b = b * pwo[start1] / pwo[start2:]
a = a * pwo[start1] / pwo[start1:]
return normalize(b, a)
def lp2hp(b, a, wo=1.0):
r"""
Transform a lowpass filter prototype to a highpass filter.
Return an analog high-pass filter with cutoff frequency `wo`
from an analog low-pass filter prototype with unity cutoff frequency, in
transfer function ('ba') representation.
Parameters
----------
b : array_like
Numerator polynomial coefficients.
a : array_like
Denominator polynomial coefficients.
wo : float
Desired cutoff, as angular frequency (e.g., rad/s).
Defaults to no change.
Returns
-------
b : array_like
Numerator polynomial coefficients of the transformed high-pass filter.
a : array_like
Denominator polynomial coefficients of the transformed high-pass filter.
See Also
--------
lp2lp, lp2bp, lp2bs, bilinear
lp2hp_zpk
Notes
-----
This is derived from the s-plane substitution
.. math:: s \rightarrow \frac{\omega_0}{s}
This maintains symmetry of the lowpass and highpass responses on a
logarithmic scale.
Examples
--------
>>> from scipy import signal
>>> import matplotlib.pyplot as plt
>>> lp = signal.lti([1.0], [1.0, 1.0])
>>> hp = signal.lti(*signal.lp2hp(lp.num, lp.den))
>>> w, mag_lp, p_lp = lp.bode()
>>> w, mag_hp, p_hp = hp.bode(w)
>>> plt.plot(w, mag_lp, label='Lowpass')
>>> plt.plot(w, mag_hp, label='Highpass')
>>> plt.semilogx()
>>> plt.grid()
>>> plt.xlabel('Frequency [rad/s]')
>>> plt.ylabel('Magnitude [dB]')
>>> plt.legend()
"""
a, b = map(atleast_1d, (a, b))
try:
wo = float(wo)
except TypeError:
wo = float(wo[0])
d = len(a)
n = len(b)
if wo != 1:
pwo = pow(wo, numpy.arange(max((d, n))))
else:
pwo = numpy.ones(max((d, n)), b.dtype.char)
if d >= n:
outa = a[::-1] * pwo
outb = resize(b, (d,))
outb[n:] = 0.0
outb[:n] = b[::-1] * pwo[:n]
else:
outb = b[::-1] * pwo
outa = resize(a, (n,))
outa[d:] = 0.0
outa[:d] = a[::-1] * pwo[:d]
return normalize(outb, outa)
def lp2bp(b, a, wo=1.0, bw=1.0):
r"""
Transform a lowpass filter prototype to a bandpass filter.
Return an analog band-pass filter with center frequency `wo` and
bandwidth `bw` from an analog low-pass filter prototype with unity
cutoff frequency, in transfer function ('ba') representation.
Parameters
----------
b : array_like
Numerator polynomial coefficients.
a : array_like
Denominator polynomial coefficients.
wo : float
Desired passband center, as angular frequency (e.g., rad/s).
Defaults to no change.
bw : float
Desired passband width, as angular frequency (e.g., rad/s).
Defaults to 1.
Returns
-------
b : array_like
Numerator polynomial coefficients of the transformed band-pass filter.
a : array_like
Denominator polynomial coefficients of the transformed band-pass filter.
See Also
--------
lp2lp, lp2hp, lp2bs, bilinear
lp2bp_zpk
Notes
-----
This is derived from the s-plane substitution
.. math:: s \rightarrow \frac{s^2 + {\omega_0}^2}{s \cdot \mathrm{BW}}
This is the "wideband" transformation, producing a passband with
geometric (log frequency) symmetry about `wo`.
Examples
--------
>>> from scipy import signal
>>> import matplotlib.pyplot as plt
>>> lp = signal.lti([1.0], [1.0, 1.0])
>>> bp = signal.lti(*signal.lp2bp(lp.num, lp.den))
>>> w, mag_lp, p_lp = lp.bode()
>>> w, mag_bp, p_bp = bp.bode(w)
>>> plt.plot(w, mag_lp, label='Lowpass')
>>> plt.plot(w, mag_bp, label='Bandpass')
>>> plt.semilogx()
>>> plt.grid()
>>> plt.xlabel('Frequency [rad/s]')
>>> plt.ylabel('Magnitude [dB]')
>>> plt.legend()
"""
a, b = map(atleast_1d, (a, b))
D = len(a) - 1
N = len(b) - 1
artype = mintypecode((a, b))
ma = max([N, D])
Np = N + ma
Dp = D + ma
bprime = numpy.empty(Np + 1, artype)
aprime = numpy.empty(Dp + 1, artype)
wosq = wo * wo
for j in range(Np + 1):
val = 0.0
for i in range(0, N + 1):
for k in range(0, i + 1):
if ma - i + 2 * k == j:
val += comb(i, k) * b[N - i] * (wosq) ** (i - k) / bw ** i
bprime[Np - j] = val
for j in range(Dp + 1):
val = 0.0
for i in range(0, D + 1):
for k in range(0, i + 1):
if ma - i + 2 * k == j:
val += comb(i, k) * a[D - i] * (wosq) ** (i - k) / bw ** i
aprime[Dp - j] = val
return normalize(bprime, aprime)
def lp2bs(b, a, wo=1.0, bw=1.0):
r"""
Transform a lowpass filter prototype to a bandstop filter.
Return an analog band-stop filter with center frequency `wo` and
bandwidth `bw` from an analog low-pass filter prototype with unity
cutoff frequency, in transfer function ('ba') representation.
Parameters
----------
b : array_like
Numerator polynomial coefficients.
a : array_like
Denominator polynomial coefficients.
wo : float
Desired stopband center, as angular frequency (e.g., rad/s).
Defaults to no change.
bw : float
Desired stopband width, as angular frequency (e.g., rad/s).
Defaults to 1.
Returns
-------
b : array_like
Numerator polynomial coefficients of the transformed band-stop filter.
a : array_like
Denominator polynomial coefficients of the transformed band-stop filter.
See Also
--------
lp2lp, lp2hp, lp2bp, bilinear
lp2bs_zpk
Notes
-----
This is derived from the s-plane substitution
.. math:: s \rightarrow \frac{s \cdot \mathrm{BW}}{s^2 + {\omega_0}^2}
This is the "wideband" transformation, producing a stopband with
geometric (log frequency) symmetry about `wo`.
Examples
--------
>>> from scipy import signal
>>> import matplotlib.pyplot as plt
>>> lp = signal.lti([1.0], [1.0, 1.5])
>>> bs = signal.lti(*signal.lp2bs(lp.num, lp.den))
>>> w, mag_lp, p_lp = lp.bode()
>>> w, mag_bs, p_bs = bs.bode(w)
>>> plt.plot(w, mag_lp, label='Lowpass')
>>> plt.plot(w, mag_bs, label='Bandstop')
>>> plt.semilogx()
>>> plt.grid()
>>> plt.xlabel('Frequency [rad/s]')
>>> plt.ylabel('Magnitude [dB]')
>>> plt.legend()
"""
a, b = map(atleast_1d, (a, b))
D = len(a) - 1
N = len(b) - 1
artype = mintypecode((a, b))
M = max([N, D])
Np = M + M
Dp = M + M
bprime = numpy.empty(Np + 1, artype)
aprime = numpy.empty(Dp + 1, artype)
wosq = wo * wo
for j in range(Np + 1):
val = 0.0
for i in range(0, N + 1):
for k in range(0, M - i + 1):
if i + 2 * k == j:
val += (comb(M - i, k) * b[N - i] *
(wosq) ** (M - i - k) * bw ** i)
bprime[Np - j] = val
for j in range(Dp + 1):
val = 0.0
for i in range(0, D + 1):
for k in range(0, M - i + 1):
if i + 2 * k == j:
val += (comb(M - i, k) * a[D - i] *
(wosq) ** (M - i - k) * bw ** i)
aprime[Dp - j] = val
return normalize(bprime, aprime)
def bilinear(b, a, fs=1.0):
r"""
Return a digital IIR filter from an analog one using a bilinear transform.
Transform a set of poles and zeros from the analog s-plane to the digital
z-plane using Tustin's method, which substitutes ``(z-1) / (z+1)`` for
``s``, maintaining the shape of the frequency response.
Parameters
----------
b : array_like
Numerator of the analog filter transfer function.
a : array_like
Denominator of the analog filter transfer function.
fs : float
Sample rate, as ordinary frequency (e.g., hertz). No prewarping is
done in this function.
Returns
-------
z : ndarray
Numerator of the transformed digital filter transfer function.
p : ndarray
Denominator of the transformed digital filter transfer function.
See Also
--------
lp2lp, lp2hp, lp2bp, lp2bs
bilinear_zpk
Examples
--------
>>> from scipy import signal
>>> import matplotlib.pyplot as plt
>>> fs = 100
>>> bf = 2 * np.pi * np.array([7, 13])
>>> filts = signal.lti(*signal.butter(4, bf, btype='bandpass',
... analog=True))
>>> filtz = signal.lti(*signal.bilinear(filts.num, filts.den, fs))
>>> wz, hz = signal.freqz(filtz.num, filtz.den)
>>> ws, hs = signal.freqs(filts.num, filts.den, worN=fs*wz)
>>> plt.semilogx(wz*fs/(2*np.pi), 20*np.log10(np.abs(hz).clip(1e-15)),
... label=r'$|H_z(e^{j \omega})|$')
>>> plt.semilogx(wz*fs/(2*np.pi), 20*np.log10(np.abs(hs).clip(1e-15)),
... label=r'$|H(j \omega)|$')
>>> plt.legend()
>>> plt.xlabel('Frequency [Hz]')
>>> plt.ylabel('Magnitude [dB]')
>>> plt.grid()
"""
fs = float(fs)
a, b = map(atleast_1d, (a, b))
D = len(a) - 1
N = len(b) - 1
artype = float
M = max([N, D])
Np = M
Dp = M
bprime = numpy.empty(Np + 1, artype)
aprime = numpy.empty(Dp + 1, artype)
for j in range(Np + 1):
val = 0.0
for i in range(N + 1):
for k in range(i + 1):
for l in range(M - i + 1):
if k + l == j:
val += (comb(i, k) * comb(M - i, l) * b[N - i] *
pow(2 * fs, i) * (-1) ** k)
bprime[j] = real(val)
for j in range(Dp + 1):
val = 0.0
for i in range(D + 1):
for k in range(i + 1):
for l in range(M - i + 1):
if k + l == j:
val += (comb(i, k) * comb(M - i, l) * a[D - i] *
pow(2 * fs, i) * (-1) ** k)
aprime[j] = real(val)
return normalize(bprime, aprime)
def _validate_gpass_gstop(gpass, gstop):
if gpass <= 0.0:
raise ValueError("gpass should be larger than 0.0")
elif gstop <= 0.0:
raise ValueError("gstop should be larger than 0.0")
elif gpass > gstop:
raise ValueError("gpass should be smaller than gstop")
def iirdesign(wp, ws, gpass, gstop, analog=False, ftype='ellip', output='ba',
fs=None):
"""Complete IIR digital and analog filter design.
Given passband and stopband frequencies and gains, construct an analog or
digital IIR filter of minimum order for a given basic type. Return the
output in numerator, denominator ('ba'), pole-zero ('zpk') or second order
sections ('sos') form.
Parameters
----------
wp, ws : float or array like, shape (2,)
Passband and stopband edge frequencies. Possible values are scalars
(for lowpass and highpass filters) or ranges (for bandpass and bandstop
filters).
For digital filters, these are in the same units as `fs`. By default,
`fs` is 2 half-cycles/sample, so these are normalized from 0 to 1,
where 1 is the Nyquist frequency. For example:
- Lowpass: wp = 0.2, ws = 0.3
- Highpass: wp = 0.3, ws = 0.2
- Bandpass: wp = [0.2, 0.5], ws = [0.1, 0.6]
- Bandstop: wp = [0.1, 0.6], ws = [0.2, 0.5]
For analog filters, `wp` and `ws` are angular frequencies (e.g., rad/s).
Note, that for bandpass and bandstop filters passband must lie strictly
inside stopband or vice versa.
gpass : float
The maximum loss in the passband (dB).
gstop : float
The minimum attenuation in the stopband (dB).
analog : bool, optional
When True, return an analog filter, otherwise a digital filter is
returned.
ftype : str, optional
The type of IIR filter to design:
- Butterworth : 'butter'
- Chebyshev I : 'cheby1'
- Chebyshev II : 'cheby2'
- Cauer/elliptic: 'ellip'
- Bessel/Thomson: 'bessel'
output : {'ba', 'zpk', 'sos'}, optional
Filter form of the output:
- second-order sections (recommended): 'sos'
- numerator/denominator (default) : 'ba'
- pole-zero : 'zpk'
In general the second-order sections ('sos') form is
recommended because inferring the coefficients for the
numerator/denominator form ('ba') suffers from numerical
instabilities. For reasons of backward compatibility the default
form is the numerator/denominator form ('ba'), where the 'b'
and the 'a' in 'ba' refer to the commonly used names of the
coefficients used.
Note: Using the second-order sections form ('sos') is sometimes
associated with additional computational costs: for
data-intense use cases it is therefore recommended to also
investigate the numerator/denominator form ('ba').
fs : float, optional
The sampling frequency of the digital system.
.. versionadded:: 1.2.0
Returns
-------
b, a : ndarray, ndarray
Numerator (`b`) and denominator (`a`) polynomials of the IIR filter.
Only returned if ``output='ba'``.
z, p, k : ndarray, ndarray, float
Zeros, poles, and system gain of the IIR filter transfer
function. Only returned if ``output='zpk'``.
sos : ndarray
Second-order sections representation of the IIR filter.
Only returned if ``output=='sos'``.
See Also
--------
butter : Filter design using order and critical points
cheby1, cheby2, ellip, bessel
buttord : Find order and critical points from passband and stopband spec
cheb1ord, cheb2ord, ellipord
iirfilter : General filter design using order and critical frequencies
Notes
-----
The ``'sos'`` output parameter was added in 0.16.0.
Examples
--------
>>> from scipy import signal
>>> import matplotlib.pyplot as plt
>>> import matplotlib.ticker
>>> wp = 0.2
>>> ws = 0.3
>>> gpass = 1
>>> gstop = 40
>>> system = signal.iirdesign(wp, ws, gpass, gstop)
>>> w, h = signal.freqz(*system)
>>> fig, ax1 = plt.subplots()
>>> ax1.set_title('Digital filter frequency response')
>>> ax1.plot(w, 20 * np.log10(abs(h)), 'b')
>>> ax1.set_ylabel('Amplitude [dB]', color='b')
>>> ax1.set_xlabel('Frequency [rad/sample]')
>>> ax1.grid()
>>> ax1.set_ylim([-120, 20])
>>> ax2 = ax1.twinx()
>>> angles = np.unwrap(np.angle(h))
>>> ax2.plot(w, angles, 'g')
>>> ax2.set_ylabel('Angle (radians)', color='g')
>>> ax2.grid()
>>> ax2.axis('tight')
>>> ax2.set_ylim([-6, 1])
>>> nticks = 8
>>> ax1.yaxis.set_major_locator(matplotlib.ticker.LinearLocator(nticks))
>>> ax2.yaxis.set_major_locator(matplotlib.ticker.LinearLocator(nticks))
"""
try:
ordfunc = filter_dict[ftype][1]
except KeyError as e:
raise ValueError("Invalid IIR filter type: %s" % ftype) from e
except IndexError as e:
raise ValueError(("%s does not have order selection. Use "
"iirfilter function.") % ftype) from e
_validate_gpass_gstop(gpass, gstop)
wp = atleast_1d(wp)
ws = atleast_1d(ws)
if wp.shape[0] != ws.shape[0] or wp.shape not in [(1,), (2,)]:
raise ValueError("wp and ws must have one or two elements each, and"
"the same shape, got %s and %s"
% (wp.shape, ws.shape))
if wp.shape[0] == 2:
if wp[0] < 0 or ws[0] < 0:
raise ValueError("Values for wp, ws can't be negative")
elif 1 < wp[1] or 1 < ws[1]:
raise ValueError("Values for wp, ws can't be larger than 1")
elif not((ws[0] < wp[0] and wp[1] < ws[1]) or
(wp[0] < ws[0] and ws[1] < wp[1])):
raise ValueError("Passband must lie strictly inside stopband"
" or vice versa")
band_type = 2 * (len(wp) - 1)
band_type += 1
if wp[0] >= ws[0]:
band_type += 1
btype = {1: 'lowpass', 2: 'highpass',
3: 'bandstop', 4: 'bandpass'}[band_type]
N, Wn = ordfunc(wp, ws, gpass, gstop, analog=analog, fs=fs)
return iirfilter(N, Wn, rp=gpass, rs=gstop, analog=analog, btype=btype,
ftype=ftype, output=output, fs=fs)
def iirfilter(N, Wn, rp=None, rs=None, btype='band', analog=False,
ftype='butter', output='ba', fs=None):
"""
IIR digital and analog filter design given order and critical points.
Design an Nth-order digital or analog filter and return the filter
coefficients.
Parameters
----------
N : int
The order of the filter.
Wn : array_like
A scalar or length-2 sequence giving the critical frequencies.
For digital filters, `Wn` are in the same units as `fs`. By default,
`fs` is 2 half-cycles/sample, so these are normalized from 0 to 1,
where 1 is the Nyquist frequency. (`Wn` is thus in
half-cycles / sample.)
For analog filters, `Wn` is an angular frequency (e.g., rad/s).
rp : float, optional
For Chebyshev and elliptic filters, provides the maximum ripple
in the passband. (dB)
rs : float, optional
For Chebyshev and elliptic filters, provides the minimum attenuation
in the stop band. (dB)
btype : {'bandpass', 'lowpass', 'highpass', 'bandstop'}, optional
The type of filter. Default is 'bandpass'.
analog : bool, optional
When True, return an analog filter, otherwise a digital filter is
returned.
ftype : str, optional
The type of IIR filter to design:
- Butterworth : 'butter'
- Chebyshev I : 'cheby1'
- Chebyshev II : 'cheby2'
- Cauer/elliptic: 'ellip'
- Bessel/Thomson: 'bessel'
output : {'ba', 'zpk', 'sos'}, optional
Filter form of the output:
- second-order sections (recommended): 'sos'
- numerator/denominator (default) : 'ba'
- pole-zero : 'zpk'
In general the second-order sections ('sos') form is
recommended because inferring the coefficients for the
numerator/denominator form ('ba') suffers from numerical
instabilities. For reasons of backward compatibility the default
form is the numerator/denominator form ('ba'), where the 'b'
and the 'a' in 'ba' refer to the commonly used names of the
coefficients used.
Note: Using the second-order sections form ('sos') is sometimes
associated with additional computational costs: for
data-intense use cases it is therefore recommended to also
investigate the numerator/denominator form ('ba').
fs : float, optional
The sampling frequency of the digital system.
.. versionadded:: 1.2.0
Returns
-------
b, a : ndarray, ndarray
Numerator (`b`) and denominator (`a`) polynomials of the IIR filter.
Only returned if ``output='ba'``.
z, p, k : ndarray, ndarray, float
Zeros, poles, and system gain of the IIR filter transfer
function. Only returned if ``output='zpk'``.
sos : ndarray
Second-order sections representation of the IIR filter.
Only returned if ``output=='sos'``.
See Also
--------
butter : Filter design using order and critical points
cheby1, cheby2, ellip, bessel
buttord : Find order and critical points from passband and stopband spec
cheb1ord, cheb2ord, ellipord
iirdesign : General filter design using passband and stopband spec
Notes
-----
The ``'sos'`` output parameter was added in 0.16.0.
Examples
--------
Generate a 17th-order Chebyshev II analog bandpass filter from 50 Hz to
200 Hz and plot the frequency response:
>>> from scipy import signal
>>> import matplotlib.pyplot as plt
>>> b, a = signal.iirfilter(17, [2*np.pi*50, 2*np.pi*200], rs=60,
... btype='band', analog=True, ftype='cheby2')
>>> w, h = signal.freqs(b, a, 1000)
>>> fig = plt.figure()
>>> ax = fig.add_subplot(1, 1, 1)
>>> ax.semilogx(w / (2*np.pi), 20 * np.log10(np.maximum(abs(h), 1e-5)))
>>> ax.set_title('Chebyshev Type II bandpass frequency response')
>>> ax.set_xlabel('Frequency [Hz]')
>>> ax.set_ylabel('Amplitude [dB]')
>>> ax.axis((10, 1000, -100, 10))
>>> ax.grid(which='both', axis='both')
>>> plt.show()
Create a digital filter with the same properties, in a system with
sampling rate of 2000 Hz, and plot the frequency response. (Second-order
sections implementation is required to ensure stability of a filter of
this order):
>>> sos = signal.iirfilter(17, [50, 200], rs=60, btype='band',
... analog=False, ftype='cheby2', fs=2000,
... output='sos')
>>> w, h = signal.sosfreqz(sos, 2000, fs=2000)
>>> fig = plt.figure()
>>> ax = fig.add_subplot(1, 1, 1)
>>> ax.semilogx(w, 20 * np.log10(np.maximum(abs(h), 1e-5)))
>>> ax.set_title('Chebyshev Type II bandpass frequency response')
>>> ax.set_xlabel('Frequency [Hz]')
>>> ax.set_ylabel('Amplitude [dB]')
>>> ax.axis((10, 1000, -100, 10))
>>> ax.grid(which='both', axis='both')
>>> plt.show()
"""
ftype, btype, output = [x.lower() for x in (ftype, btype, output)]
Wn = asarray(Wn)
if fs is not None:
if analog:
raise ValueError("fs cannot be specified for an analog filter")
Wn = 2*Wn/fs
try:
btype = band_dict[btype]
except KeyError as e:
raise ValueError("'%s' is an invalid bandtype for filter." % btype) from e
try:
typefunc = filter_dict[ftype][0]
except KeyError as e:
raise ValueError("'%s' is not a valid basic IIR filter." % ftype) from e
if output not in ['ba', 'zpk', 'sos']:
raise ValueError("'%s' is not a valid output form." % output)
if rp is not None and rp < 0:
raise ValueError("passband ripple (rp) must be positive")
if rs is not None and rs < 0:
raise ValueError("stopband attenuation (rs) must be positive")
# Get analog lowpass prototype
if typefunc == buttap:
z, p, k = typefunc(N)
elif typefunc == besselap:
z, p, k = typefunc(N, norm=bessel_norms[ftype])
elif typefunc == cheb1ap:
if rp is None:
raise ValueError("passband ripple (rp) must be provided to "
"design a Chebyshev I filter.")
z, p, k = typefunc(N, rp)
elif typefunc == cheb2ap:
if rs is None:
raise ValueError("stopband attenuation (rs) must be provided to "
"design an Chebyshev II filter.")
z, p, k = typefunc(N, rs)
elif typefunc == ellipap:
if rs is None or rp is None:
raise ValueError("Both rp and rs must be provided to design an "
"elliptic filter.")
z, p, k = typefunc(N, rp, rs)
else:
raise NotImplementedError("'%s' not implemented in iirfilter." % ftype)
# Pre-warp frequencies for digital filter design
if not analog:
if numpy.any(Wn <= 0) or numpy.any(Wn >= 1):
if fs is not None:
raise ValueError("Digital filter critical frequencies "
"must be 0 < Wn < fs/2 (fs={} -> fs/2={})".format(fs, fs/2))
raise ValueError("Digital filter critical frequencies "
"must be 0 < Wn < 1")
fs = 2.0
warped = 2 * fs * tan(pi * Wn / fs)
else:
warped = Wn
# transform to lowpass, bandpass, highpass, or bandstop
if btype in ('lowpass', 'highpass'):
if numpy.size(Wn) != 1:
raise ValueError('Must specify a single critical frequency Wn for lowpass or highpass filter')
if btype == 'lowpass':
z, p, k = lp2lp_zpk(z, p, k, wo=warped)
elif btype == 'highpass':
z, p, k = lp2hp_zpk(z, p, k, wo=warped)
elif btype in ('bandpass', 'bandstop'):
try:
bw = warped[1] - warped[0]
wo = sqrt(warped[0] * warped[1])
except IndexError as e:
raise ValueError('Wn must specify start and stop frequencies for bandpass or bandstop '
'filter') from e
if btype == 'bandpass':
z, p, k = lp2bp_zpk(z, p, k, wo=wo, bw=bw)
elif btype == 'bandstop':
z, p, k = lp2bs_zpk(z, p, k, wo=wo, bw=bw)
else:
raise NotImplementedError("'%s' not implemented in iirfilter." % btype)
# Find discrete equivalent if necessary
if not analog:
z, p, k = bilinear_zpk(z, p, k, fs=fs)
# Transform to proper out type (pole-zero, state-space, numer-denom)
if output == 'zpk':
return z, p, k
elif output == 'ba':
return zpk2tf(z, p, k)
elif output == 'sos':
return zpk2sos(z, p, k)
def _relative_degree(z, p):
"""
Return relative degree of transfer function from zeros and poles
"""
degree = len(p) - len(z)
if degree < 0:
raise ValueError("Improper transfer function. "
"Must have at least as many poles as zeros.")
else:
return degree
def bilinear_zpk(z, p, k, fs):
r"""
Return a digital IIR filter from an analog one using a bilinear transform.
Transform a set of poles and zeros from the analog s-plane to the digital
z-plane using Tustin's method, which substitutes ``(z-1) / (z+1)`` for
``s``, maintaining the shape of the frequency response.
Parameters
----------
z : array_like
Zeros of the analog filter transfer function.
p : array_like
Poles of the analog filter transfer function.
k : float
System gain of the analog filter transfer function.
fs : float
Sample rate, as ordinary frequency (e.g., hertz). No prewarping is
done in this function.
Returns
-------
z : ndarray
Zeros of the transformed digital filter transfer function.
p : ndarray
Poles of the transformed digital filter transfer function.
k : float
System gain of the transformed digital filter.
See Also
--------
lp2lp_zpk, lp2hp_zpk, lp2bp_zpk, lp2bs_zpk
bilinear
Notes
-----
.. versionadded:: 1.1.0
Examples
--------
>>> from scipy import signal
>>> import matplotlib.pyplot as plt
>>> fs = 100
>>> bf = 2 * np.pi * np.array([7, 13])
>>> filts = signal.lti(*signal.butter(4, bf, btype='bandpass', analog=True,
... output='zpk'))
>>> filtz = signal.lti(*signal.bilinear_zpk(filts.zeros, filts.poles,
... filts.gain, fs))
>>> wz, hz = signal.freqz_zpk(filtz.zeros, filtz.poles, filtz.gain)
>>> ws, hs = signal.freqs_zpk(filts.zeros, filts.poles, filts.gain,
... worN=fs*wz)
>>> plt.semilogx(wz*fs/(2*np.pi), 20*np.log10(np.abs(hz).clip(1e-15)),
... label=r'$|H_z(e^{j \omega})|$')
>>> plt.semilogx(wz*fs/(2*np.pi), 20*np.log10(np.abs(hs).clip(1e-15)),
... label=r'$|H(j \omega)|$')
>>> plt.legend()
>>> plt.xlabel('Frequency [Hz]')
>>> plt.ylabel('Magnitude [dB]')
>>> plt.grid()
"""
z = atleast_1d(z)
p = atleast_1d(p)
degree = _relative_degree(z, p)
fs2 = 2.0*fs
# Bilinear transform the poles and zeros
z_z = (fs2 + z) / (fs2 - z)
p_z = (fs2 + p) / (fs2 - p)
# Any zeros that were at infinity get moved to the Nyquist frequency
z_z = append(z_z, -ones(degree))
# Compensate for gain change
k_z = k * real(prod(fs2 - z) / prod(fs2 - p))
return z_z, p_z, k_z
def lp2lp_zpk(z, p, k, wo=1.0):
r"""
Transform a lowpass filter prototype to a different frequency.
Return an analog low-pass filter with cutoff frequency `wo`
from an analog low-pass filter prototype with unity cutoff frequency,
using zeros, poles, and gain ('zpk') representation.
Parameters
----------
z : array_like
Zeros of the analog filter transfer function.
p : array_like
Poles of the analog filter transfer function.
k : float
System gain of the analog filter transfer function.
wo : float
Desired cutoff, as angular frequency (e.g., rad/s).
Defaults to no change.
Returns
-------
z : ndarray
Zeros of the transformed low-pass filter transfer function.
p : ndarray
Poles of the transformed low-pass filter transfer function.
k : float
System gain of the transformed low-pass filter.
See Also
--------
lp2hp_zpk, lp2bp_zpk, lp2bs_zpk, bilinear
lp2lp
Notes
-----
This is derived from the s-plane substitution
.. math:: s \rightarrow \frac{s}{\omega_0}
.. versionadded:: 1.1.0
"""
z = atleast_1d(z)
p = atleast_1d(p)
wo = float(wo) # Avoid int wraparound
degree = _relative_degree(z, p)
# Scale all points radially from origin to shift cutoff frequency
z_lp = wo * z
p_lp = wo * p
# Each shifted pole decreases gain by wo, each shifted zero increases it.
# Cancel out the net change to keep overall gain the same
k_lp = k * wo**degree
return z_lp, p_lp, k_lp
def lp2hp_zpk(z, p, k, wo=1.0):
r"""
Transform a lowpass filter prototype to a highpass filter.
Return an analog high-pass filter with cutoff frequency `wo`
from an analog low-pass filter prototype with unity cutoff frequency,
using zeros, poles, and gain ('zpk') representation.
Parameters
----------
z : array_like
Zeros of the analog filter transfer function.
p : array_like
Poles of the analog filter transfer function.
k : float
System gain of the analog filter transfer function.
wo : float
Desired cutoff, as angular frequency (e.g., rad/s).
Defaults to no change.
Returns
-------
z : ndarray
Zeros of the transformed high-pass filter transfer function.
p : ndarray
Poles of the transformed high-pass filter transfer function.
k : float
System gain of the transformed high-pass filter.
See Also
--------
lp2lp_zpk, lp2bp_zpk, lp2bs_zpk, bilinear
lp2hp
Notes
-----
This is derived from the s-plane substitution
.. math:: s \rightarrow \frac{\omega_0}{s}
This maintains symmetry of the lowpass and highpass responses on a
logarithmic scale.
.. versionadded:: 1.1.0
"""
z = atleast_1d(z)
p = atleast_1d(p)
wo = float(wo)
degree = _relative_degree(z, p)
# Invert positions radially about unit circle to convert LPF to HPF
# Scale all points radially from origin to shift cutoff frequency
z_hp = wo / z
p_hp = wo / p
# If lowpass had zeros at infinity, inverting moves them to origin.
z_hp = append(z_hp, zeros(degree))
# Cancel out gain change caused by inversion
k_hp = k * real(prod(-z) / prod(-p))
return z_hp, p_hp, k_hp
def lp2bp_zpk(z, p, k, wo=1.0, bw=1.0):
r"""
Transform a lowpass filter prototype to a bandpass filter.
Return an analog band-pass filter with center frequency `wo` and
bandwidth `bw` from an analog low-pass filter prototype with unity
cutoff frequency, using zeros, poles, and gain ('zpk') representation.
Parameters
----------
z : array_like
Zeros of the analog filter transfer function.
p : array_like
Poles of the analog filter transfer function.
k : float
System gain of the analog filter transfer function.
wo : float
Desired passband center, as angular frequency (e.g., rad/s).
Defaults to no change.
bw : float
Desired passband width, as angular frequency (e.g., rad/s).
Defaults to 1.
Returns
-------
z : ndarray
Zeros of the transformed band-pass filter transfer function.
p : ndarray
Poles of the transformed band-pass filter transfer function.
k : float
System gain of the transformed band-pass filter.
See Also
--------
lp2lp_zpk, lp2hp_zpk, lp2bs_zpk, bilinear
lp2bp
Notes
-----
This is derived from the s-plane substitution
.. math:: s \rightarrow \frac{s^2 + {\omega_0}^2}{s \cdot \mathrm{BW}}
This is the "wideband" transformation, producing a passband with
geometric (log frequency) symmetry about `wo`.
.. versionadded:: 1.1.0
"""
z = atleast_1d(z)
p = atleast_1d(p)
wo = float(wo)
bw = float(bw)
degree = _relative_degree(z, p)
# Scale poles and zeros to desired bandwidth
z_lp = z * bw/2
p_lp = p * bw/2
# Square root needs to produce complex result, not NaN
z_lp = z_lp.astype(complex)
p_lp = p_lp.astype(complex)
# Duplicate poles and zeros and shift from baseband to +wo and -wo
z_bp = concatenate((z_lp + sqrt(z_lp**2 - wo**2),
z_lp - sqrt(z_lp**2 - wo**2)))
p_bp = concatenate((p_lp + sqrt(p_lp**2 - wo**2),
p_lp - sqrt(p_lp**2 - wo**2)))
# Move degree zeros to origin, leaving degree zeros at infinity for BPF
z_bp = append(z_bp, zeros(degree))
# Cancel out gain change from frequency scaling
k_bp = k * bw**degree
return z_bp, p_bp, k_bp
def lp2bs_zpk(z, p, k, wo=1.0, bw=1.0):
r"""
Transform a lowpass filter prototype to a bandstop filter.
Return an analog band-stop filter with center frequency `wo` and
stopband width `bw` from an analog low-pass filter prototype with unity
cutoff frequency, using zeros, poles, and gain ('zpk') representation.
Parameters
----------
z : array_like
Zeros of the analog filter transfer function.
p : array_like
Poles of the analog filter transfer function.
k : float
System gain of the analog filter transfer function.
wo : float
Desired stopband center, as angular frequency (e.g., rad/s).
Defaults to no change.
bw : float
Desired stopband width, as angular frequency (e.g., rad/s).
Defaults to 1.
Returns
-------
z : ndarray
Zeros of the transformed band-stop filter transfer function.
p : ndarray
Poles of the transformed band-stop filter transfer function.
k : float
System gain of the transformed band-stop filter.
See Also
--------
lp2lp_zpk, lp2hp_zpk, lp2bp_zpk, bilinear
lp2bs
Notes
-----
This is derived from the s-plane substitution
.. math:: s \rightarrow \frac{s \cdot \mathrm{BW}}{s^2 + {\omega_0}^2}
This is the "wideband" transformation, producing a stopband with
geometric (log frequency) symmetry about `wo`.
.. versionadded:: 1.1.0
"""
z = atleast_1d(z)
p = atleast_1d(p)
wo = float(wo)
bw = float(bw)
degree = _relative_degree(z, p)
# Invert to a highpass filter with desired bandwidth
z_hp = (bw/2) / z
p_hp = (bw/2) / p
# Square root needs to produce complex result, not NaN
z_hp = z_hp.astype(complex)
p_hp = p_hp.astype(complex)
# Duplicate poles and zeros and shift from baseband to +wo and -wo
z_bs = concatenate((z_hp + sqrt(z_hp**2 - wo**2),
z_hp - sqrt(z_hp**2 - wo**2)))
p_bs = concatenate((p_hp + sqrt(p_hp**2 - wo**2),
p_hp - sqrt(p_hp**2 - wo**2)))
# Move any zeros that were at infinity to the center of the stopband
z_bs = append(z_bs, full(degree, +1j*wo))
z_bs = append(z_bs, full(degree, -1j*wo))
# Cancel out gain change caused by inversion
k_bs = k * real(prod(-z) / prod(-p))
return z_bs, p_bs, k_bs
def butter(N, Wn, btype='low', analog=False, output='ba', fs=None):
"""
Butterworth digital and analog filter design.
Design an Nth-order digital or analog Butterworth filter and return
the filter coefficients.
Parameters
----------
N : int
The order of the filter.
Wn : array_like
The critical frequency or frequencies. For lowpass and highpass
filters, Wn is a scalar; for bandpass and bandstop filters,
Wn is a length-2 sequence.
For a Butterworth filter, this is the point at which the gain
drops to 1/sqrt(2) that of the passband (the "-3 dB point").
For digital filters, `Wn` are in the same units as `fs`. By default,
`fs` is 2 half-cycles/sample, so these are normalized from 0 to 1,
where 1 is the Nyquist frequency. (`Wn` is thus in
half-cycles / sample.)
For analog filters, `Wn` is an angular frequency (e.g. rad/s).
btype : {'lowpass', 'highpass', 'bandpass', 'bandstop'}, optional
The type of filter. Default is 'lowpass'.
analog : bool, optional
When True, return an analog filter, otherwise a digital filter is
returned.
output : {'ba', 'zpk', 'sos'}, optional
Type of output: numerator/denominator ('ba'), pole-zero ('zpk'), or
second-order sections ('sos'). Default is 'ba' for backwards
compatibility, but 'sos' should be used for general-purpose filtering.
fs : float, optional
The sampling frequency of the digital system.
.. versionadded:: 1.2.0
Returns
-------
b, a : ndarray, ndarray
Numerator (`b`) and denominator (`a`) polynomials of the IIR filter.
Only returned if ``output='ba'``.
z, p, k : ndarray, ndarray, float
Zeros, poles, and system gain of the IIR filter transfer
function. Only returned if ``output='zpk'``.
sos : ndarray
Second-order sections representation of the IIR filter.
Only returned if ``output=='sos'``.
See Also
--------
buttord, buttap
Notes
-----
The Butterworth filter has maximally flat frequency response in the
passband.
The ``'sos'`` output parameter was added in 0.16.0.
If the transfer function form ``[b, a]`` is requested, numerical
problems can occur since the conversion between roots and
the polynomial coefficients is a numerically sensitive operation,
even for N >= 4. It is recommended to work with the SOS
representation.
Examples
--------
Design an analog filter and plot its frequency response, showing the
critical points:
>>> from scipy import signal
>>> import matplotlib.pyplot as plt
>>> b, a = signal.butter(4, 100, 'low', analog=True)
>>> w, h = signal.freqs(b, a)
>>> plt.semilogx(w, 20 * np.log10(abs(h)))
>>> plt.title('Butterworth filter frequency response')
>>> plt.xlabel('Frequency [radians / second]')
>>> plt.ylabel('Amplitude [dB]')
>>> plt.margins(0, 0.1)
>>> plt.grid(which='both', axis='both')
>>> plt.axvline(100, color='green') # cutoff frequency
>>> plt.show()
Generate a signal made up of 10 Hz and 20 Hz, sampled at 1 kHz
>>> t = np.linspace(0, 1, 1000, False) # 1 second
>>> sig = np.sin(2*np.pi*10*t) + np.sin(2*np.pi*20*t)
>>> fig, (ax1, ax2) = plt.subplots(2, 1, sharex=True)
>>> ax1.plot(t, sig)
>>> ax1.set_title('10 Hz and 20 Hz sinusoids')
>>> ax1.axis([0, 1, -2, 2])
Design a digital high-pass filter at 15 Hz to remove the 10 Hz tone, and
apply it to the signal. (It's recommended to use second-order sections
format when filtering, to avoid numerical error with transfer function
(``ba``) format):
>>> sos = signal.butter(10, 15, 'hp', fs=1000, output='sos')
>>> filtered = signal.sosfilt(sos, sig)
>>> ax2.plot(t, filtered)
>>> ax2.set_title('After 15 Hz high-pass filter')
>>> ax2.axis([0, 1, -2, 2])
>>> ax2.set_xlabel('Time [seconds]')
>>> plt.tight_layout()
>>> plt.show()
"""
return iirfilter(N, Wn, btype=btype, analog=analog,
output=output, ftype='butter', fs=fs)
def cheby1(N, rp, Wn, btype='low', analog=False, output='ba', fs=None):
"""
Chebyshev type I digital and analog filter design.
Design an Nth-order digital or analog Chebyshev type I filter and
return the filter coefficients.
Parameters
----------
N : int
The order of the filter.
rp : float
The maximum ripple allowed below unity gain in the passband.
Specified in decibels, as a positive number.
Wn : array_like
A scalar or length-2 sequence giving the critical frequencies.
For Type I filters, this is the point in the transition band at which
the gain first drops below -`rp`.
For digital filters, `Wn` are in the same units as `fs`. By default,
`fs` is 2 half-cycles/sample, so these are normalized from 0 to 1,
where 1 is the Nyquist frequency. (`Wn` is thus in
half-cycles / sample.)
For analog filters, `Wn` is an angular frequency (e.g., rad/s).
btype : {'lowpass', 'highpass', 'bandpass', 'bandstop'}, optional
The type of filter. Default is 'lowpass'.
analog : bool, optional
When True, return an analog filter, otherwise a digital filter is
returned.
output : {'ba', 'zpk', 'sos'}, optional
Type of output: numerator/denominator ('ba'), pole-zero ('zpk'), or
second-order sections ('sos'). Default is 'ba' for backwards
compatibility, but 'sos' should be used for general-purpose filtering.
fs : float, optional
The sampling frequency of the digital system.
.. versionadded:: 1.2.0
Returns
-------
b, a : ndarray, ndarray
Numerator (`b`) and denominator (`a`) polynomials of the IIR filter.
Only returned if ``output='ba'``.
z, p, k : ndarray, ndarray, float
Zeros, poles, and system gain of the IIR filter transfer
function. Only returned if ``output='zpk'``.
sos : ndarray
Second-order sections representation of the IIR filter.
Only returned if ``output=='sos'``.
See Also
--------
cheb1ord, cheb1ap
Notes
-----
The Chebyshev type I filter maximizes the rate of cutoff between the
frequency response's passband and stopband, at the expense of ripple in
the passband and increased ringing in the step response.
Type I filters roll off faster than Type II (`cheby2`), but Type II
filters do not have any ripple in the passband.
The equiripple passband has N maxima or minima (for example, a
5th-order filter has 3 maxima and 2 minima). Consequently, the DC gain is
unity for odd-order filters, or -rp dB for even-order filters.
The ``'sos'`` output parameter was added in 0.16.0.
Examples
--------
Design an analog filter and plot its frequency response, showing the
critical points:
>>> from scipy import signal
>>> import matplotlib.pyplot as plt
>>> b, a = signal.cheby1(4, 5, 100, 'low', analog=True)
>>> w, h = signal.freqs(b, a)
>>> plt.semilogx(w, 20 * np.log10(abs(h)))
>>> plt.title('Chebyshev Type I frequency response (rp=5)')
>>> plt.xlabel('Frequency [radians / second]')
>>> plt.ylabel('Amplitude [dB]')
>>> plt.margins(0, 0.1)
>>> plt.grid(which='both', axis='both')
>>> plt.axvline(100, color='green') # cutoff frequency
>>> plt.axhline(-5, color='green') # rp
>>> plt.show()
Generate a signal made up of 10 Hz and 20 Hz, sampled at 1 kHz
>>> t = np.linspace(0, 1, 1000, False) # 1 second
>>> sig = np.sin(2*np.pi*10*t) + np.sin(2*np.pi*20*t)
>>> fig, (ax1, ax2) = plt.subplots(2, 1, sharex=True)
>>> ax1.plot(t, sig)
>>> ax1.set_title('10 Hz and 20 Hz sinusoids')
>>> ax1.axis([0, 1, -2, 2])
Design a digital high-pass filter at 15 Hz to remove the 10 Hz tone, and
apply it to the signal. (It's recommended to use second-order sections
format when filtering, to avoid numerical error with transfer function
(``ba``) format):
>>> sos = signal.cheby1(10, 1, 15, 'hp', fs=1000, output='sos')
>>> filtered = signal.sosfilt(sos, sig)
>>> ax2.plot(t, filtered)
>>> ax2.set_title('After 15 Hz high-pass filter')
>>> ax2.axis([0, 1, -2, 2])
>>> ax2.set_xlabel('Time [seconds]')
>>> plt.tight_layout()
>>> plt.show()
"""
return iirfilter(N, Wn, rp=rp, btype=btype, analog=analog,
output=output, ftype='cheby1', fs=fs)
def cheby2(N, rs, Wn, btype='low', analog=False, output='ba', fs=None):
"""
Chebyshev type II digital and analog filter design.
Design an Nth-order digital or analog Chebyshev type II filter and
return the filter coefficients.
Parameters
----------
N : int
The order of the filter.
rs : float
The minimum attenuation required in the stop band.
Specified in decibels, as a positive number.
Wn : array_like
A scalar or length-2 sequence giving the critical frequencies.
For Type II filters, this is the point in the transition band at which
the gain first reaches -`rs`.
For digital filters, `Wn` are in the same units as `fs`. By default,
`fs` is 2 half-cycles/sample, so these are normalized from 0 to 1,
where 1 is the Nyquist frequency. (`Wn` is thus in
half-cycles / sample.)
For analog filters, `Wn` is an angular frequency (e.g., rad/s).
btype : {'lowpass', 'highpass', 'bandpass', 'bandstop'}, optional
The type of filter. Default is 'lowpass'.
analog : bool, optional
When True, return an analog filter, otherwise a digital filter is
returned.
output : {'ba', 'zpk', 'sos'}, optional
Type of output: numerator/denominator ('ba'), pole-zero ('zpk'), or
second-order sections ('sos'). Default is 'ba' for backwards
compatibility, but 'sos' should be used for general-purpose filtering.
fs : float, optional
The sampling frequency of the digital system.
.. versionadded:: 1.2.0
Returns
-------
b, a : ndarray, ndarray
Numerator (`b`) and denominator (`a`) polynomials of the IIR filter.
Only returned if ``output='ba'``.
z, p, k : ndarray, ndarray, float
Zeros, poles, and system gain of the IIR filter transfer
function. Only returned if ``output='zpk'``.
sos : ndarray
Second-order sections representation of the IIR filter.
Only returned if ``output=='sos'``.
See Also
--------
cheb2ord, cheb2ap
Notes
-----
The Chebyshev type II filter maximizes the rate of cutoff between the
frequency response's passband and stopband, at the expense of ripple in
the stopband and increased ringing in the step response.
Type II filters do not roll off as fast as Type I (`cheby1`).
The ``'sos'`` output parameter was added in 0.16.0.
Examples
--------
Design an analog filter and plot its frequency response, showing the
critical points:
>>> from scipy import signal
>>> import matplotlib.pyplot as plt
>>> b, a = signal.cheby2(4, 40, 100, 'low', analog=True)
>>> w, h = signal.freqs(b, a)
>>> plt.semilogx(w, 20 * np.log10(abs(h)))
>>> plt.title('Chebyshev Type II frequency response (rs=40)')
>>> plt.xlabel('Frequency [radians / second]')
>>> plt.ylabel('Amplitude [dB]')
>>> plt.margins(0, 0.1)
>>> plt.grid(which='both', axis='both')
>>> plt.axvline(100, color='green') # cutoff frequency
>>> plt.axhline(-40, color='green') # rs
>>> plt.show()
Generate a signal made up of 10 Hz and 20 Hz, sampled at 1 kHz
>>> t = np.linspace(0, 1, 1000, False) # 1 second
>>> sig = np.sin(2*np.pi*10*t) + np.sin(2*np.pi*20*t)
>>> fig, (ax1, ax2) = plt.subplots(2, 1, sharex=True)
>>> ax1.plot(t, sig)
>>> ax1.set_title('10 Hz and 20 Hz sinusoids')
>>> ax1.axis([0, 1, -2, 2])
Design a digital high-pass filter at 17 Hz to remove the 10 Hz tone, and
apply it to the signal. (It's recommended to use second-order sections
format when filtering, to avoid numerical error with transfer function
(``ba``) format):
>>> sos = signal.cheby2(12, 20, 17, 'hp', fs=1000, output='sos')
>>> filtered = signal.sosfilt(sos, sig)
>>> ax2.plot(t, filtered)
>>> ax2.set_title('After 17 Hz high-pass filter')
>>> ax2.axis([0, 1, -2, 2])
>>> ax2.set_xlabel('Time [seconds]')
>>> plt.show()
"""
return iirfilter(N, Wn, rs=rs, btype=btype, analog=analog,
output=output, ftype='cheby2', fs=fs)
def ellip(N, rp, rs, Wn, btype='low', analog=False, output='ba', fs=None):
"""
Elliptic (Cauer) digital and analog filter design.
Design an Nth-order digital or analog elliptic filter and return
the filter coefficients.
Parameters
----------
N : int
The order of the filter.
rp : float
The maximum ripple allowed below unity gain in the passband.
Specified in decibels, as a positive number.
rs : float
The minimum attenuation required in the stop band.
Specified in decibels, as a positive number.
Wn : array_like
A scalar or length-2 sequence giving the critical frequencies.
For elliptic filters, this is the point in the transition band at
which the gain first drops below -`rp`.
For digital filters, `Wn` are in the same units as `fs`. By default,
`fs` is 2 half-cycles/sample, so these are normalized from 0 to 1,
where 1 is the Nyquist frequency. (`Wn` is thus in
half-cycles / sample.)
For analog filters, `Wn` is an angular frequency (e.g., rad/s).
btype : {'lowpass', 'highpass', 'bandpass', 'bandstop'}, optional
The type of filter. Default is 'lowpass'.
analog : bool, optional
When True, return an analog filter, otherwise a digital filter is
returned.
output : {'ba', 'zpk', 'sos'}, optional
Type of output: numerator/denominator ('ba'), pole-zero ('zpk'), or
second-order sections ('sos'). Default is 'ba' for backwards
compatibility, but 'sos' should be used for general-purpose filtering.
fs : float, optional
The sampling frequency of the digital system.
.. versionadded:: 1.2.0
Returns
-------
b, a : ndarray, ndarray
Numerator (`b`) and denominator (`a`) polynomials of the IIR filter.
Only returned if ``output='ba'``.
z, p, k : ndarray, ndarray, float
Zeros, poles, and system gain of the IIR filter transfer
function. Only returned if ``output='zpk'``.
sos : ndarray
Second-order sections representation of the IIR filter.
Only returned if ``output=='sos'``.
See Also
--------
ellipord, ellipap
Notes
-----
Also known as Cauer or Zolotarev filters, the elliptical filter maximizes
the rate of transition between the frequency response's passband and
stopband, at the expense of ripple in both, and increased ringing in the
step response.
As `rp` approaches 0, the elliptical filter becomes a Chebyshev
type II filter (`cheby2`). As `rs` approaches 0, it becomes a Chebyshev
type I filter (`cheby1`). As both approach 0, it becomes a Butterworth
filter (`butter`).
The equiripple passband has N maxima or minima (for example, a
5th-order filter has 3 maxima and 2 minima). Consequently, the DC gain is
unity for odd-order filters, or -rp dB for even-order filters.
The ``'sos'`` output parameter was added in 0.16.0.
Examples
--------
Design an analog filter and plot its frequency response, showing the
critical points:
>>> from scipy import signal
>>> import matplotlib.pyplot as plt
>>> b, a = signal.ellip(4, 5, 40, 100, 'low', analog=True)
>>> w, h = signal.freqs(b, a)
>>> plt.semilogx(w, 20 * np.log10(abs(h)))
>>> plt.title('Elliptic filter frequency response (rp=5, rs=40)')
>>> plt.xlabel('Frequency [radians / second]')
>>> plt.ylabel('Amplitude [dB]')
>>> plt.margins(0, 0.1)
>>> plt.grid(which='both', axis='both')
>>> plt.axvline(100, color='green') # cutoff frequency
>>> plt.axhline(-40, color='green') # rs
>>> plt.axhline(-5, color='green') # rp
>>> plt.show()
Generate a signal made up of 10 Hz and 20 Hz, sampled at 1 kHz
>>> t = np.linspace(0, 1, 1000, False) # 1 second
>>> sig = np.sin(2*np.pi*10*t) + np.sin(2*np.pi*20*t)
>>> fig, (ax1, ax2) = plt.subplots(2, 1, sharex=True)
>>> ax1.plot(t, sig)
>>> ax1.set_title('10 Hz and 20 Hz sinusoids')
>>> ax1.axis([0, 1, -2, 2])
Design a digital high-pass filter at 17 Hz to remove the 10 Hz tone, and
apply it to the signal. (It's recommended to use second-order sections
format when filtering, to avoid numerical error with transfer function
(``ba``) format):
>>> sos = signal.ellip(8, 1, 100, 17, 'hp', fs=1000, output='sos')
>>> filtered = signal.sosfilt(sos, sig)
>>> ax2.plot(t, filtered)
>>> ax2.set_title('After 17 Hz high-pass filter')
>>> ax2.axis([0, 1, -2, 2])
>>> ax2.set_xlabel('Time [seconds]')
>>> plt.tight_layout()
>>> plt.show()
"""
return iirfilter(N, Wn, rs=rs, rp=rp, btype=btype, analog=analog,
output=output, ftype='elliptic', fs=fs)
def bessel(N, Wn, btype='low', analog=False, output='ba', norm='phase',
fs=None):
"""
Bessel/Thomson digital and analog filter design.
Design an Nth-order digital or analog Bessel filter and return the
filter coefficients.
Parameters
----------
N : int
The order of the filter.
Wn : array_like
A scalar or length-2 sequence giving the critical frequencies (defined
by the `norm` parameter).
For analog filters, `Wn` is an angular frequency (e.g., rad/s).
For digital filters, `Wn` are in the same units as `fs`. By default,
`fs` is 2 half-cycles/sample, so these are normalized from 0 to 1,
where 1 is the Nyquist frequency. (`Wn` is thus in
half-cycles / sample.)
btype : {'lowpass', 'highpass', 'bandpass', 'bandstop'}, optional
The type of filter. Default is 'lowpass'.
analog : bool, optional
When True, return an analog filter, otherwise a digital filter is
returned. (See Notes.)
output : {'ba', 'zpk', 'sos'}, optional
Type of output: numerator/denominator ('ba'), pole-zero ('zpk'), or
second-order sections ('sos'). Default is 'ba'.
norm : {'phase', 'delay', 'mag'}, optional
Critical frequency normalization:
``phase``
The filter is normalized such that the phase response reaches its
midpoint at angular (e.g. rad/s) frequency `Wn`. This happens for
both low-pass and high-pass filters, so this is the
"phase-matched" case.
The magnitude response asymptotes are the same as a Butterworth
filter of the same order with a cutoff of `Wn`.
This is the default, and matches MATLAB's implementation.
``delay``
The filter is normalized such that the group delay in the passband
is 1/`Wn` (e.g., seconds). This is the "natural" type obtained by
solving Bessel polynomials.
``mag``
The filter is normalized such that the gain magnitude is -3 dB at
angular frequency `Wn`.
.. versionadded:: 0.18.0
fs : float, optional
The sampling frequency of the digital system.
.. versionadded:: 1.2.0
Returns
-------
b, a : ndarray, ndarray
Numerator (`b`) and denominator (`a`) polynomials of the IIR filter.
Only returned if ``output='ba'``.
z, p, k : ndarray, ndarray, float
Zeros, poles, and system gain of the IIR filter transfer
function. Only returned if ``output='zpk'``.
sos : ndarray
Second-order sections representation of the IIR filter.
Only returned if ``output=='sos'``.
Notes
-----
Also known as a Thomson filter, the analog Bessel filter has maximally
flat group delay and maximally linear phase response, with very little
ringing in the step response. [1]_
The Bessel is inherently an analog filter. This function generates digital
Bessel filters using the bilinear transform, which does not preserve the
phase response of the analog filter. As such, it is only approximately
correct at frequencies below about fs/4. To get maximally-flat group
delay at higher frequencies, the analog Bessel filter must be transformed
using phase-preserving techniques.
See `besselap` for implementation details and references.
The ``'sos'`` output parameter was added in 0.16.0.
Examples
--------
Plot the phase-normalized frequency response, showing the relationship
to the Butterworth's cutoff frequency (green):
>>> from scipy import signal
>>> import matplotlib.pyplot as plt
>>> b, a = signal.butter(4, 100, 'low', analog=True)
>>> w, h = signal.freqs(b, a)
>>> plt.semilogx(w, 20 * np.log10(np.abs(h)), color='silver', ls='dashed')
>>> b, a = signal.bessel(4, 100, 'low', analog=True, norm='phase')
>>> w, h = signal.freqs(b, a)
>>> plt.semilogx(w, 20 * np.log10(np.abs(h)))
>>> plt.title('Bessel filter magnitude response (with Butterworth)')
>>> plt.xlabel('Frequency [radians / second]')
>>> plt.ylabel('Amplitude [dB]')
>>> plt.margins(0, 0.1)
>>> plt.grid(which='both', axis='both')
>>> plt.axvline(100, color='green') # cutoff frequency
>>> plt.show()
and the phase midpoint:
>>> plt.figure()
>>> plt.semilogx(w, np.unwrap(np.angle(h)))
>>> plt.axvline(100, color='green') # cutoff frequency
>>> plt.axhline(-np.pi, color='red') # phase midpoint
>>> plt.title('Bessel filter phase response')
>>> plt.xlabel('Frequency [radians / second]')
>>> plt.ylabel('Phase [radians]')
>>> plt.margins(0, 0.1)
>>> plt.grid(which='both', axis='both')
>>> plt.show()
Plot the magnitude-normalized frequency response, showing the -3 dB cutoff:
>>> b, a = signal.bessel(3, 10, 'low', analog=True, norm='mag')
>>> w, h = signal.freqs(b, a)
>>> plt.semilogx(w, 20 * np.log10(np.abs(h)))
>>> plt.axhline(-3, color='red') # -3 dB magnitude
>>> plt.axvline(10, color='green') # cutoff frequency
>>> plt.title('Magnitude-normalized Bessel filter frequency response')
>>> plt.xlabel('Frequency [radians / second]')
>>> plt.ylabel('Amplitude [dB]')
>>> plt.margins(0, 0.1)
>>> plt.grid(which='both', axis='both')
>>> plt.show()
Plot the delay-normalized filter, showing the maximally-flat group delay
at 0.1 seconds:
>>> b, a = signal.bessel(5, 1/0.1, 'low', analog=True, norm='delay')
>>> w, h = signal.freqs(b, a)
>>> plt.figure()
>>> plt.semilogx(w[1:], -np.diff(np.unwrap(np.angle(h)))/np.diff(w))
>>> plt.axhline(0.1, color='red') # 0.1 seconds group delay
>>> plt.title('Bessel filter group delay')
>>> plt.xlabel('Frequency [radians / second]')
>>> plt.ylabel('Group delay [seconds]')
>>> plt.margins(0, 0.1)
>>> plt.grid(which='both', axis='both')
>>> plt.show()
References
----------
.. [1] Thomson, W.E., "Delay Networks having Maximally Flat Frequency
Characteristics", Proceedings of the Institution of Electrical
Engineers, Part III, November 1949, Vol. 96, No. 44, pp. 487-490.
"""
return iirfilter(N, Wn, btype=btype, analog=analog,
output=output, ftype='bessel_'+norm, fs=fs)
def maxflat():
pass
def yulewalk():
pass
def band_stop_obj(wp, ind, passb, stopb, gpass, gstop, type):
"""
Band Stop Objective Function for order minimization.
Returns the non-integer order for an analog band stop filter.
Parameters
----------
wp : scalar
Edge of passband `passb`.
ind : int, {0, 1}
Index specifying which `passb` edge to vary (0 or 1).
passb : ndarray
Two element sequence of fixed passband edges.
stopb : ndarray
Two element sequence of fixed stopband edges.
gstop : float
Amount of attenuation in stopband in dB.
gpass : float
Amount of ripple in the passband in dB.
type : {'butter', 'cheby', 'ellip'}
Type of filter.
Returns
-------
n : scalar
Filter order (possibly non-integer).
"""
_validate_gpass_gstop(gpass, gstop)
passbC = passb.copy()
passbC[ind] = wp
nat = (stopb * (passbC[0] - passbC[1]) /
(stopb ** 2 - passbC[0] * passbC[1]))
nat = min(abs(nat))
if type == 'butter':
GSTOP = 10 ** (0.1 * abs(gstop))
GPASS = 10 ** (0.1 * abs(gpass))
n = (log10((GSTOP - 1.0) / (GPASS - 1.0)) / (2 * log10(nat)))
elif type == 'cheby':
GSTOP = 10 ** (0.1 * abs(gstop))
GPASS = 10 ** (0.1 * abs(gpass))
n = arccosh(sqrt((GSTOP - 1.0) / (GPASS - 1.0))) / arccosh(nat)
elif type == 'ellip':
GSTOP = 10 ** (0.1 * gstop)
GPASS = 10 ** (0.1 * gpass)
arg1 = sqrt((GPASS - 1.0) / (GSTOP - 1.0))
arg0 = 1.0 / nat
d0 = special.ellipk([arg0 ** 2, 1 - arg0 ** 2])
d1 = special.ellipk([arg1 ** 2, 1 - arg1 ** 2])
n = (d0[0] * d1[1] / (d0[1] * d1[0]))
else:
raise ValueError("Incorrect type: %s" % type)
return n
def buttord(wp, ws, gpass, gstop, analog=False, fs=None):
"""Butterworth filter order selection.
Return the order of the lowest order digital or analog Butterworth filter
that loses no more than `gpass` dB in the passband and has at least
`gstop` dB attenuation in the stopband.
Parameters
----------
wp, ws : float
Passband and stopband edge frequencies.
For digital filters, these are in the same units as `fs`. By default,
`fs` is 2 half-cycles/sample, so these are normalized from 0 to 1,
where 1 is the Nyquist frequency. (`wp` and `ws` are thus in
half-cycles / sample.) For example:
- Lowpass: wp = 0.2, ws = 0.3
- Highpass: wp = 0.3, ws = 0.2
- Bandpass: wp = [0.2, 0.5], ws = [0.1, 0.6]
- Bandstop: wp = [0.1, 0.6], ws = [0.2, 0.5]
For analog filters, `wp` and `ws` are angular frequencies (e.g., rad/s).
gpass : float
The maximum loss in the passband (dB).
gstop : float
The minimum attenuation in the stopband (dB).
analog : bool, optional
When True, return an analog filter, otherwise a digital filter is
returned.
fs : float, optional
The sampling frequency of the digital system.
.. versionadded:: 1.2.0
Returns
-------
ord : int
The lowest order for a Butterworth filter which meets specs.
wn : ndarray or float
The Butterworth natural frequency (i.e. the "3dB frequency"). Should
be used with `butter` to give filter results. If `fs` is specified,
this is in the same units, and `fs` must also be passed to `butter`.
See Also
--------
butter : Filter design using order and critical points
cheb1ord : Find order and critical points from passband and stopband spec
cheb2ord, ellipord
iirfilter : General filter design using order and critical frequencies
iirdesign : General filter design using passband and stopband spec
Examples
--------
Design an analog bandpass filter with passband within 3 dB from 20 to
50 rad/s, while rejecting at least -40 dB below 14 and above 60 rad/s.
Plot its frequency response, showing the passband and stopband
constraints in gray.
>>> from scipy import signal
>>> import matplotlib.pyplot as plt
>>> N, Wn = signal.buttord([20, 50], [14, 60], 3, 40, True)
>>> b, a = signal.butter(N, Wn, 'band', True)
>>> w, h = signal.freqs(b, a, np.logspace(1, 2, 500))
>>> plt.semilogx(w, 20 * np.log10(abs(h)))
>>> plt.title('Butterworth bandpass filter fit to constraints')
>>> plt.xlabel('Frequency [radians / second]')
>>> plt.ylabel('Amplitude [dB]')
>>> plt.grid(which='both', axis='both')
>>> plt.fill([1, 14, 14, 1], [-40, -40, 99, 99], '0.9', lw=0) # stop
>>> plt.fill([20, 20, 50, 50], [-99, -3, -3, -99], '0.9', lw=0) # pass
>>> plt.fill([60, 60, 1e9, 1e9], [99, -40, -40, 99], '0.9', lw=0) # stop
>>> plt.axis([10, 100, -60, 3])
>>> plt.show()
"""
_validate_gpass_gstop(gpass, gstop)
wp = atleast_1d(wp)
ws = atleast_1d(ws)
if fs is not None:
if analog:
raise ValueError("fs cannot be specified for an analog filter")
wp = 2*wp/fs
ws = 2*ws/fs
filter_type = 2 * (len(wp) - 1)
filter_type += 1
if wp[0] >= ws[0]:
filter_type += 1
# Pre-warp frequencies for digital filter design
if not analog:
passb = tan(pi * wp / 2.0)
stopb = tan(pi * ws / 2.0)
else:
passb = wp * 1.0
stopb = ws * 1.0
if filter_type == 1: # low
nat = stopb / passb
elif filter_type == 2: # high
nat = passb / stopb
elif filter_type == 3: # stop
wp0 = optimize.fminbound(band_stop_obj, passb[0], stopb[0] - 1e-12,
args=(0, passb, stopb, gpass, gstop,
'butter'),
disp=0)
passb[0] = wp0
wp1 = optimize.fminbound(band_stop_obj, stopb[1] + 1e-12, passb[1],
args=(1, passb, stopb, gpass, gstop,
'butter'),
disp=0)
passb[1] = wp1
nat = ((stopb * (passb[0] - passb[1])) /
(stopb ** 2 - passb[0] * passb[1]))
elif filter_type == 4: # pass
nat = ((stopb ** 2 - passb[0] * passb[1]) /
(stopb * (passb[0] - passb[1])))
nat = min(abs(nat))
GSTOP = 10 ** (0.1 * abs(gstop))
GPASS = 10 ** (0.1 * abs(gpass))
ord = int(ceil(log10((GSTOP - 1.0) / (GPASS - 1.0)) / (2 * log10(nat))))
# Find the Butterworth natural frequency WN (or the "3dB" frequency")
# to give exactly gpass at passb.
try:
W0 = (GPASS - 1.0) ** (-1.0 / (2.0 * ord))
except ZeroDivisionError:
W0 = 1.0
print("Warning, order is zero...check input parameters.")
# now convert this frequency back from lowpass prototype
# to the original analog filter
if filter_type == 1: # low
WN = W0 * passb
elif filter_type == 2: # high
WN = passb / W0
elif filter_type == 3: # stop
WN = numpy.empty(2, float)
discr = sqrt((passb[1] - passb[0]) ** 2 +
4 * W0 ** 2 * passb[0] * passb[1])
WN[0] = ((passb[1] - passb[0]) + discr) / (2 * W0)
WN[1] = ((passb[1] - passb[0]) - discr) / (2 * W0)
WN = numpy.sort(abs(WN))
elif filter_type == 4: # pass
W0 = numpy.array([-W0, W0], float)
WN = (-W0 * (passb[1] - passb[0]) / 2.0 +
sqrt(W0 ** 2 / 4.0 * (passb[1] - passb[0]) ** 2 +
passb[0] * passb[1]))
WN = numpy.sort(abs(WN))
else:
raise ValueError("Bad type: %s" % filter_type)
if not analog:
wn = (2.0 / pi) * arctan(WN)
else:
wn = WN
if len(wn) == 1:
wn = wn[0]
if fs is not None:
wn = wn*fs/2
return ord, wn
def cheb1ord(wp, ws, gpass, gstop, analog=False, fs=None):
"""Chebyshev type I filter order selection.
Return the order of the lowest order digital or analog Chebyshev Type I
filter that loses no more than `gpass` dB in the passband and has at
least `gstop` dB attenuation in the stopband.
Parameters
----------
wp, ws : float
Passband and stopband edge frequencies.
For digital filters, these are in the same units as `fs`. By default,
`fs` is 2 half-cycles/sample, so these are normalized from 0 to 1,
where 1 is the Nyquist frequency. (`wp` and `ws` are thus in
half-cycles / sample.) For example:
- Lowpass: wp = 0.2, ws = 0.3
- Highpass: wp = 0.3, ws = 0.2
- Bandpass: wp = [0.2, 0.5], ws = [0.1, 0.6]
- Bandstop: wp = [0.1, 0.6], ws = [0.2, 0.5]
For analog filters, `wp` and `ws` are angular frequencies (e.g., rad/s).
gpass : float
The maximum loss in the passband (dB).
gstop : float
The minimum attenuation in the stopband (dB).
analog : bool, optional
When True, return an analog filter, otherwise a digital filter is
returned.
fs : float, optional
The sampling frequency of the digital system.
.. versionadded:: 1.2.0
Returns
-------
ord : int
The lowest order for a Chebyshev type I filter that meets specs.
wn : ndarray or float
The Chebyshev natural frequency (the "3dB frequency") for use with
`cheby1` to give filter results. If `fs` is specified,
this is in the same units, and `fs` must also be passed to `cheby1`.
See Also
--------
cheby1 : Filter design using order and critical points
buttord : Find order and critical points from passband and stopband spec
cheb2ord, ellipord
iirfilter : General filter design using order and critical frequencies
iirdesign : General filter design using passband and stopband spec
Examples
--------
Design a digital lowpass filter such that the passband is within 3 dB up
to 0.2*(fs/2), while rejecting at least -40 dB above 0.3*(fs/2). Plot its
frequency response, showing the passband and stopband constraints in gray.
>>> from scipy import signal
>>> import matplotlib.pyplot as plt
>>> N, Wn = signal.cheb1ord(0.2, 0.3, 3, 40)
>>> b, a = signal.cheby1(N, 3, Wn, 'low')
>>> w, h = signal.freqz(b, a)
>>> plt.semilogx(w / np.pi, 20 * np.log10(abs(h)))
>>> plt.title('Chebyshev I lowpass filter fit to constraints')
>>> plt.xlabel('Normalized frequency')
>>> plt.ylabel('Amplitude [dB]')
>>> plt.grid(which='both', axis='both')
>>> plt.fill([.01, 0.2, 0.2, .01], [-3, -3, -99, -99], '0.9', lw=0) # stop
>>> plt.fill([0.3, 0.3, 2, 2], [ 9, -40, -40, 9], '0.9', lw=0) # pass
>>> plt.axis([0.08, 1, -60, 3])
>>> plt.show()
"""
_validate_gpass_gstop(gpass, gstop)
wp = atleast_1d(wp)
ws = atleast_1d(ws)
if fs is not None:
if analog:
raise ValueError("fs cannot be specified for an analog filter")
wp = 2*wp/fs
ws = 2*ws/fs
filter_type = 2 * (len(wp) - 1)
if wp[0] < ws[0]:
filter_type += 1
else:
filter_type += 2
# Pre-warp frequencies for digital filter design
if not analog:
passb = tan(pi * wp / 2.0)
stopb = tan(pi * ws / 2.0)
else:
passb = wp * 1.0
stopb = ws * 1.0
if filter_type == 1: # low
nat = stopb / passb
elif filter_type == 2: # high
nat = passb / stopb
elif filter_type == 3: # stop
wp0 = optimize.fminbound(band_stop_obj, passb[0], stopb[0] - 1e-12,
args=(0, passb, stopb, gpass, gstop, 'cheby'),
disp=0)
passb[0] = wp0
wp1 = optimize.fminbound(band_stop_obj, stopb[1] + 1e-12, passb[1],
args=(1, passb, stopb, gpass, gstop, 'cheby'),
disp=0)
passb[1] = wp1
nat = ((stopb * (passb[0] - passb[1])) /
(stopb ** 2 - passb[0] * passb[1]))
elif filter_type == 4: # pass
nat = ((stopb ** 2 - passb[0] * passb[1]) /
(stopb * (passb[0] - passb[1])))
nat = min(abs(nat))
GSTOP = 10 ** (0.1 * abs(gstop))
GPASS = 10 ** (0.1 * abs(gpass))
ord = int(ceil(arccosh(sqrt((GSTOP - 1.0) / (GPASS - 1.0))) /
arccosh(nat)))
# Natural frequencies are just the passband edges
if not analog:
wn = (2.0 / pi) * arctan(passb)
else:
wn = passb
if len(wn) == 1:
wn = wn[0]
if fs is not None:
wn = wn*fs/2
return ord, wn
def cheb2ord(wp, ws, gpass, gstop, analog=False, fs=None):
"""Chebyshev type II filter order selection.
Return the order of the lowest order digital or analog Chebyshev Type II
filter that loses no more than `gpass` dB in the passband and has at least
`gstop` dB attenuation in the stopband.
Parameters
----------
wp, ws : float
Passband and stopband edge frequencies.
For digital filters, these are in the same units as `fs`. By default,
`fs` is 2 half-cycles/sample, so these are normalized from 0 to 1,
where 1 is the Nyquist frequency. (`wp` and `ws` are thus in
half-cycles / sample.) For example:
- Lowpass: wp = 0.2, ws = 0.3
- Highpass: wp = 0.3, ws = 0.2
- Bandpass: wp = [0.2, 0.5], ws = [0.1, 0.6]
- Bandstop: wp = [0.1, 0.6], ws = [0.2, 0.5]
For analog filters, `wp` and `ws` are angular frequencies (e.g., rad/s).
gpass : float
The maximum loss in the passband (dB).
gstop : float
The minimum attenuation in the stopband (dB).
analog : bool, optional
When True, return an analog filter, otherwise a digital filter is
returned.
fs : float, optional
The sampling frequency of the digital system.
.. versionadded:: 1.2.0
Returns
-------
ord : int
The lowest order for a Chebyshev type II filter that meets specs.
wn : ndarray or float
The Chebyshev natural frequency (the "3dB frequency") for use with
`cheby2` to give filter results. If `fs` is specified,
this is in the same units, and `fs` must also be passed to `cheby2`.
See Also
--------
cheby2 : Filter design using order and critical points
buttord : Find order and critical points from passband and stopband spec
cheb1ord, ellipord
iirfilter : General filter design using order and critical frequencies
iirdesign : General filter design using passband and stopband spec
Examples
--------
Design a digital bandstop filter which rejects -60 dB from 0.2*(fs/2) to
0.5*(fs/2), while staying within 3 dB below 0.1*(fs/2) or above
0.6*(fs/2). Plot its frequency response, showing the passband and
stopband constraints in gray.
>>> from scipy import signal
>>> import matplotlib.pyplot as plt
>>> N, Wn = signal.cheb2ord([0.1, 0.6], [0.2, 0.5], 3, 60)
>>> b, a = signal.cheby2(N, 60, Wn, 'stop')
>>> w, h = signal.freqz(b, a)
>>> plt.semilogx(w / np.pi, 20 * np.log10(abs(h)))
>>> plt.title('Chebyshev II bandstop filter fit to constraints')
>>> plt.xlabel('Normalized frequency')
>>> plt.ylabel('Amplitude [dB]')
>>> plt.grid(which='both', axis='both')
>>> plt.fill([.01, .1, .1, .01], [-3, -3, -99, -99], '0.9', lw=0) # stop
>>> plt.fill([.2, .2, .5, .5], [ 9, -60, -60, 9], '0.9', lw=0) # pass
>>> plt.fill([.6, .6, 2, 2], [-99, -3, -3, -99], '0.9', lw=0) # stop
>>> plt.axis([0.06, 1, -80, 3])
>>> plt.show()
"""
_validate_gpass_gstop(gpass, gstop)
wp = atleast_1d(wp)
ws = atleast_1d(ws)
if fs is not None:
if analog:
raise ValueError("fs cannot be specified for an analog filter")
wp = 2*wp/fs
ws = 2*ws/fs
filter_type = 2 * (len(wp) - 1)
if wp[0] < ws[0]:
filter_type += 1
else:
filter_type += 2
# Pre-warp frequencies for digital filter design
if not analog:
passb = tan(pi * wp / 2.0)
stopb = tan(pi * ws / 2.0)
else:
passb = wp * 1.0
stopb = ws * 1.0
if filter_type == 1: # low
nat = stopb / passb
elif filter_type == 2: # high
nat = passb / stopb
elif filter_type == 3: # stop
wp0 = optimize.fminbound(band_stop_obj, passb[0], stopb[0] - 1e-12,
args=(0, passb, stopb, gpass, gstop, 'cheby'),
disp=0)
passb[0] = wp0
wp1 = optimize.fminbound(band_stop_obj, stopb[1] + 1e-12, passb[1],
args=(1, passb, stopb, gpass, gstop, 'cheby'),
disp=0)
passb[1] = wp1
nat = ((stopb * (passb[0] - passb[1])) /
(stopb ** 2 - passb[0] * passb[1]))
elif filter_type == 4: # pass
nat = ((stopb ** 2 - passb[0] * passb[1]) /
(stopb * (passb[0] - passb[1])))
nat = min(abs(nat))
GSTOP = 10 ** (0.1 * abs(gstop))
GPASS = 10 ** (0.1 * abs(gpass))
ord = int(ceil(arccosh(sqrt((GSTOP - 1.0) / (GPASS - 1.0))) /
arccosh(nat)))
# Find frequency where analog response is -gpass dB.
# Then convert back from low-pass prototype to the original filter.
new_freq = cosh(1.0 / ord * arccosh(sqrt((GSTOP - 1.0) / (GPASS - 1.0))))
new_freq = 1.0 / new_freq
if filter_type == 1:
nat = passb / new_freq
elif filter_type == 2:
nat = passb * new_freq
elif filter_type == 3:
nat = numpy.empty(2, float)
nat[0] = (new_freq / 2.0 * (passb[0] - passb[1]) +
sqrt(new_freq ** 2 * (passb[1] - passb[0]) ** 2 / 4.0 +
passb[1] * passb[0]))
nat[1] = passb[1] * passb[0] / nat[0]
elif filter_type == 4:
nat = numpy.empty(2, float)
nat[0] = (1.0 / (2.0 * new_freq) * (passb[0] - passb[1]) +
sqrt((passb[1] - passb[0]) ** 2 / (4.0 * new_freq ** 2) +
passb[1] * passb[0]))
nat[1] = passb[0] * passb[1] / nat[0]
if not analog:
wn = (2.0 / pi) * arctan(nat)
else:
wn = nat
if len(wn) == 1:
wn = wn[0]
if fs is not None:
wn = wn*fs/2
return ord, wn
def ellipord(wp, ws, gpass, gstop, analog=False, fs=None):
"""Elliptic (Cauer) filter order selection.
Return the order of the lowest order digital or analog elliptic filter
that loses no more than `gpass` dB in the passband and has at least
`gstop` dB attenuation in the stopband.
Parameters
----------
wp, ws : float
Passband and stopband edge frequencies.
For digital filters, these are in the same units as `fs`. By default,
`fs` is 2 half-cycles/sample, so these are normalized from 0 to 1,
where 1 is the Nyquist frequency. (`wp` and `ws` are thus in
half-cycles / sample.) For example:
- Lowpass: wp = 0.2, ws = 0.3
- Highpass: wp = 0.3, ws = 0.2
- Bandpass: wp = [0.2, 0.5], ws = [0.1, 0.6]
- Bandstop: wp = [0.1, 0.6], ws = [0.2, 0.5]
For analog filters, `wp` and `ws` are angular frequencies (e.g., rad/s).
gpass : float
The maximum loss in the passband (dB).
gstop : float
The minimum attenuation in the stopband (dB).
analog : bool, optional
When True, return an analog filter, otherwise a digital filter is
returned.
fs : float, optional
The sampling frequency of the digital system.
.. versionadded:: 1.2.0
Returns
-------
ord : int
The lowest order for an Elliptic (Cauer) filter that meets specs.
wn : ndarray or float
The Chebyshev natural frequency (the "3dB frequency") for use with
`ellip` to give filter results. If `fs` is specified,
this is in the same units, and `fs` must also be passed to `ellip`.
See Also
--------
ellip : Filter design using order and critical points
buttord : Find order and critical points from passband and stopband spec
cheb1ord, cheb2ord
iirfilter : General filter design using order and critical frequencies
iirdesign : General filter design using passband and stopband spec
Examples
--------
Design an analog highpass filter such that the passband is within 3 dB
above 30 rad/s, while rejecting -60 dB at 10 rad/s. Plot its
frequency response, showing the passband and stopband constraints in gray.
>>> from scipy import signal
>>> import matplotlib.pyplot as plt
>>> N, Wn = signal.ellipord(30, 10, 3, 60, True)
>>> b, a = signal.ellip(N, 3, 60, Wn, 'high', True)
>>> w, h = signal.freqs(b, a, np.logspace(0, 3, 500))
>>> plt.semilogx(w, 20 * np.log10(abs(h)))
>>> plt.title('Elliptical highpass filter fit to constraints')
>>> plt.xlabel('Frequency [radians / second]')
>>> plt.ylabel('Amplitude [dB]')
>>> plt.grid(which='both', axis='both')
>>> plt.fill([.1, 10, 10, .1], [1e4, 1e4, -60, -60], '0.9', lw=0) # stop
>>> plt.fill([30, 30, 1e9, 1e9], [-99, -3, -3, -99], '0.9', lw=0) # pass
>>> plt.axis([1, 300, -80, 3])
>>> plt.show()
"""
_validate_gpass_gstop(gpass, gstop)
wp = atleast_1d(wp)
ws = atleast_1d(ws)
if fs is not None:
if analog:
raise ValueError("fs cannot be specified for an analog filter")
wp = 2*wp/fs
ws = 2*ws/fs
filter_type = 2 * (len(wp) - 1)
filter_type += 1
if wp[0] >= ws[0]:
filter_type += 1
# Pre-warp frequencies for digital filter design
if not analog:
passb = tan(pi * wp / 2.0)
stopb = tan(pi * ws / 2.0)
else:
passb = wp * 1.0
stopb = ws * 1.0
if filter_type == 1: # low
nat = stopb / passb
elif filter_type == 2: # high
nat = passb / stopb
elif filter_type == 3: # stop
wp0 = optimize.fminbound(band_stop_obj, passb[0], stopb[0] - 1e-12,
args=(0, passb, stopb, gpass, gstop, 'ellip'),
disp=0)
passb[0] = wp0
wp1 = optimize.fminbound(band_stop_obj, stopb[1] + 1e-12, passb[1],
args=(1, passb, stopb, gpass, gstop, 'ellip'),
disp=0)
passb[1] = wp1
nat = ((stopb * (passb[0] - passb[1])) /
(stopb ** 2 - passb[0] * passb[1]))
elif filter_type == 4: # pass
nat = ((stopb ** 2 - passb[0] * passb[1]) /
(stopb * (passb[0] - passb[1])))
nat = min(abs(nat))
GSTOP = 10 ** (0.1 * gstop)
GPASS = 10 ** (0.1 * gpass)
arg1 = sqrt((GPASS - 1.0) / (GSTOP - 1.0))
arg0 = 1.0 / nat
d0 = special.ellipk([arg0 ** 2, 1 - arg0 ** 2])
d1 = special.ellipk([arg1 ** 2, 1 - arg1 ** 2])
ord = int(ceil(d0[0] * d1[1] / (d0[1] * d1[0])))
if not analog:
wn = arctan(passb) * 2.0 / pi
else:
wn = passb
if len(wn) == 1:
wn = wn[0]
if fs is not None:
wn = wn*fs/2
return ord, wn
def buttap(N):
"""Return (z,p,k) for analog prototype of Nth-order Butterworth filter.
The filter will have an angular (e.g., rad/s) cutoff frequency of 1.
See Also
--------
butter : Filter design function using this prototype
"""
if abs(int(N)) != N:
raise ValueError("Filter order must be a nonnegative integer")
z = numpy.array([])
m = numpy.arange(-N+1, N, 2)
# Middle value is 0 to ensure an exactly real pole
p = -numpy.exp(1j * pi * m / (2 * N))
k = 1
return z, p, k
def cheb1ap(N, rp):
"""
Return (z,p,k) for Nth-order Chebyshev type I analog lowpass filter.
The returned filter prototype has `rp` decibels of ripple in the passband.
The filter's angular (e.g. rad/s) cutoff frequency is normalized to 1,
defined as the point at which the gain first drops below ``-rp``.
See Also
--------
cheby1 : Filter design function using this prototype
"""
if abs(int(N)) != N:
raise ValueError("Filter order must be a nonnegative integer")
elif N == 0:
# Avoid divide-by-zero error
# Even order filters have DC gain of -rp dB
return numpy.array([]), numpy.array([]), 10**(-rp/20)
z = numpy.array([])
# Ripple factor (epsilon)
eps = numpy.sqrt(10 ** (0.1 * rp) - 1.0)
mu = 1.0 / N * arcsinh(1 / eps)
# Arrange poles in an ellipse on the left half of the S-plane
m = numpy.arange(-N+1, N, 2)
theta = pi * m / (2*N)
p = -sinh(mu + 1j*theta)
k = numpy.prod(-p, axis=0).real
if N % 2 == 0:
k = k / sqrt((1 + eps * eps))
return z, p, k
def cheb2ap(N, rs):
"""
Return (z,p,k) for Nth-order Chebyshev type I analog lowpass filter.
The returned filter prototype has `rs` decibels of ripple in the stopband.
The filter's angular (e.g. rad/s) cutoff frequency is normalized to 1,
defined as the point at which the gain first reaches ``-rs``.
See Also
--------
cheby2 : Filter design function using this prototype
"""
if abs(int(N)) != N:
raise ValueError("Filter order must be a nonnegative integer")
elif N == 0:
# Avoid divide-by-zero warning
return numpy.array([]), numpy.array([]), 1
# Ripple factor (epsilon)
de = 1.0 / sqrt(10 ** (0.1 * rs) - 1)
mu = arcsinh(1.0 / de) / N
if N % 2:
m = numpy.concatenate((numpy.arange(-N+1, 0, 2),
numpy.arange(2, N, 2)))
else:
m = numpy.arange(-N+1, N, 2)
z = -conjugate(1j / sin(m * pi / (2.0 * N)))
# Poles around the unit circle like Butterworth
p = -exp(1j * pi * numpy.arange(-N+1, N, 2) / (2 * N))
# Warp into Chebyshev II
p = sinh(mu) * p.real + 1j * cosh(mu) * p.imag
p = 1.0 / p
k = (numpy.prod(-p, axis=0) / numpy.prod(-z, axis=0)).real
return z, p, k
EPSILON = 2e-16
def _vratio(u, ineps, mp):
[s, c, d, phi] = special.ellipj(u, mp)
ret = abs(ineps - s / c)
return ret
def _kratio(m, k_ratio):
m = float(m)
if m < 0:
m = 0.0
if m > 1:
m = 1.0
if abs(m) > EPSILON and (abs(m) + EPSILON) < 1:
k = special.ellipk([m, 1 - m])
r = k[0] / k[1] - k_ratio
elif abs(m) > EPSILON:
r = -k_ratio
else:
r = 1e20
return abs(r)
def ellipap(N, rp, rs):
"""Return (z,p,k) of Nth-order elliptic analog lowpass filter.
The filter is a normalized prototype that has `rp` decibels of ripple
in the passband and a stopband `rs` decibels down.
The filter's angular (e.g., rad/s) cutoff frequency is normalized to 1,
defined as the point at which the gain first drops below ``-rp``.
See Also
--------
ellip : Filter design function using this prototype
References
----------
.. [1] Lutova, Tosic, and Evans, "Filter Design for Signal Processing",
Chapters 5 and 12.
"""
if abs(int(N)) != N:
raise ValueError("Filter order must be a nonnegative integer")
elif N == 0:
# Avoid divide-by-zero warning
# Even order filters have DC gain of -rp dB
return numpy.array([]), numpy.array([]), 10**(-rp/20)
elif N == 1:
p = -sqrt(1.0 / (10 ** (0.1 * rp) - 1.0))
k = -p
z = []
return asarray(z), asarray(p), k
eps = numpy.sqrt(10 ** (0.1 * rp) - 1)
ck1 = eps / numpy.sqrt(10 ** (0.1 * rs) - 1)
ck1p = numpy.sqrt(1 - ck1 * ck1)
if ck1p == 1:
raise ValueError("Cannot design a filter with given rp and rs"
" specifications.")
val = special.ellipk([ck1 * ck1, ck1p * ck1p])
if abs(1 - ck1p * ck1p) < EPSILON:
krat = 0
else:
krat = N * val[0] / val[1]
m = optimize.fmin(_kratio, [0.5], args=(krat,), maxfun=250, maxiter=250,
disp=0)
if m < 0 or m > 1:
m = optimize.fminbound(_kratio, 0, 1, args=(krat,), maxfun=250,
disp=0)
capk = special.ellipk(m)
j = numpy.arange(1 - N % 2, N, 2)
jj = len(j)
[s, c, d, phi] = special.ellipj(j * capk / N, m * numpy.ones(jj))
snew = numpy.compress(abs(s) > EPSILON, s, axis=-1)
z = 1.0 / (sqrt(m) * snew)
z = 1j * z
z = numpy.concatenate((z, conjugate(z)))
r = optimize.fmin(_vratio, special.ellipk(m), args=(1. / eps, ck1p * ck1p),
maxfun=250, maxiter=250, disp=0)
v0 = capk * r / (N * val[0])
[sv, cv, dv, phi] = special.ellipj(v0, 1 - m)
p = -(c * d * sv * cv + 1j * s * dv) / (1 - (d * sv) ** 2.0)
if N % 2:
newp = numpy.compress(abs(p.imag) > EPSILON *
numpy.sqrt(numpy.sum(p * numpy.conjugate(p),
axis=0).real),
p, axis=-1)
p = numpy.concatenate((p, conjugate(newp)))
else:
p = numpy.concatenate((p, conjugate(p)))
k = (numpy.prod(-p, axis=0) / numpy.prod(-z, axis=0)).real
if N % 2 == 0:
k = k / numpy.sqrt((1 + eps * eps))
return z, p, k
# TODO: Make this a real public function scipy.misc.ff
def _falling_factorial(x, n):
r"""
Return the factorial of `x` to the `n` falling.
This is defined as:
.. math:: x^\underline n = (x)_n = x (x-1) \cdots (x-n+1)
This can more efficiently calculate ratios of factorials, since:
n!/m! == falling_factorial(n, n-m)
where n >= m
skipping the factors that cancel out
the usual factorial n! == ff(n, n)
"""
val = 1
for k in range(x - n + 1, x + 1):
val *= k
return val
def _bessel_poly(n, reverse=False):
"""
Return the coefficients of Bessel polynomial of degree `n`
If `reverse` is true, a reverse Bessel polynomial is output.
Output is a list of coefficients:
[1] = 1
[1, 1] = 1*s + 1
[1, 3, 3] = 1*s^2 + 3*s + 3
[1, 6, 15, 15] = 1*s^3 + 6*s^2 + 15*s + 15
[1, 10, 45, 105, 105] = 1*s^4 + 10*s^3 + 45*s^2 + 105*s + 105
etc.
Output is a Python list of arbitrary precision long ints, so n is only
limited by your hardware's memory.
Sequence is http://oeis.org/A001498, and output can be confirmed to
match http://oeis.org/A001498/b001498.txt :
>>> i = 0
>>> for n in range(51):
... for x in _bessel_poly(n, reverse=True):
... print(i, x)
... i += 1
"""
if abs(int(n)) != n:
raise ValueError("Polynomial order must be a nonnegative integer")
else:
n = int(n) # np.int32 doesn't work, for instance
out = []
for k in range(n + 1):
num = _falling_factorial(2*n - k, n)
den = 2**(n - k) * math.factorial(k)
out.append(num // den)
if reverse:
return out[::-1]
else:
return out
def _campos_zeros(n):
"""
Return approximate zero locations of Bessel polynomials y_n(x) for order
`n` using polynomial fit (Campos-Calderon 2011)
"""
if n == 1:
return asarray([-1+0j])
s = npp_polyval(n, [0, 0, 2, 0, -3, 1])
b3 = npp_polyval(n, [16, -8]) / s
b2 = npp_polyval(n, [-24, -12, 12]) / s
b1 = npp_polyval(n, [8, 24, -12, -2]) / s
b0 = npp_polyval(n, [0, -6, 0, 5, -1]) / s
r = npp_polyval(n, [0, 0, 2, 1])
a1 = npp_polyval(n, [-6, -6]) / r
a2 = 6 / r
k = np.arange(1, n+1)
x = npp_polyval(k, [0, a1, a2])
y = npp_polyval(k, [b0, b1, b2, b3])
return x + 1j*y
def _aberth(f, fp, x0, tol=1e-15, maxiter=50):
"""
Given a function `f`, its first derivative `fp`, and a set of initial
guesses `x0`, simultaneously find the roots of the polynomial using the
Aberth-Ehrlich method.
``len(x0)`` should equal the number of roots of `f`.
(This is not a complete implementation of Bini's algorithm.)
"""
N = len(x0)
x = array(x0, complex)
beta = np.empty_like(x0)
for iteration in range(maxiter):
alpha = -f(x) / fp(x) # Newton's method
# Model "repulsion" between zeros
for k in range(N):
beta[k] = np.sum(1/(x[k] - x[k+1:]))
beta[k] += np.sum(1/(x[k] - x[:k]))
x += alpha / (1 + alpha * beta)
if not all(np.isfinite(x)):
raise RuntimeError('Root-finding calculation failed')
# Mekwi: The iterative process can be stopped when |hn| has become
# less than the largest error one is willing to permit in the root.
if all(abs(alpha) <= tol):
break
else:
raise Exception('Zeros failed to converge')
return x
def _bessel_zeros(N):
"""
Find zeros of ordinary Bessel polynomial of order `N`, by root-finding of
modified Bessel function of the second kind
"""
if N == 0:
return asarray([])
# Generate starting points
x0 = _campos_zeros(N)
# Zeros are the same for exp(1/x)*K_{N+0.5}(1/x) and Nth-order ordinary
# Bessel polynomial y_N(x)
def f(x):
return special.kve(N+0.5, 1/x)
# First derivative of above
def fp(x):
return (special.kve(N-0.5, 1/x)/(2*x**2) -
special.kve(N+0.5, 1/x)/(x**2) +
special.kve(N+1.5, 1/x)/(2*x**2))
# Starting points converge to true zeros
x = _aberth(f, fp, x0)
# Improve precision using Newton's method on each
for i in range(len(x)):
x[i] = optimize.newton(f, x[i], fp, tol=1e-15)
# Average complex conjugates to make them exactly symmetrical
x = np.mean((x, x[::-1].conj()), 0)
# Zeros should sum to -1
if abs(np.sum(x) + 1) > 1e-15:
raise RuntimeError('Generated zeros are inaccurate')
return x
def _norm_factor(p, k):
"""
Numerically find frequency shift to apply to delay-normalized filter such
that -3 dB point is at 1 rad/sec.
`p` is an array_like of polynomial poles
`k` is a float gain
First 10 values are listed in "Bessel Scale Factors" table,
"Bessel Filters Polynomials, Poles and Circuit Elements 2003, C. Bond."
"""
p = asarray(p, dtype=complex)
def G(w):
"""
Gain of filter
"""
return abs(k / prod(1j*w - p))
def cutoff(w):
"""
When gain = -3 dB, return 0
"""
return G(w) - 1/np.sqrt(2)
return optimize.newton(cutoff, 1.5)
def besselap(N, norm='phase'):
"""
Return (z,p,k) for analog prototype of an Nth-order Bessel filter.
Parameters
----------
N : int
The order of the filter.
norm : {'phase', 'delay', 'mag'}, optional
Frequency normalization:
``phase``
The filter is normalized such that the phase response reaches its
midpoint at an angular (e.g., rad/s) cutoff frequency of 1. This
happens for both low-pass and high-pass filters, so this is the
"phase-matched" case. [6]_
The magnitude response asymptotes are the same as a Butterworth
filter of the same order with a cutoff of `Wn`.
This is the default, and matches MATLAB's implementation.
``delay``
The filter is normalized such that the group delay in the passband
is 1 (e.g., 1 second). This is the "natural" type obtained by
solving Bessel polynomials
``mag``
The filter is normalized such that the gain magnitude is -3 dB at
angular frequency 1. This is called "frequency normalization" by
Bond. [1]_
.. versionadded:: 0.18.0
Returns
-------
z : ndarray
Zeros of the transfer function. Is always an empty array.
p : ndarray
Poles of the transfer function.
k : scalar
Gain of the transfer function. For phase-normalized, this is always 1.
See Also
--------
bessel : Filter design function using this prototype
Notes
-----
To find the pole locations, approximate starting points are generated [2]_
for the zeros of the ordinary Bessel polynomial [3]_, then the
Aberth-Ehrlich method [4]_ [5]_ is used on the Kv(x) Bessel function to
calculate more accurate zeros, and these locations are then inverted about
the unit circle.
References
----------
.. [1] C.R. Bond, "Bessel Filter Constants",
http://www.crbond.com/papers/bsf.pdf
.. [2] Campos and Calderon, "Approximate closed-form formulas for the
zeros of the Bessel Polynomials", :arXiv:`1105.0957`.
.. [3] Thomson, W.E., "Delay Networks having Maximally Flat Frequency
Characteristics", Proceedings of the Institution of Electrical
Engineers, Part III, November 1949, Vol. 96, No. 44, pp. 487-490.
.. [4] Aberth, "Iteration Methods for Finding all Zeros of a Polynomial
Simultaneously", Mathematics of Computation, Vol. 27, No. 122,
April 1973
.. [5] Ehrlich, "A modified Newton method for polynomials", Communications
of the ACM, Vol. 10, Issue 2, pp. 107-108, Feb. 1967,
:DOI:`10.1145/363067.363115`
.. [6] Miller and Bohn, "A Bessel Filter Crossover, and Its Relation to
Others", RaneNote 147, 1998, https://www.ranecommercial.com/legacy/note147.html
"""
if abs(int(N)) != N:
raise ValueError("Filter order must be a nonnegative integer")
if N == 0:
p = []
k = 1
else:
# Find roots of reverse Bessel polynomial
p = 1/_bessel_zeros(N)
a_last = _falling_factorial(2*N, N) // 2**N
# Shift them to a different normalization if required
if norm in ('delay', 'mag'):
# Normalized for group delay of 1
k = a_last
if norm == 'mag':
# -3 dB magnitude point is at 1 rad/sec
norm_factor = _norm_factor(p, k)
p /= norm_factor
k = norm_factor**-N * a_last
elif norm == 'phase':
# Phase-matched (1/2 max phase shift at 1 rad/sec)
# Asymptotes are same as Butterworth filter
p *= 10**(-math.log10(a_last)/N)
k = 1
else:
raise ValueError('normalization not understood')
return asarray([]), asarray(p, dtype=complex), float(k)
def iirnotch(w0, Q, fs=2.0):
"""
Design second-order IIR notch digital filter.
A notch filter is a band-stop filter with a narrow bandwidth
(high quality factor). It rejects a narrow frequency band and
leaves the rest of the spectrum little changed.
Parameters
----------
w0 : float
Frequency to remove from a signal. If `fs` is specified, this is in
the same units as `fs`. By default, it is a normalized scalar that must
satisfy ``0 < w0 < 1``, with ``w0 = 1`` corresponding to half of the
sampling frequency.
Q : float
Quality factor. Dimensionless parameter that characterizes
notch filter -3 dB bandwidth ``bw`` relative to its center
frequency, ``Q = w0/bw``.
fs : float, optional
The sampling frequency of the digital system.
.. versionadded:: 1.2.0
Returns
-------
b, a : ndarray, ndarray
Numerator (``b``) and denominator (``a``) polynomials
of the IIR filter.
See Also
--------
iirpeak
Notes
-----
.. versionadded:: 0.19.0
References
----------
.. [1] Sophocles J. Orfanidis, "Introduction To Signal Processing",
Prentice-Hall, 1996
Examples
--------
Design and plot filter to remove the 60 Hz component from a
signal sampled at 200 Hz, using a quality factor Q = 30
>>> from scipy import signal
>>> import matplotlib.pyplot as plt
>>> fs = 200.0 # Sample frequency (Hz)
>>> f0 = 60.0 # Frequency to be removed from signal (Hz)
>>> Q = 30.0 # Quality factor
>>> # Design notch filter
>>> b, a = signal.iirnotch(f0, Q, fs)
>>> # Frequency response
>>> freq, h = signal.freqz(b, a, fs=fs)
>>> # Plot
>>> fig, ax = plt.subplots(2, 1, figsize=(8, 6))
>>> ax[0].plot(freq, 20*np.log10(abs(h)), color='blue')
>>> ax[0].set_title("Frequency Response")
>>> ax[0].set_ylabel("Amplitude (dB)", color='blue')
>>> ax[0].set_xlim([0, 100])
>>> ax[0].set_ylim([-25, 10])
>>> ax[0].grid()
>>> ax[1].plot(freq, np.unwrap(np.angle(h))*180/np.pi, color='green')
>>> ax[1].set_ylabel("Angle (degrees)", color='green')
>>> ax[1].set_xlabel("Frequency (Hz)")
>>> ax[1].set_xlim([0, 100])
>>> ax[1].set_yticks([-90, -60, -30, 0, 30, 60, 90])
>>> ax[1].set_ylim([-90, 90])
>>> ax[1].grid()
>>> plt.show()
"""
return _design_notch_peak_filter(w0, Q, "notch", fs)
def iirpeak(w0, Q, fs=2.0):
"""
Design second-order IIR peak (resonant) digital filter.
A peak filter is a band-pass filter with a narrow bandwidth
(high quality factor). It rejects components outside a narrow
frequency band.
Parameters
----------
w0 : float
Frequency to be retained in a signal. If `fs` is specified, this is in
the same units as `fs`. By default, it is a normalized scalar that must
satisfy ``0 < w0 < 1``, with ``w0 = 1`` corresponding to half of the
sampling frequency.
Q : float
Quality factor. Dimensionless parameter that characterizes
peak filter -3 dB bandwidth ``bw`` relative to its center
frequency, ``Q = w0/bw``.
fs : float, optional
The sampling frequency of the digital system.
.. versionadded:: 1.2.0
Returns
-------
b, a : ndarray, ndarray
Numerator (``b``) and denominator (``a``) polynomials
of the IIR filter.
See Also
--------
iirnotch
Notes
-----
.. versionadded:: 0.19.0
References
----------
.. [1] Sophocles J. Orfanidis, "Introduction To Signal Processing",
Prentice-Hall, 1996
Examples
--------
Design and plot filter to remove the frequencies other than the 300 Hz
component from a signal sampled at 1000 Hz, using a quality factor Q = 30
>>> from scipy import signal
>>> import matplotlib.pyplot as plt
>>> fs = 1000.0 # Sample frequency (Hz)
>>> f0 = 300.0 # Frequency to be retained (Hz)
>>> Q = 30.0 # Quality factor
>>> # Design peak filter
>>> b, a = signal.iirpeak(f0, Q, fs)
>>> # Frequency response
>>> freq, h = signal.freqz(b, a, fs=fs)
>>> # Plot
>>> fig, ax = plt.subplots(2, 1, figsize=(8, 6))
>>> ax[0].plot(freq, 20*np.log10(np.maximum(abs(h), 1e-5)), color='blue')
>>> ax[0].set_title("Frequency Response")
>>> ax[0].set_ylabel("Amplitude (dB)", color='blue')
>>> ax[0].set_xlim([0, 500])
>>> ax[0].set_ylim([-50, 10])
>>> ax[0].grid()
>>> ax[1].plot(freq, np.unwrap(np.angle(h))*180/np.pi, color='green')
>>> ax[1].set_ylabel("Angle (degrees)", color='green')
>>> ax[1].set_xlabel("Frequency (Hz)")
>>> ax[1].set_xlim([0, 500])
>>> ax[1].set_yticks([-90, -60, -30, 0, 30, 60, 90])
>>> ax[1].set_ylim([-90, 90])
>>> ax[1].grid()
>>> plt.show()
"""
return _design_notch_peak_filter(w0, Q, "peak", fs)
def _design_notch_peak_filter(w0, Q, ftype, fs=2.0):
"""
Design notch or peak digital filter.
Parameters
----------
w0 : float
Normalized frequency to remove from a signal. If `fs` is specified,
this is in the same units as `fs`. By default, it is a normalized
scalar that must satisfy ``0 < w0 < 1``, with ``w0 = 1``
corresponding to half of the sampling frequency.
Q : float
Quality factor. Dimensionless parameter that characterizes
notch filter -3 dB bandwidth ``bw`` relative to its center
frequency, ``Q = w0/bw``.
ftype : str
The type of IIR filter to design:
- notch filter : ``notch``
- peak filter : ``peak``
fs : float, optional
The sampling frequency of the digital system.
.. versionadded:: 1.2.0:
Returns
-------
b, a : ndarray, ndarray
Numerator (``b``) and denominator (``a``) polynomials
of the IIR filter.
"""
# Guarantee that the inputs are floats
w0 = float(w0)
Q = float(Q)
w0 = 2*w0/fs
# Checks if w0 is within the range
if w0 > 1.0 or w0 < 0.0:
raise ValueError("w0 should be such that 0 < w0 < 1")
# Get bandwidth
bw = w0/Q
# Normalize inputs
bw = bw*np.pi
w0 = w0*np.pi
# Compute -3dB attenuation
gb = 1/np.sqrt(2)
if ftype == "notch":
# Compute beta: formula 11.3.4 (p.575) from reference [1]
beta = (np.sqrt(1.0-gb**2.0)/gb)*np.tan(bw/2.0)
elif ftype == "peak":
# Compute beta: formula 11.3.19 (p.579) from reference [1]
beta = (gb/np.sqrt(1.0-gb**2.0))*np.tan(bw/2.0)
else:
raise ValueError("Unknown ftype.")
# Compute gain: formula 11.3.6 (p.575) from reference [1]
gain = 1.0/(1.0+beta)
# Compute numerator b and denominator a
# formulas 11.3.7 (p.575) and 11.3.21 (p.579)
# from reference [1]
if ftype == "notch":
b = gain*np.array([1.0, -2.0*np.cos(w0), 1.0])
else:
b = (1.0-gain)*np.array([1.0, 0.0, -1.0])
a = np.array([1.0, -2.0*gain*np.cos(w0), (2.0*gain-1.0)])
return b, a
def iircomb(w0, Q, ftype='notch', fs=2.0):
"""
Design IIR notching or peaking digital comb filter.
A notching comb filter is a band-stop filter with a narrow bandwidth
(high quality factor). It rejects a narrow frequency band and
leaves the rest of the spectrum little changed.
A peaking comb filter is a band-pass filter with a narrow bandwidth
(high quality factor). It rejects components outside a narrow
frequency band.
Parameters
----------
w0 : float
Frequency to attenuate (notching) or boost (peaking). If `fs` is
specified, this is in the same units as `fs`. By default, it is
a normalized scalar that must satisfy ``0 < w0 < 1``, with
``w0 = 1`` corresponding to half of the sampling frequency.
Q : float
Quality factor. Dimensionless parameter that characterizes
notch filter -3 dB bandwidth ``bw`` relative to its center
frequency, ``Q = w0/bw``.
ftype : {'notch', 'peak'}
The type of comb filter generated by the function. If 'notch', then
it returns a filter with notches at frequencies ``0``, ``w0``,
``2 * w0``, etc. If 'peak', then it returns a filter with peaks at
frequencies ``0.5 * w0``, ``1.5 * w0``, ``2.5 * w0```, etc.
Default is 'notch'.
fs : float, optional
The sampling frequency of the signal. Default is 2.0.
Returns
-------
b, a : ndarray, ndarray
Numerator (``b``) and denominator (``a``) polynomials
of the IIR filter.
Raises
------
ValueError
If `w0` is less than or equal to 0 or greater than or equal to
``fs/2``, if `fs` is not divisible by `w0`, if `ftype`
is not 'notch' or 'peak'
See Also
--------
iirnotch
iirpeak
Notes
-----
For implementation details, see [1]_. The TF implementation of the
comb filter is numerically stable even at higher orders due to the
use of a single repeated pole, which won't suffer from precision loss.
References
----------
.. [1] Sophocles J. Orfanidis, "Introduction To Signal Processing",
Prentice-Hall, 1996
Examples
--------
Design and plot notching comb filter at 20 Hz for a
signal sampled at 200 Hz, using quality factor Q = 30
>>> from scipy import signal
>>> import matplotlib.pyplot as plt
>>> fs = 200.0 # Sample frequency (Hz)
>>> f0 = 20.0 # Frequency to be removed from signal (Hz)
>>> Q = 30.0 # Quality factor
>>> # Design notching comb filter
>>> b, a = signal.iircomb(f0, Q, ftype='notch', fs=fs)
>>> # Frequency response
>>> freq, h = signal.freqz(b, a, fs=fs)
>>> response = abs(h)
>>> # To avoid divide by zero when graphing
>>> response[response == 0] = 1e-20
>>> # Plot
>>> fig, ax = plt.subplots(2, 1, figsize=(8, 6))
>>> ax[0].plot(freq, 20*np.log10(abs(response)), color='blue')
>>> ax[0].set_title("Frequency Response")
>>> ax[0].set_ylabel("Amplitude (dB)", color='blue')
>>> ax[0].set_xlim([0, 100])
>>> ax[0].set_ylim([-30, 10])
>>> ax[0].grid()
>>> ax[1].plot(freq, np.unwrap(np.angle(h))*180/np.pi, color='green')
>>> ax[1].set_ylabel("Angle (degrees)", color='green')
>>> ax[1].set_xlabel("Frequency (Hz)")
>>> ax[1].set_xlim([0, 100])
>>> ax[1].set_yticks([-90, -60, -30, 0, 30, 60, 90])
>>> ax[1].set_ylim([-90, 90])
>>> ax[1].grid()
>>> plt.show()
Design and plot peaking comb filter at 250 Hz for a
signal sampled at 1000 Hz, using quality factor Q = 30
>>> fs = 1000.0 # Sample frequency (Hz)
>>> f0 = 250.0 # Frequency to be retained (Hz)
>>> Q = 30.0 # Quality factor
>>> # Design peaking filter
>>> b, a = signal.iircomb(f0, Q, ftype='peak', fs=fs)
>>> # Frequency response
>>> freq, h = signal.freqz(b, a, fs=fs)
>>> response = abs(h)
>>> # To avoid divide by zero when graphing
>>> response[response == 0] = 1e-20
>>> # Plot
>>> fig, ax = plt.subplots(2, 1, figsize=(8, 6))
>>> ax[0].plot(freq, 20*np.log10(np.maximum(abs(h), 1e-5)), color='blue')
>>> ax[0].set_title("Frequency Response")
>>> ax[0].set_ylabel("Amplitude (dB)", color='blue')
>>> ax[0].set_xlim([0, 500])
>>> ax[0].set_ylim([-80, 10])
>>> ax[0].grid()
>>> ax[1].plot(freq, np.unwrap(np.angle(h))*180/np.pi, color='green')
>>> ax[1].set_ylabel("Angle (degrees)", color='green')
>>> ax[1].set_xlabel("Frequency (Hz)")
>>> ax[1].set_xlim([0, 500])
>>> ax[1].set_yticks([-90, -60, -30, 0, 30, 60, 90])
>>> ax[1].set_ylim([-90, 90])
>>> ax[1].grid()
>>> plt.show()
"""
# Convert w0, Q, and fs to float
w0 = float(w0)
Q = float(Q)
fs = float(fs)
# Check for invalid cutoff frequency or filter type
ftype = ftype.lower()
filter_types = ['notch', 'peak']
if not 0 < w0 < fs / 2:
raise ValueError("w0 must be between 0 and {}"
" (nyquist), but given {}.".format(fs / 2, w0))
if np.round(fs % w0) != 0:
raise ValueError('fs must be divisible by w0.')
if ftype not in filter_types:
raise ValueError('ftype must be either notch or peak.')
# Compute the order of the filter
N = int(fs // w0)
# Compute frequency in radians and filter bandwith
# Eq. 11.3.1 (p. 574) from reference [1]
w0 = (2 * np.pi * w0) / fs
w_delta = w0 / Q
# Define base gain values depending on notch or peak filter
# Compute -3dB attenuation
# Eqs. 11.4.1 and 11.4.2 (p. 582) from reference [1]
if ftype == 'notch':
G0, G = [1, 0]
elif ftype == 'peak':
G0, G = [0, 1]
GB = 1 / np.sqrt(2)
# Compute beta
# Eq. 11.5.3 (p. 591) from reference [1]
beta = np.sqrt((GB**2 - G0**2) / (G**2 - GB**2)) * np.tan(N * w_delta / 4)
# Compute filter coefficients
# Eq 11.5.1 (p. 590) variables a, b, c from reference [1]
ax = (1 - beta) / (1 + beta)
bx = (G0 + G * beta) / (1 + beta)
cx = (G0 - G * beta) / (1 + beta)
# Compute numerator coefficients
# Eq 11.5.1 (p. 590) or Eq 11.5.4 (p. 591) from reference [1]
# b - cz^-N or b + cz^-N
b = np.zeros(N + 1)
b[0] = bx
b[-1] = cx
if ftype == 'notch':
b[-1] = -cx
# Compute denominator coefficients
# Eq 11.5.1 (p. 590) or Eq 11.5.4 (p. 591) from reference [1]
# 1 - az^-N or 1 + az^-N
a = np.zeros(N + 1)
a[0] = 1
a[-1] = ax
if ftype == 'notch':
a[-1] = -ax
return b, a
def _hz_to_erb(hz):
"""
Utility for converting from frequency (Hz) to the
Equivalent Rectangular Bandwith (ERB) scale
ERB = frequency / EarQ + minBW
"""
EarQ = 9.26449
minBW = 24.7
return hz / EarQ + minBW
def gammatone(freq, ftype, order=None, numtaps=None, fs=None):
"""
Gammatone filter design.
This function computes the coefficients of an FIR or IIR gammatone
digital filter [1]_.
Parameters
----------
freq : float
Center frequency of the filter (expressed in the same units
as `fs`).
ftype : {'fir', 'iir'}
The type of filter the function generates. If 'fir', the function
will generate an Nth order FIR gammatone filter. If 'iir', the
function will generate an 8th order digital IIR filter, modeled as
as 4th order gammatone filter.
order : int, optional
The order of the filter. Only used when ``ftype='fir'``.
Default is 4 to model the human auditory system. Must be between
0 and 24.
numtaps : int, optional
Length of the filter. Only used when ``ftype='fir'``.
Default is ``fs*0.015`` if `fs` is greater than 1000,
15 if `fs` is less than or equal to 1000.
fs : float, optional
The sampling frequency of the signal. `freq` must be between
0 and ``fs/2``. Default is 2.
Returns
-------
b, a : ndarray, ndarray
Numerator (``b``) and denominator (``a``) polynomials of the filter.
Raises
------
ValueError
If `freq` is less than or equal to 0 or greater than or equal to
``fs/2``, if `ftype` is not 'fir' or 'iir', if `order` is less than
or equal to 0 or greater than 24 when ``ftype='fir'``
See Also
--------
firwin
iirfilter
References
----------
.. [1] Slaney, Malcolm, "An Efficient Implementation of the
Patterson-Holdsworth Auditory Filter Bank", Apple Computer
Technical Report 35, 1993, pp.3-8, 34-39.
Examples
--------
16-sample 4th order FIR Gammatone filter centered at 440 Hz
>>> from scipy import signal
>>> signal.gammatone(440, 'fir', numtaps=16, fs=16000)
(array([ 0.00000000e+00, 2.22196719e-07, 1.64942101e-06, 4.99298227e-06,
1.01993969e-05, 1.63125770e-05, 2.14648940e-05, 2.29947263e-05,
1.76776931e-05, 2.04980537e-06, -2.72062858e-05, -7.28455299e-05,
-1.36651076e-04, -2.19066855e-04, -3.18905076e-04, -4.33156712e-04]),
[1.0])
IIR Gammatone filter centered at 440 Hz
>>> from scipy import signal
>>> import matplotlib.pyplot as plt
>>> b, a = signal.gammatone(440, 'iir', fs=16000)
>>> w, h = signal.freqz(b, a)
>>> plt.plot(w / ((2 * np.pi) / 16000), 20 * np.log10(abs(h)))
>>> plt.xscale('log')
>>> plt.title('Gammatone filter frequency response')
>>> plt.xlabel('Frequency')
>>> plt.ylabel('Amplitude [dB]')
>>> plt.margins(0, 0.1)
>>> plt.grid(which='both', axis='both')
>>> plt.axvline(440, color='green') # cutoff frequency
>>> plt.show()
"""
# Converts freq to float
freq = float(freq)
# Set sampling rate if not passed
if fs is None:
fs = 2
fs = float(fs)
# Check for invalid cutoff frequency or filter type
ftype = ftype.lower()
filter_types = ['fir', 'iir']
if not 0 < freq < fs / 2:
raise ValueError("The frequency must be between 0 and {}"
" (nyquist), but given {}.".format(fs / 2, freq))
if ftype not in filter_types:
raise ValueError('ftype must be either fir or iir.')
# Calculate FIR gammatone filter
if ftype == 'fir':
# Set order and numtaps if not passed
if order is None:
order = 4
order = operator.index(order)
if numtaps is None:
numtaps = max(int(fs * 0.015), 15)
numtaps = operator.index(numtaps)
# Check for invalid order
if not 0 < order <= 24:
raise ValueError("Invalid order: order must be > 0 and <= 24.")
# Gammatone impulse response settings
t = np.arange(numtaps) / fs
bw = 1.019 * _hz_to_erb(freq)
# Calculate the FIR gammatone filter
b = (t ** (order - 1)) * np.exp(-2 * np.pi * bw * t)
b *= np.cos(2 * np.pi * freq * t)
# Scale the FIR filter so the frequency response is 1 at cutoff
scale_factor = 2 * (2 * np.pi * bw) ** (order)
scale_factor /= float_factorial(order - 1)
scale_factor /= fs
b *= scale_factor
a = [1.0]
# Calculate IIR gammatone filter
elif ftype == 'iir':
# Raise warning if order and/or numtaps is passed
if order is not None:
warnings.warn('order is not used for IIR gammatone filter.')
if numtaps is not None:
warnings.warn('numtaps is not used for IIR gammatone filter.')
# Gammatone impulse response settings
T = 1./fs
bw = 2 * np.pi * 1.019 * _hz_to_erb(freq)
fr = 2 * freq * np.pi * T
bwT = bw * T
# Calculate the gain to normalize the volume at the center frequency
g1 = -2 * np.exp(2j * fr) * T
g2 = 2 * np.exp(-(bwT) + 1j * fr) * T
g3 = np.sqrt(3 + 2 ** (3 / 2)) * np.sin(fr)
g4 = np.sqrt(3 - 2 ** (3 / 2)) * np.sin(fr)
g5 = np.exp(2j * fr)
g = g1 + g2 * (np.cos(fr) - g4)
g *= (g1 + g2 * (np.cos(fr) + g4))
g *= (g1 + g2 * (np.cos(fr) - g3))
g *= (g1 + g2 * (np.cos(fr) + g3))
g /= ((-2 / np.exp(2 * bwT) - 2 * g5 + 2 * (1 + g5) / np.exp(bwT)) ** 4)
g = np.abs(g)
# Create empty filter coefficient lists
b = np.empty(5)
a = np.empty(9)
# Calculate the numerator coefficients
b[0] = (T ** 4) / g
b[1] = -4 * T ** 4 * np.cos(fr) / np.exp(bw * T) / g
b[2] = 6 * T ** 4 * np.cos(2 * fr) / np.exp(2 * bw * T) / g
b[3] = -4 * T ** 4 * np.cos(3 * fr) / np.exp(3 * bw * T) / g
b[4] = T ** 4 * np.cos(4 * fr) / np.exp(4 * bw * T) / g
# Calculate the denominator coefficients
a[0] = 1
a[1] = -8 * np.cos(fr) / np.exp(bw * T)
a[2] = 4 * (4 + 3 * np.cos(2 * fr)) / np.exp(2 * bw * T)
a[3] = -8 * (6 * np.cos(fr) + np.cos(3 * fr))
a[3] /= np.exp(3 * bw * T)
a[4] = 2 * (18 + 16 * np.cos(2 * fr) + np.cos(4 * fr))
a[4] /= np.exp(4 * bw * T)
a[5] = -8 * (6 * np.cos(fr) + np.cos(3 * fr))
a[5] /= np.exp(5 * bw * T)
a[6] = 4 * (4 + 3 * np.cos(2 * fr)) / np.exp(6 * bw * T)
a[7] = -8 * np.cos(fr) / np.exp(7 * bw * T)
a[8] = np.exp(-8 * bw * T)
return b, a
filter_dict = {'butter': [buttap, buttord],
'butterworth': [buttap, buttord],
'cauer': [ellipap, ellipord],
'elliptic': [ellipap, ellipord],
'ellip': [ellipap, ellipord],
'bessel': [besselap],
'bessel_phase': [besselap],
'bessel_delay': [besselap],
'bessel_mag': [besselap],
'cheby1': [cheb1ap, cheb1ord],
'chebyshev1': [cheb1ap, cheb1ord],
'chebyshevi': [cheb1ap, cheb1ord],
'cheby2': [cheb2ap, cheb2ord],
'chebyshev2': [cheb2ap, cheb2ord],
'chebyshevii': [cheb2ap, cheb2ord],
}
band_dict = {'band': 'bandpass',
'bandpass': 'bandpass',
'pass': 'bandpass',
'bp': 'bandpass',
'bs': 'bandstop',
'bandstop': 'bandstop',
'bands': 'bandstop',
'stop': 'bandstop',
'l': 'lowpass',
'low': 'lowpass',
'lowpass': 'lowpass',
'lp': 'lowpass',
'high': 'highpass',
'highpass': 'highpass',
'h': 'highpass',
'hp': 'highpass',
}
bessel_norms = {'bessel': 'phase',
'bessel_phase': 'phase',
'bessel_delay': 'delay',
'bessel_mag': 'mag'}
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