freqz(b, a=1, worN=512, whole=False, plot=None, fs=6.283185307179586, include_nyquist=False)
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
Using Matplotlib's matplotlib.pyplot.plot
function as the callable for :None:None:`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:
An integer value is given for :None:None:`worN`.
:None:None:`worN` is fast to compute via FFT (i.e., :None:None:`next_fast_len(worN) <scipy.fft.next_fast_len>` equals :None:None:`worN`).
The denominator coefficients are a single value ( a.shape[0] == 1
).
:None:None:`worN` is at least as long as the numerator coefficients ( worN >= b.shape[0]
).
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.
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.
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.
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 :None:None:`fs`.
Normally, frequencies are computed from 0 to the Nyquist frequency, fs/2 (upper-half of unit-circle). If :None:None:`whole` is True, compute frequencies from 0 to fs. Ignored if worN is array_like.
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
.
The sampling frequency of the digital system. Defaults to 2*pi radians/sample (so w is from 0 to pi).
If :None:None:`whole` is False and :None:None:`worN` is an integer, setting :None:None:`include_nyquist` to True will include the last frequency (Nyquist frequency) and is otherwise ignored.
The frequencies at which h was computed, in the same units as :None:None:`fs`. By default, w is normalized to the range [0, pi) (radians/sample).
The frequency response, as complex numbers.
Compute the frequency response of a digital filter.
>>> 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:
>>> rng = np.random.default_rng()
... b = rng.random((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)
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)See :
The following pages refer to to this document either explicitly or contain code examples using this.
scipy.signal._filter_design.sosfreqz
scipy.signal._filter_design.gammatone
scipy.signal._filter_design.freqs
scipy.signal._fir_filter_design.minimum_phase
scipy.signal._fir_filter_design.kaiserord
scipy.signal._filter_design.iirnotch
scipy.signal._filter_design.freqz
scipy.signal._filter_design.iirpeak
scipy.signal._fir_filter_design.remez
scipy.signal._filter_design.group_delay
scipy.signal._fir_filter_design.firls
scipy.signal._filter_design.freqz_zpk
scipy.signal._filter_design.freqs_zpk
scipy.signal._filter_design.iircomb
scipy.signal._filter_design.bilinear
scipy.signal._filter_design.iirdesign
scipy.signal._filter_design.cheb1ord
scipy.signal.windows._windows.dpss
scipy.signal._filter_design.cheb2ord
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