Vous êtes un professionnel et vous avez besoin d'une formation ? Machine Learning
avec Scikit-Learn Voir le programme détaillé

Module « scipy.signal »

Fonction czt - module scipy.signal

Signature de la fonction czt

def czt(x, m=None, w=None, a=(1+0j), *, axis=-1) 

Description

help(scipy.signal.czt)

Compute the frequency response around a spiral in the Z plane.

Parameters
----------
x : array
    The signal to transform.
m : int, optional
    The number of output points desired.  Default is the length of the
    input data.
w : complex, optional
    The ratio between points in each step.  This must be precise or the
    accumulated error will degrade the tail of the output sequence.
    Defaults to equally spaced points around the entire unit circle.
a : complex, optional
    The starting point in the complex plane.  Default is 1+0j.
axis : int, optional
    Axis over which to compute the FFT. If not given, the last axis is
    used.

Returns
-------
out : ndarray
    An array of the same dimensions as `x`, but with the length of the
    transformed axis set to `m`.

See Also
--------
CZT : Class that creates a callable chirp z-transform function.
zoom_fft : Convenience function for partial FFT calculations.

Notes
-----
The defaults are chosen such that ``signal.czt(x)`` is equivalent to
``fft.fft(x)`` and, if ``m > len(x)``, that ``signal.czt(x, m)`` is
equivalent to ``fft.fft(x, m)``.

If the transform needs to be repeated, use `CZT` to construct a
specialized transform function which can be reused without
recomputing constants.

An example application is in system identification, repeatedly evaluating
small slices of the z-transform of a system, around where a pole is
expected to exist, to refine the estimate of the pole's true location. [1]_

References
----------
.. [1] Steve Alan Shilling, "A study of the chirp z-transform and its
       applications", pg 20 (1970)
       https://krex.k-state.edu/dspace/bitstream/handle/2097/7844/LD2668R41972S43.pdf

Examples
--------
Generate a sinusoid:

>>> import numpy as np
>>> f1, f2, fs = 8, 10, 200  # Hz
>>> t = np.linspace(0, 1, fs, endpoint=False)
>>> x = np.sin(2*np.pi*t*f2)
>>> import matplotlib.pyplot as plt
>>> plt.plot(t, x)
>>> plt.axis([0, 1, -1.1, 1.1])
>>> plt.show()

Its discrete Fourier transform has all of its energy in a single frequency
bin:

>>> from scipy.fft import rfft, rfftfreq
>>> from scipy.signal import czt, czt_points
>>> plt.plot(rfftfreq(fs, 1/fs), abs(rfft(x)))
>>> plt.margins(0, 0.1)
>>> plt.show()

However, if the sinusoid is logarithmically-decaying:

>>> x = np.exp(-t*f1) * np.sin(2*np.pi*t*f2)
>>> plt.plot(t, x)
>>> plt.axis([0, 1, -1.1, 1.1])
>>> plt.show()

the DFT will have spectral leakage:

>>> plt.plot(rfftfreq(fs, 1/fs), abs(rfft(x)))
>>> plt.margins(0, 0.1)
>>> plt.show()

While the DFT always samples the z-transform around the unit circle, the
chirp z-transform allows us to sample the Z-transform along any
logarithmic spiral, such as a circle with radius smaller than unity:

>>> M = fs // 2  # Just positive frequencies, like rfft
>>> a = np.exp(-f1/fs)  # Starting point of the circle, radius < 1
>>> w = np.exp(-1j*np.pi/M)  # "Step size" of circle
>>> points = czt_points(M + 1, w, a)  # M + 1 to include Nyquist
>>> plt.plot(points.real, points.imag, '.')
>>> plt.gca().add_patch(plt.Circle((0,0), radius=1, fill=False, alpha=.3))
>>> plt.axis('equal'); plt.axis([-1.05, 1.05, -0.05, 1.05])
>>> plt.show()

With the correct radius, this transforms the decaying sinusoid (and others
with the same decay rate) without spectral leakage:

>>> z_vals = czt(x, M + 1, w, a)  # Include Nyquist for comparison to rfft
>>> freqs = np.angle(points)*fs/(2*np.pi)  # angle = omega, radius = sigma
>>> plt.plot(freqs, abs(z_vals))
>>> plt.margins(0, 0.1)
>>> plt.show()

Vous êtes un professionnel et vous avez besoin d'une formation ? RAG (Retrieval-Augmented Generation)
et Fine Tuning d'un LLM Voir le programme détaillé