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Vehicle-Anti-Theft-Face-Rec.../venv/Lib/site-packages/scipy/signal/windows/windows.py

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Fixed database typo and removed unnecessary class identifier.
2020-10-14 14:10:37 +00:00
"""The suite of window functions."""
import operator
import warnings
import numpy as np
from scipy import linalg, special, fft as sp_fft
__all__ = ['boxcar', 'triang', 'parzen', 'bohman', 'blackman', 'nuttall',
'blackmanharris', 'flattop', 'bartlett', 'hanning', 'barthann',
'hamming', 'kaiser', 'gaussian', 'general_cosine','general_gaussian',
'general_hamming', 'chebwin', 'slepian', 'cosine', 'hann',
'exponential', 'tukey', 'dpss', 'get_window']
def _len_guards(M):
"""Handle small or incorrect window lengths"""
if int(M) != M or M < 0:
raise ValueError('Window length M must be a non-negative integer')
return M <= 1
def _extend(M, sym):
"""Extend window by 1 sample if needed for DFT-even symmetry"""
if not sym:
return M + 1, True
else:
return M, False
def _truncate(w, needed):
"""Truncate window by 1 sample if needed for DFT-even symmetry"""
if needed:
return w[:-1]
else:
return w
def general_cosine(M, a, sym=True):
r"""
Generic weighted sum of cosine terms window
Parameters
----------
M : int
Number of points in the output window
a : array_like
Sequence of weighting coefficients. This uses the convention of being
centered on the origin, so these will typically all be positive
numbers, not alternating sign.
sym : bool, optional
When True (default), generates a symmetric window, for use in filter
design.
When False, generates a periodic window, for use in spectral analysis.
References
----------
.. [1] A. Nuttall, "Some windows with very good sidelobe behavior," IEEE
Transactions on Acoustics, Speech, and Signal Processing, vol. 29,
no. 1, pp. 84-91, Feb 1981. :doi:`10.1109/TASSP.1981.1163506`.
.. [2] Heinzel G. et al., "Spectrum and spectral density estimation by the
Discrete Fourier transform (DFT), including a comprehensive list of
window functions and some new flat-top windows", February 15, 2002
https://holometer.fnal.gov/GH_FFT.pdf
Examples
--------
Heinzel describes a flat-top window named "HFT90D" with formula: [2]_
.. math:: w_j = 1 - 1.942604 \cos(z) + 1.340318 \cos(2z)
- 0.440811 \cos(3z) + 0.043097 \cos(4z)
where
.. math:: z = \frac{2 \pi j}{N}, j = 0...N - 1
Since this uses the convention of starting at the origin, to reproduce the
window, we need to convert every other coefficient to a positive number:
>>> HFT90D = [1, 1.942604, 1.340318, 0.440811, 0.043097]
The paper states that the highest sidelobe is at -90.2 dB. Reproduce
Figure 42 by plotting the window and its frequency response, and confirm
the sidelobe level in red:
>>> from scipy.signal.windows import general_cosine
>>> from scipy.fft import fft, fftshift
>>> import matplotlib.pyplot as plt
>>> window = general_cosine(1000, HFT90D, sym=False)
>>> plt.plot(window)
>>> plt.title("HFT90D window")
>>> plt.ylabel("Amplitude")
>>> plt.xlabel("Sample")
>>> plt.figure()
>>> A = fft(window, 10000) / (len(window)/2.0)
>>> freq = np.linspace(-0.5, 0.5, len(A))
>>> response = np.abs(fftshift(A / abs(A).max()))
>>> response = 20 * np.log10(np.maximum(response, 1e-10))
>>> plt.plot(freq, response)
>>> plt.axis([-50/1000, 50/1000, -140, 0])
>>> plt.title("Frequency response of the HFT90D window")
>>> plt.ylabel("Normalized magnitude [dB]")
>>> plt.xlabel("Normalized frequency [cycles per sample]")
>>> plt.axhline(-90.2, color='red')
>>> plt.show()
"""
if _len_guards(M):
return np.ones(M)
M, needs_trunc = _extend(M, sym)
fac = np.linspace(-np.pi, np.pi, M)
w = np.zeros(M)
for k in range(len(a)):
w += a[k] * np.cos(k * fac)
return _truncate(w, needs_trunc)
def boxcar(M, sym=True):
"""Return a boxcar or rectangular window.
Also known as a rectangular window or Dirichlet window, this is equivalent
to no window at all.
Parameters
----------
M : int
Number of points in the output window. If zero or less, an empty
array is returned.
sym : bool, optional
Whether the window is symmetric. (Has no effect for boxcar.)
Returns
-------
w : ndarray
The window, with the maximum value normalized to 1.
Examples
--------
Plot the window and its frequency response:
>>> from scipy import signal
>>> from scipy.fft import fft, fftshift
>>> import matplotlib.pyplot as plt
>>> window = signal.boxcar(51)
>>> plt.plot(window)
>>> plt.title("Boxcar window")
>>> plt.ylabel("Amplitude")
>>> plt.xlabel("Sample")
>>> plt.figure()
>>> A = fft(window, 2048) / (len(window)/2.0)
>>> freq = np.linspace(-0.5, 0.5, len(A))
>>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max())))
>>> plt.plot(freq, response)
>>> plt.axis([-0.5, 0.5, -120, 0])
>>> plt.title("Frequency response of the boxcar window")
>>> plt.ylabel("Normalized magnitude [dB]")
>>> plt.xlabel("Normalized frequency [cycles per sample]")
"""
if _len_guards(M):
return np.ones(M)
M, needs_trunc = _extend(M, sym)
w = np.ones(M, float)
return _truncate(w, needs_trunc)
def triang(M, sym=True):
"""Return a triangular window.
Parameters
----------
M : int
Number of points in the output window. If zero or less, an empty
array is returned.
sym : bool, optional
When True (default), generates a symmetric window, for use in filter
design.
When False, generates a periodic window, for use in spectral analysis.
Returns
-------
w : ndarray
The window, with the maximum value normalized to 1 (though the value 1
does not appear if `M` is even and `sym` is True).
See Also
--------
bartlett : A triangular window that touches zero
Examples
--------
Plot the window and its frequency response:
>>> from scipy import signal
>>> from scipy.fft import fft, fftshift
>>> import matplotlib.pyplot as plt
>>> window = signal.triang(51)
>>> plt.plot(window)
>>> plt.title("Triangular window")
>>> plt.ylabel("Amplitude")
>>> plt.xlabel("Sample")
>>> plt.figure()
>>> A = fft(window, 2048) / (len(window)/2.0)
>>> freq = np.linspace(-0.5, 0.5, len(A))
>>> response = np.abs(fftshift(A / abs(A).max()))
>>> response = 20 * np.log10(np.maximum(response, 1e-10))
>>> plt.plot(freq, response)
>>> plt.axis([-0.5, 0.5, -120, 0])
>>> plt.title("Frequency response of the triangular window")
>>> plt.ylabel("Normalized magnitude [dB]")
>>> plt.xlabel("Normalized frequency [cycles per sample]")
"""
if _len_guards(M):
return np.ones(M)
M, needs_trunc = _extend(M, sym)
n = np.arange(1, (M + 1) // 2 + 1)
if M % 2 == 0:
w = (2 * n - 1.0) / M
w = np.r_[w, w[::-1]]
else:
w = 2 * n / (M + 1.0)
w = np.r_[w, w[-2::-1]]
return _truncate(w, needs_trunc)
def parzen(M, sym=True):
"""Return a Parzen window.
Parameters
----------
M : int
Number of points in the output window. If zero or less, an empty
array is returned.
sym : bool, optional
When True (default), generates a symmetric window, for use in filter
design.
When False, generates a periodic window, for use in spectral analysis.
Returns
-------
w : ndarray
The window, with the maximum value normalized to 1 (though the value 1
does not appear if `M` is even and `sym` is True).
References
----------
.. [1] E. Parzen, "Mathematical Considerations in the Estimation of
Spectra", Technometrics, Vol. 3, No. 2 (May, 1961), pp. 167-190
Examples
--------
Plot the window and its frequency response:
>>> from scipy import signal
>>> from scipy.fft import fft, fftshift
>>> import matplotlib.pyplot as plt
>>> window = signal.parzen(51)
>>> plt.plot(window)
>>> plt.title("Parzen window")
>>> plt.ylabel("Amplitude")
>>> plt.xlabel("Sample")
>>> plt.figure()
>>> A = fft(window, 2048) / (len(window)/2.0)
>>> freq = np.linspace(-0.5, 0.5, len(A))
>>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max())))
>>> plt.plot(freq, response)
>>> plt.axis([-0.5, 0.5, -120, 0])
>>> plt.title("Frequency response of the Parzen window")
>>> plt.ylabel("Normalized magnitude [dB]")
>>> plt.xlabel("Normalized frequency [cycles per sample]")
"""
if _len_guards(M):
return np.ones(M)
M, needs_trunc = _extend(M, sym)
n = np.arange(-(M - 1) / 2.0, (M - 1) / 2.0 + 0.5, 1.0)
na = np.extract(n < -(M - 1) / 4.0, n)
nb = np.extract(abs(n) <= (M - 1) / 4.0, n)
wa = 2 * (1 - np.abs(na) / (M / 2.0)) ** 3.0
wb = (1 - 6 * (np.abs(nb) / (M / 2.0)) ** 2.0 +
6 * (np.abs(nb) / (M / 2.0)) ** 3.0)
w = np.r_[wa, wb, wa[::-1]]
return _truncate(w, needs_trunc)
def bohman(M, sym=True):
"""Return a Bohman window.
Parameters
----------
M : int
Number of points in the output window. If zero or less, an empty
array is returned.
sym : bool, optional
When True (default), generates a symmetric window, for use in filter
design.
When False, generates a periodic window, for use in spectral analysis.
Returns
-------
w : ndarray
The window, with the maximum value normalized to 1 (though the value 1
does not appear if `M` is even and `sym` is True).
Examples
--------
Plot the window and its frequency response:
>>> from scipy import signal
>>> from scipy.fft import fft, fftshift
>>> import matplotlib.pyplot as plt
>>> window = signal.bohman(51)
>>> plt.plot(window)
>>> plt.title("Bohman window")
>>> plt.ylabel("Amplitude")
>>> plt.xlabel("Sample")
>>> plt.figure()
>>> A = fft(window, 2048) / (len(window)/2.0)
>>> freq = np.linspace(-0.5, 0.5, len(A))
>>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max())))
>>> plt.plot(freq, response)
>>> plt.axis([-0.5, 0.5, -120, 0])
>>> plt.title("Frequency response of the Bohman window")
>>> plt.ylabel("Normalized magnitude [dB]")
>>> plt.xlabel("Normalized frequency [cycles per sample]")
"""
if _len_guards(M):
return np.ones(M)
M, needs_trunc = _extend(M, sym)
fac = np.abs(np.linspace(-1, 1, M)[1:-1])
w = (1 - fac) * np.cos(np.pi * fac) + 1.0 / np.pi * np.sin(np.pi * fac)
w = np.r_[0, w, 0]
return _truncate(w, needs_trunc)
def blackman(M, sym=True):
r"""
Return a Blackman window.
The Blackman window is a taper formed by using the first three terms of
a summation of cosines. It was designed to have close to the minimal
leakage possible. It is close to optimal, only slightly worse than a
Kaiser window.
Parameters
----------
M : int
Number of points in the output window. If zero or less, an empty
array is returned.
sym : bool, optional
When True (default), generates a symmetric window, for use in filter
design.
When False, generates a periodic window, for use in spectral analysis.
Returns
-------
w : ndarray
The window, with the maximum value normalized to 1 (though the value 1
does not appear if `M` is even and `sym` is True).
Notes
-----
The Blackman window is defined as
.. math:: w(n) = 0.42 - 0.5 \cos(2\pi n/M) + 0.08 \cos(4\pi n/M)
The "exact Blackman" window was designed to null out the third and fourth
sidelobes, but has discontinuities at the boundaries, resulting in a
6 dB/oct fall-off. This window is an approximation of the "exact" window,
which does not null the sidelobes as well, but is smooth at the edges,
improving the fall-off rate to 18 dB/oct. [3]_
Most references to the Blackman window come from the signal processing
literature, where it is used as one of many windowing functions for
smoothing values. It is also known as an apodization (which means
"removing the foot", i.e. smoothing discontinuities at the beginning
and end of the sampled signal) or tapering function. It is known as a
"near optimal" tapering function, almost as good (by some measures)
as the Kaiser window.
References
----------
.. [1] Blackman, R.B. and Tukey, J.W., (1958) The measurement of power
spectra, Dover Publications, New York.
.. [2] Oppenheim, A.V., and R.W. Schafer. Discrete-Time Signal Processing.
Upper Saddle River, NJ: Prentice-Hall, 1999, pp. 468-471.
.. [3] Harris, Fredric J. (Jan 1978). "On the use of Windows for Harmonic
Analysis with the Discrete Fourier Transform". Proceedings of the
IEEE 66 (1): 51-83. :doi:`10.1109/PROC.1978.10837`.
Examples
--------
Plot the window and its frequency response:
>>> from scipy import signal
>>> from scipy.fft import fft, fftshift
>>> import matplotlib.pyplot as plt
>>> window = signal.blackman(51)
>>> plt.plot(window)
>>> plt.title("Blackman window")
>>> plt.ylabel("Amplitude")
>>> plt.xlabel("Sample")
>>> plt.figure()
>>> A = fft(window, 2048) / (len(window)/2.0)
>>> freq = np.linspace(-0.5, 0.5, len(A))
>>> response = np.abs(fftshift(A / abs(A).max()))
>>> response = 20 * np.log10(np.maximum(response, 1e-10))
>>> plt.plot(freq, response)
>>> plt.axis([-0.5, 0.5, -120, 0])
>>> plt.title("Frequency response of the Blackman window")
>>> plt.ylabel("Normalized magnitude [dB]")
>>> plt.xlabel("Normalized frequency [cycles per sample]")
"""
# Docstring adapted from NumPy's blackman function
return general_cosine(M, [0.42, 0.50, 0.08], sym)
def nuttall(M, sym=True):
"""Return a minimum 4-term Blackman-Harris window according to Nuttall.
This variation is called "Nuttall4c" by Heinzel. [2]_
Parameters
----------
M : int
Number of points in the output window. If zero or less, an empty
array is returned.
sym : bool, optional
When True (default), generates a symmetric window, for use in filter
design.
When False, generates a periodic window, for use in spectral analysis.
Returns
-------
w : ndarray
The window, with the maximum value normalized to 1 (though the value 1
does not appear if `M` is even and `sym` is True).
References
----------
.. [1] A. Nuttall, "Some windows with very good sidelobe behavior," IEEE
Transactions on Acoustics, Speech, and Signal Processing, vol. 29,
no. 1, pp. 84-91, Feb 1981. :doi:`10.1109/TASSP.1981.1163506`.
.. [2] Heinzel G. et al., "Spectrum and spectral density estimation by the
Discrete Fourier transform (DFT), including a comprehensive list of
window functions and some new flat-top windows", February 15, 2002
https://holometer.fnal.gov/GH_FFT.pdf
Examples
--------
Plot the window and its frequency response:
>>> from scipy import signal
>>> from scipy.fft import fft, fftshift
>>> import matplotlib.pyplot as plt
>>> window = signal.nuttall(51)
>>> plt.plot(window)
>>> plt.title("Nuttall window")
>>> plt.ylabel("Amplitude")
>>> plt.xlabel("Sample")
>>> plt.figure()
>>> A = fft(window, 2048) / (len(window)/2.0)
>>> freq = np.linspace(-0.5, 0.5, len(A))
>>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max())))
>>> plt.plot(freq, response)
>>> plt.axis([-0.5, 0.5, -120, 0])
>>> plt.title("Frequency response of the Nuttall window")
>>> plt.ylabel("Normalized magnitude [dB]")
>>> plt.xlabel("Normalized frequency [cycles per sample]")
"""
return general_cosine(M, [0.3635819, 0.4891775, 0.1365995, 0.0106411], sym)
def blackmanharris(M, sym=True):
"""Return a minimum 4-term Blackman-Harris window.
Parameters
----------
M : int
Number of points in the output window. If zero or less, an empty
array is returned.
sym : bool, optional
When True (default), generates a symmetric window, for use in filter
design.
When False, generates a periodic window, for use in spectral analysis.
Returns
-------
w : ndarray
The window, with the maximum value normalized to 1 (though the value 1
does not appear if `M` is even and `sym` is True).
Examples
--------
Plot the window and its frequency response:
>>> from scipy import signal
>>> from scipy.fft import fft, fftshift
>>> import matplotlib.pyplot as plt
>>> window = signal.blackmanharris(51)
>>> plt.plot(window)
>>> plt.title("Blackman-Harris window")
>>> plt.ylabel("Amplitude")
>>> plt.xlabel("Sample")
>>> plt.figure()
>>> A = fft(window, 2048) / (len(window)/2.0)
>>> freq = np.linspace(-0.5, 0.5, len(A))
>>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max())))
>>> plt.plot(freq, response)
>>> plt.axis([-0.5, 0.5, -120, 0])
>>> plt.title("Frequency response of the Blackman-Harris window")
>>> plt.ylabel("Normalized magnitude [dB]")
>>> plt.xlabel("Normalized frequency [cycles per sample]")
"""
return general_cosine(M, [0.35875, 0.48829, 0.14128, 0.01168], sym)
def flattop(M, sym=True):
"""Return a flat top window.
Parameters
----------
M : int
Number of points in the output window. If zero or less, an empty
array is returned.
sym : bool, optional
When True (default), generates a symmetric window, for use in filter
design.
When False, generates a periodic window, for use in spectral analysis.
Returns
-------
w : ndarray
The window, with the maximum value normalized to 1 (though the value 1
does not appear if `M` is even and `sym` is True).
Notes
-----
Flat top windows are used for taking accurate measurements of signal
amplitude in the frequency domain, with minimal scalloping error from the
center of a frequency bin to its edges, compared to others. This is a
5th-order cosine window, with the 5 terms optimized to make the main lobe
maximally flat. [1]_
References
----------
.. [1] D'Antona, Gabriele, and A. Ferrero, "Digital Signal Processing for
Measurement Systems", Springer Media, 2006, p. 70
:doi:`10.1007/0-387-28666-7`.
Examples
--------
Plot the window and its frequency response:
>>> from scipy import signal
>>> from scipy.fft import fft, fftshift
>>> import matplotlib.pyplot as plt
>>> window = signal.flattop(51)
>>> plt.plot(window)
>>> plt.title("Flat top window")
>>> plt.ylabel("Amplitude")
>>> plt.xlabel("Sample")
>>> plt.figure()
>>> A = fft(window, 2048) / (len(window)/2.0)
>>> freq = np.linspace(-0.5, 0.5, len(A))
>>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max())))
>>> plt.plot(freq, response)
>>> plt.axis([-0.5, 0.5, -120, 0])
>>> plt.title("Frequency response of the flat top window")
>>> plt.ylabel("Normalized magnitude [dB]")
>>> plt.xlabel("Normalized frequency [cycles per sample]")
"""
a = [0.21557895, 0.41663158, 0.277263158, 0.083578947, 0.006947368]
return general_cosine(M, a, sym)
def bartlett(M, sym=True):
r"""
Return a Bartlett window.
The Bartlett window is very similar to a triangular window, except
that the end points are at zero. It is often used in signal
processing for tapering a signal, without generating too much
ripple in the frequency domain.
Parameters
----------
M : int
Number of points in the output window. If zero or less, an empty
array is returned.
sym : bool, optional
When True (default), generates a symmetric window, for use in filter
design.
When False, generates a periodic window, for use in spectral analysis.
Returns
-------
w : ndarray
The triangular window, with the first and last samples equal to zero
and the maximum value normalized to 1 (though the value 1 does not
appear if `M` is even and `sym` is True).
See Also
--------
triang : A triangular window that does not touch zero at the ends
Notes
-----
The Bartlett window is defined as
.. math:: w(n) = \frac{2}{M-1} \left(
\frac{M-1}{2} - \left|n - \frac{M-1}{2}\right|
\right)
Most references to the Bartlett window come from the signal
processing literature, where it is used as one of many windowing
functions for smoothing values. Note that convolution with this
window produces linear interpolation. It is also known as an
apodization (which means"removing the foot", i.e. smoothing
discontinuities at the beginning and end of the sampled signal) or
tapering function. The Fourier transform of the Bartlett is the product
of two sinc functions.
Note the excellent discussion in Kanasewich. [2]_
References
----------
.. [1] M.S. Bartlett, "Periodogram Analysis and Continuous Spectra",
Biometrika 37, 1-16, 1950.
.. [2] E.R. Kanasewich, "Time Sequence Analysis in Geophysics",
The University of Alberta Press, 1975, pp. 109-110.
.. [3] A.V. Oppenheim and R.W. Schafer, "Discrete-Time Signal
Processing", Prentice-Hall, 1999, pp. 468-471.
.. [4] Wikipedia, "Window function",
https://en.wikipedia.org/wiki/Window_function
.. [5] W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling,
"Numerical Recipes", Cambridge University Press, 1986, page 429.
Examples
--------
Plot the window and its frequency response:
>>> from scipy import signal
>>> from scipy.fft import fft, fftshift
>>> import matplotlib.pyplot as plt
>>> window = signal.bartlett(51)
>>> plt.plot(window)
>>> plt.title("Bartlett window")
>>> plt.ylabel("Amplitude")
>>> plt.xlabel("Sample")
>>> plt.figure()
>>> A = fft(window, 2048) / (len(window)/2.0)
>>> freq = np.linspace(-0.5, 0.5, len(A))
>>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max())))
>>> plt.plot(freq, response)
>>> plt.axis([-0.5, 0.5, -120, 0])
>>> plt.title("Frequency response of the Bartlett window")
>>> plt.ylabel("Normalized magnitude [dB]")
>>> plt.xlabel("Normalized frequency [cycles per sample]")
"""
# Docstring adapted from NumPy's bartlett function
if _len_guards(M):
return np.ones(M)
M, needs_trunc = _extend(M, sym)
n = np.arange(0, M)
w = np.where(np.less_equal(n, (M - 1) / 2.0),
2.0 * n / (M - 1), 2.0 - 2.0 * n / (M - 1))
return _truncate(w, needs_trunc)
def hann(M, sym=True):
r"""
Return a Hann window.
The Hann window is a taper formed by using a raised cosine or sine-squared
with ends that touch zero.
Parameters
----------
M : int
Number of points in the output window. If zero or less, an empty
array is returned.
sym : bool, optional
When True (default), generates a symmetric window, for use in filter
design.
When False, generates a periodic window, for use in spectral analysis.
Returns
-------
w : ndarray
The window, with the maximum value normalized to 1 (though the value 1
does not appear if `M` is even and `sym` is True).
Notes
-----
The Hann window is defined as
.. math:: w(n) = 0.5 - 0.5 \cos\left(\frac{2\pi{n}}{M-1}\right)
\qquad 0 \leq n \leq M-1
The window was named for Julius von Hann, an Austrian meteorologist. It is
also known as the Cosine Bell. It is sometimes erroneously referred to as
the "Hanning" window, from the use of "hann" as a verb in the original
paper and confusion with the very similar Hamming window.
Most references to the Hann window come from the signal processing
literature, where it is used as one of many windowing functions for
smoothing values. It is also known as an apodization (which means
"removing the foot", i.e. smoothing discontinuities at the beginning
and end of the sampled signal) or tapering function.
References
----------
.. [1] Blackman, R.B. and Tukey, J.W., (1958) The measurement of power
spectra, Dover Publications, New York.
.. [2] E.R. Kanasewich, "Time Sequence Analysis in Geophysics",
The University of Alberta Press, 1975, pp. 106-108.
.. [3] Wikipedia, "Window function",
https://en.wikipedia.org/wiki/Window_function
.. [4] W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling,
"Numerical Recipes", Cambridge University Press, 1986, page 425.
Examples
--------
Plot the window and its frequency response:
>>> from scipy import signal
>>> from scipy.fft import fft, fftshift
>>> import matplotlib.pyplot as plt
>>> window = signal.hann(51)
>>> plt.plot(window)
>>> plt.title("Hann window")
>>> plt.ylabel("Amplitude")
>>> plt.xlabel("Sample")
>>> plt.figure()
>>> A = fft(window, 2048) / (len(window)/2.0)
>>> freq = np.linspace(-0.5, 0.5, len(A))
>>> response = np.abs(fftshift(A / abs(A).max()))
>>> response = 20 * np.log10(np.maximum(response, 1e-10))
>>> plt.plot(freq, response)
>>> plt.axis([-0.5, 0.5, -120, 0])
>>> plt.title("Frequency response of the Hann window")
>>> plt.ylabel("Normalized magnitude [dB]")
>>> plt.xlabel("Normalized frequency [cycles per sample]")
"""
# Docstring adapted from NumPy's hanning function
return general_hamming(M, 0.5, sym)
@np.deprecate(new_name='scipy.signal.windows.hann')
def hanning(*args, **kwargs):
return hann(*args, **kwargs)
def tukey(M, alpha=0.5, sym=True):
r"""Return a Tukey window, also known as a tapered cosine window.
Parameters
----------
M : int
Number of points in the output window. If zero or less, an empty
array is returned.
alpha : float, optional
Shape parameter of the Tukey window, representing the fraction of the
window inside the cosine tapered region.
If zero, the Tukey window is equivalent to a rectangular window.
If one, the Tukey window is equivalent to a Hann window.
sym : bool, optional
When True (default), generates a symmetric window, for use in filter
design.
When False, generates a periodic window, for use in spectral analysis.
Returns
-------
w : ndarray
The window, with the maximum value normalized to 1 (though the value 1
does not appear if `M` is even and `sym` is True).
References
----------
.. [1] Harris, Fredric J. (Jan 1978). "On the use of Windows for Harmonic
Analysis with the Discrete Fourier Transform". Proceedings of the
IEEE 66 (1): 51-83. :doi:`10.1109/PROC.1978.10837`
.. [2] Wikipedia, "Window function",
https://en.wikipedia.org/wiki/Window_function#Tukey_window
Examples
--------
Plot the window and its frequency response:
>>> from scipy import signal
>>> from scipy.fft import fft, fftshift
>>> import matplotlib.pyplot as plt
>>> window = signal.tukey(51)
>>> plt.plot(window)
>>> plt.title("Tukey window")
>>> plt.ylabel("Amplitude")
>>> plt.xlabel("Sample")
>>> plt.ylim([0, 1.1])
>>> plt.figure()
>>> A = fft(window, 2048) / (len(window)/2.0)
>>> freq = np.linspace(-0.5, 0.5, len(A))
>>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max())))
>>> plt.plot(freq, response)
>>> plt.axis([-0.5, 0.5, -120, 0])
>>> plt.title("Frequency response of the Tukey window")
>>> plt.ylabel("Normalized magnitude [dB]")
>>> plt.xlabel("Normalized frequency [cycles per sample]")
"""
if _len_guards(M):
return np.ones(M)
if alpha <= 0:
return np.ones(M, 'd')
elif alpha >= 1.0:
return hann(M, sym=sym)
M, needs_trunc = _extend(M, sym)
n = np.arange(0, M)
width = int(np.floor(alpha*(M-1)/2.0))
n1 = n[0:width+1]
n2 = n[width+1:M-width-1]
n3 = n[M-width-1:]
w1 = 0.5 * (1 + np.cos(np.pi * (-1 + 2.0*n1/alpha/(M-1))))
w2 = np.ones(n2.shape)
w3 = 0.5 * (1 + np.cos(np.pi * (-2.0/alpha + 1 + 2.0*n3/alpha/(M-1))))
w = np.concatenate((w1, w2, w3))
return _truncate(w, needs_trunc)
def barthann(M, sym=True):
"""Return a modified Bartlett-Hann window.
Parameters
----------
M : int
Number of points in the output window. If zero or less, an empty
array is returned.
sym : bool, optional
When True (default), generates a symmetric window, for use in filter
design.
When False, generates a periodic window, for use in spectral analysis.
Returns
-------
w : ndarray
The window, with the maximum value normalized to 1 (though the value 1
does not appear if `M` is even and `sym` is True).
Examples
--------
Plot the window and its frequency response:
>>> from scipy import signal
>>> from scipy.fft import fft, fftshift
>>> import matplotlib.pyplot as plt
>>> window = signal.barthann(51)
>>> plt.plot(window)
>>> plt.title("Bartlett-Hann window")
>>> plt.ylabel("Amplitude")
>>> plt.xlabel("Sample")
>>> plt.figure()
>>> A = fft(window, 2048) / (len(window)/2.0)
>>> freq = np.linspace(-0.5, 0.5, len(A))
>>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max())))
>>> plt.plot(freq, response)
>>> plt.axis([-0.5, 0.5, -120, 0])
>>> plt.title("Frequency response of the Bartlett-Hann window")
>>> plt.ylabel("Normalized magnitude [dB]")
>>> plt.xlabel("Normalized frequency [cycles per sample]")
"""
if _len_guards(M):
return np.ones(M)
M, needs_trunc = _extend(M, sym)
n = np.arange(0, M)
fac = np.abs(n / (M - 1.0) - 0.5)
w = 0.62 - 0.48 * fac + 0.38 * np.cos(2 * np.pi * fac)
return _truncate(w, needs_trunc)
def general_hamming(M, alpha, sym=True):
r"""Return a generalized Hamming window.
The generalized Hamming window is constructed by multiplying a rectangular
window by one period of a cosine function [1]_.
Parameters
----------
M : int
Number of points in the output window. If zero or less, an empty
array is returned.
alpha : float
The window coefficient, :math:`\alpha`
sym : bool, optional
When True (default), generates a symmetric window, for use in filter
design.
When False, generates a periodic window, for use in spectral analysis.
Returns
-------
w : ndarray
The window, with the maximum value normalized to 1 (though the value 1
does not appear if `M` is even and `sym` is True).
Notes
-----
The generalized Hamming window is defined as
.. math:: w(n) = \alpha - \left(1 - \alpha\right) \cos\left(\frac{2\pi{n}}{M-1}\right)
\qquad 0 \leq n \leq M-1
Both the common Hamming window and Hann window are special cases of the
generalized Hamming window with :math:`\alpha` = 0.54 and :math:`\alpha` =
0.5, respectively [2]_.
See Also
--------
hamming, hann
Examples
--------
The Sentinel-1A/B Instrument Processing Facility uses generalized Hamming
windows in the processing of spaceborne Synthetic Aperture Radar (SAR)
data [3]_. The facility uses various values for the :math:`\alpha`
parameter based on operating mode of the SAR instrument. Some common
:math:`\alpha` values include 0.75, 0.7 and 0.52 [4]_. As an example, we
plot these different windows.
>>> from scipy.signal.windows import general_hamming
>>> from scipy.fft import fft, fftshift
>>> import matplotlib.pyplot as plt
>>> fig1, spatial_plot = plt.subplots()
>>> spatial_plot.set_title("Generalized Hamming Windows")
>>> spatial_plot.set_ylabel("Amplitude")
>>> spatial_plot.set_xlabel("Sample")
>>> fig2, freq_plot = plt.subplots()
>>> freq_plot.set_title("Frequency Responses")
>>> freq_plot.set_ylabel("Normalized magnitude [dB]")
>>> freq_plot.set_xlabel("Normalized frequency [cycles per sample]")
>>> for alpha in [0.75, 0.7, 0.52]:
... window = general_hamming(41, alpha)
... spatial_plot.plot(window, label="{:.2f}".format(alpha))
... A = fft(window, 2048) / (len(window)/2.0)
... freq = np.linspace(-0.5, 0.5, len(A))
... response = 20 * np.log10(np.abs(fftshift(A / abs(A).max())))
... freq_plot.plot(freq, response, label="{:.2f}".format(alpha))
>>> freq_plot.legend(loc="upper right")
>>> spatial_plot.legend(loc="upper right")
References
----------
.. [1] DSPRelated, "Generalized Hamming Window Family",
https://www.dsprelated.com/freebooks/sasp/Generalized_Hamming_Window_Family.html
.. [2] Wikipedia, "Window function",
https://en.wikipedia.org/wiki/Window_function
.. [3] Riccardo Piantanida ESA, "Sentinel-1 Level 1 Detailed Algorithm
Definition",
https://sentinel.esa.int/documents/247904/1877131/Sentinel-1-Level-1-Detailed-Algorithm-Definition
.. [4] Matthieu Bourbigot ESA, "Sentinel-1 Product Definition",
https://sentinel.esa.int/documents/247904/1877131/Sentinel-1-Product-Definition
"""
return general_cosine(M, [alpha, 1. - alpha], sym)
def hamming(M, sym=True):
r"""Return a Hamming window.
The Hamming window is a taper formed by using a raised cosine with
non-zero endpoints, optimized to minimize the nearest side lobe.
Parameters
----------
M : int
Number of points in the output window. If zero or less, an empty
array is returned.
sym : bool, optional
When True (default), generates a symmetric window, for use in filter
design.
When False, generates a periodic window, for use in spectral analysis.
Returns
-------
w : ndarray
The window, with the maximum value normalized to 1 (though the value 1
does not appear if `M` is even and `sym` is True).
Notes
-----
The Hamming window is defined as
.. math:: w(n) = 0.54 - 0.46 \cos\left(\frac{2\pi{n}}{M-1}\right)
\qquad 0 \leq n \leq M-1
The Hamming was named for R. W. Hamming, an associate of J. W. Tukey and
is described in Blackman and Tukey. It was recommended for smoothing the
truncated autocovariance function in the time domain.
Most references to the Hamming window come from the signal processing
literature, where it is used as one of many windowing functions for
smoothing values. It is also known as an apodization (which means
"removing the foot", i.e. smoothing discontinuities at the beginning
and end of the sampled signal) or tapering function.
References
----------
.. [1] Blackman, R.B. and Tukey, J.W., (1958) The measurement of power
spectra, Dover Publications, New York.
.. [2] E.R. Kanasewich, "Time Sequence Analysis in Geophysics", The
University of Alberta Press, 1975, pp. 109-110.
.. [3] Wikipedia, "Window function",
https://en.wikipedia.org/wiki/Window_function
.. [4] W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling,
"Numerical Recipes", Cambridge University Press, 1986, page 425.
Examples
--------
Plot the window and its frequency response:
>>> from scipy import signal
>>> from scipy.fft import fft, fftshift
>>> import matplotlib.pyplot as plt
>>> window = signal.hamming(51)
>>> plt.plot(window)
>>> plt.title("Hamming window")
>>> plt.ylabel("Amplitude")
>>> plt.xlabel("Sample")
>>> plt.figure()
>>> A = fft(window, 2048) / (len(window)/2.0)
>>> freq = np.linspace(-0.5, 0.5, len(A))
>>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max())))
>>> plt.plot(freq, response)
>>> plt.axis([-0.5, 0.5, -120, 0])
>>> plt.title("Frequency response of the Hamming window")
>>> plt.ylabel("Normalized magnitude [dB]")
>>> plt.xlabel("Normalized frequency [cycles per sample]")
"""
# Docstring adapted from NumPy's hamming function
return general_hamming(M, 0.54, sym)
def kaiser(M, beta, sym=True):
r"""Return a Kaiser window.
The Kaiser window is a taper formed by using a Bessel function.
Parameters
----------
M : int
Number of points in the output window. If zero or less, an empty
array is returned.
beta : float
Shape parameter, determines trade-off between main-lobe width and
side lobe level. As beta gets large, the window narrows.
sym : bool, optional
When True (default), generates a symmetric window, for use in filter
design.
When False, generates a periodic window, for use in spectral analysis.
Returns
-------
w : ndarray
The window, with the maximum value normalized to 1 (though the value 1
does not appear if `M` is even and `sym` is True).
Notes
-----
The Kaiser window is defined as
.. math:: w(n) = I_0\left( \beta \sqrt{1-\frac{4n^2}{(M-1)^2}}
\right)/I_0(\beta)
with
.. math:: \quad -\frac{M-1}{2} \leq n \leq \frac{M-1}{2},
where :math:`I_0` is the modified zeroth-order Bessel function.
The Kaiser was named for Jim Kaiser, who discovered a simple approximation
to the DPSS window based on Bessel functions.
The Kaiser window is a very good approximation to the Digital Prolate
Spheroidal Sequence, or Slepian window, which is the transform which
maximizes the energy in the main lobe of the window relative to total
energy.
The Kaiser can approximate other windows by varying the beta parameter.
(Some literature uses alpha = beta/pi.) [4]_
==== =======================
beta Window shape
==== =======================
0 Rectangular
5 Similar to a Hamming
6 Similar to a Hann
8.6 Similar to a Blackman
==== =======================
A beta value of 14 is probably a good starting point. Note that as beta
gets large, the window narrows, and so the number of samples needs to be
large enough to sample the increasingly narrow spike, otherwise NaNs will
be returned.
Most references to the Kaiser window come from the signal processing
literature, where it is used as one of many windowing functions for
smoothing values. It is also known as an apodization (which means
"removing the foot", i.e. smoothing discontinuities at the beginning
and end of the sampled signal) or tapering function.
References
----------
.. [1] J. F. Kaiser, "Digital Filters" - Ch 7 in "Systems analysis by
digital computer", Editors: F.F. Kuo and J.F. Kaiser, p 218-285.
John Wiley and Sons, New York, (1966).
.. [2] E.R. Kanasewich, "Time Sequence Analysis in Geophysics", The
University of Alberta Press, 1975, pp. 177-178.
.. [3] Wikipedia, "Window function",
https://en.wikipedia.org/wiki/Window_function
.. [4] F. J. Harris, "On the use of windows for harmonic analysis with the
discrete Fourier transform," Proceedings of the IEEE, vol. 66,
no. 1, pp. 51-83, Jan. 1978. :doi:`10.1109/PROC.1978.10837`.
Examples
--------
Plot the window and its frequency response:
>>> from scipy import signal
>>> from scipy.fft import fft, fftshift
>>> import matplotlib.pyplot as plt
>>> window = signal.kaiser(51, beta=14)
>>> plt.plot(window)
>>> plt.title(r"Kaiser window ($\beta$=14)")
>>> plt.ylabel("Amplitude")
>>> plt.xlabel("Sample")
>>> plt.figure()
>>> A = fft(window, 2048) / (len(window)/2.0)
>>> freq = np.linspace(-0.5, 0.5, len(A))
>>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max())))
>>> plt.plot(freq, response)
>>> plt.axis([-0.5, 0.5, -120, 0])
>>> plt.title(r"Frequency response of the Kaiser window ($\beta$=14)")
>>> plt.ylabel("Normalized magnitude [dB]")
>>> plt.xlabel("Normalized frequency [cycles per sample]")
"""
# Docstring adapted from NumPy's kaiser function
if _len_guards(M):
return np.ones(M)
M, needs_trunc = _extend(M, sym)
n = np.arange(0, M)
alpha = (M - 1) / 2.0
w = (special.i0(beta * np.sqrt(1 - ((n - alpha) / alpha) ** 2.0)) /
special.i0(beta))
return _truncate(w, needs_trunc)
def gaussian(M, std, sym=True):
r"""Return a Gaussian window.
Parameters
----------
M : int
Number of points in the output window. If zero or less, an empty
array is returned.
std : float
The standard deviation, sigma.
sym : bool, optional
When True (default), generates a symmetric window, for use in filter
design.
When False, generates a periodic window, for use in spectral analysis.
Returns
-------
w : ndarray
The window, with the maximum value normalized to 1 (though the value 1
does not appear if `M` is even and `sym` is True).
Notes
-----
The Gaussian window is defined as
.. math:: w(n) = e^{ -\frac{1}{2}\left(\frac{n}{\sigma}\right)^2 }
Examples
--------
Plot the window and its frequency response:
>>> from scipy import signal
>>> from scipy.fft import fft, fftshift
>>> import matplotlib.pyplot as plt
>>> window = signal.gaussian(51, std=7)
>>> plt.plot(window)
>>> plt.title(r"Gaussian window ($\sigma$=7)")
>>> plt.ylabel("Amplitude")
>>> plt.xlabel("Sample")
>>> plt.figure()
>>> A = fft(window, 2048) / (len(window)/2.0)
>>> freq = np.linspace(-0.5, 0.5, len(A))
>>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max())))
>>> plt.plot(freq, response)
>>> plt.axis([-0.5, 0.5, -120, 0])
>>> plt.title(r"Frequency response of the Gaussian window ($\sigma$=7)")
>>> plt.ylabel("Normalized magnitude [dB]")
>>> plt.xlabel("Normalized frequency [cycles per sample]")
"""
if _len_guards(M):
return np.ones(M)
M, needs_trunc = _extend(M, sym)
n = np.arange(0, M) - (M - 1.0) / 2.0
sig2 = 2 * std * std
w = np.exp(-n ** 2 / sig2)
return _truncate(w, needs_trunc)
def general_gaussian(M, p, sig, sym=True):
r"""Return a window with a generalized Gaussian shape.
Parameters
----------
M : int
Number of points in the output window. If zero or less, an empty
array is returned.
p : float
Shape parameter. p = 1 is identical to `gaussian`, p = 0.5 is
the same shape as the Laplace distribution.
sig : float
The standard deviation, sigma.
sym : bool, optional
When True (default), generates a symmetric window, for use in filter
design.
When False, generates a periodic window, for use in spectral analysis.
Returns
-------
w : ndarray
The window, with the maximum value normalized to 1 (though the value 1
does not appear if `M` is even and `sym` is True).
Notes
-----
The generalized Gaussian window is defined as
.. math:: w(n) = e^{ -\frac{1}{2}\left|\frac{n}{\sigma}\right|^{2p} }
the half-power point is at
.. math:: (2 \log(2))^{1/(2 p)} \sigma
Examples
--------
Plot the window and its frequency response:
>>> from scipy import signal
>>> from scipy.fft import fft, fftshift
>>> import matplotlib.pyplot as plt
>>> window = signal.general_gaussian(51, p=1.5, sig=7)
>>> plt.plot(window)
>>> plt.title(r"Generalized Gaussian window (p=1.5, $\sigma$=7)")
>>> plt.ylabel("Amplitude")
>>> plt.xlabel("Sample")
>>> plt.figure()
>>> A = fft(window, 2048) / (len(window)/2.0)
>>> freq = np.linspace(-0.5, 0.5, len(A))
>>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max())))
>>> plt.plot(freq, response)
>>> plt.axis([-0.5, 0.5, -120, 0])
>>> plt.title(r"Freq. resp. of the gen. Gaussian "
... r"window (p=1.5, $\sigma$=7)")
>>> plt.ylabel("Normalized magnitude [dB]")
>>> plt.xlabel("Normalized frequency [cycles per sample]")
"""
if _len_guards(M):
return np.ones(M)
M, needs_trunc = _extend(M, sym)
n = np.arange(0, M) - (M - 1.0) / 2.0
w = np.exp(-0.5 * np.abs(n / sig) ** (2 * p))
return _truncate(w, needs_trunc)
# `chebwin` contributed by Kumar Appaiah.
def chebwin(M, at, sym=True):
r"""Return a Dolph-Chebyshev window.
Parameters
----------
M : int
Number of points in the output window. If zero or less, an empty
array is returned.
at : float
Attenuation (in dB).
sym : bool, optional
When True (default), generates a symmetric window, for use in filter
design.
When False, generates a periodic window, for use in spectral analysis.
Returns
-------
w : ndarray
The window, with the maximum value always normalized to 1
Notes
-----
This window optimizes for the narrowest main lobe width for a given order
`M` and sidelobe equiripple attenuation `at`, using Chebyshev
polynomials. It was originally developed by Dolph to optimize the
directionality of radio antenna arrays.
Unlike most windows, the Dolph-Chebyshev is defined in terms of its
frequency response:
.. math:: W(k) = \frac
{\cos\{M \cos^{-1}[\beta \cos(\frac{\pi k}{M})]\}}
{\cosh[M \cosh^{-1}(\beta)]}
where
.. math:: \beta = \cosh \left [\frac{1}{M}
\cosh^{-1}(10^\frac{A}{20}) \right ]
and 0 <= abs(k) <= M-1. A is the attenuation in decibels (`at`).
The time domain window is then generated using the IFFT, so
power-of-two `M` are the fastest to generate, and prime number `M` are
the slowest.
The equiripple condition in the frequency domain creates impulses in the
time domain, which appear at the ends of the window.
References
----------
.. [1] C. Dolph, "A current distribution for broadside arrays which
optimizes the relationship between beam width and side-lobe level",
Proceedings of the IEEE, Vol. 34, Issue 6
.. [2] Peter Lynch, "The Dolph-Chebyshev Window: A Simple Optimal Filter",
American Meteorological Society (April 1997)
http://mathsci.ucd.ie/~plynch/Publications/Dolph.pdf
.. [3] F. J. Harris, "On the use of windows for harmonic analysis with the
discrete Fourier transforms", Proceedings of the IEEE, Vol. 66,
No. 1, January 1978
Examples
--------
Plot the window and its frequency response:
>>> from scipy import signal
>>> from scipy.fft import fft, fftshift
>>> import matplotlib.pyplot as plt
>>> window = signal.chebwin(51, at=100)
>>> plt.plot(window)
>>> plt.title("Dolph-Chebyshev window (100 dB)")
>>> plt.ylabel("Amplitude")
>>> plt.xlabel("Sample")
>>> plt.figure()
>>> A = fft(window, 2048) / (len(window)/2.0)
>>> freq = np.linspace(-0.5, 0.5, len(A))
>>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max())))
>>> plt.plot(freq, response)
>>> plt.axis([-0.5, 0.5, -120, 0])
>>> plt.title("Frequency response of the Dolph-Chebyshev window (100 dB)")
>>> plt.ylabel("Normalized magnitude [dB]")
>>> plt.xlabel("Normalized frequency [cycles per sample]")
"""
if np.abs(at) < 45:
warnings.warn("This window is not suitable for spectral analysis "
"for attenuation values lower than about 45dB because "
"the equivalent noise bandwidth of a Chebyshev window "
"does not grow monotonically with increasing sidelobe "
"attenuation when the attenuation is smaller than "
"about 45 dB.")
if _len_guards(M):
return np.ones(M)
M, needs_trunc = _extend(M, sym)
# compute the parameter beta
order = M - 1.0
beta = np.cosh(1.0 / order * np.arccosh(10 ** (np.abs(at) / 20.)))
k = np.r_[0:M] * 1.0
x = beta * np.cos(np.pi * k / M)
# Find the window's DFT coefficients
# Use analytic definition of Chebyshev polynomial instead of expansion
# from scipy.special. Using the expansion in scipy.special leads to errors.
p = np.zeros(x.shape)
p[x > 1] = np.cosh(order * np.arccosh(x[x > 1]))
p[x < -1] = (2 * (M % 2) - 1) * np.cosh(order * np.arccosh(-x[x < -1]))
p[np.abs(x) <= 1] = np.cos(order * np.arccos(x[np.abs(x) <= 1]))
# Appropriate IDFT and filling up
# depending on even/odd M
if M % 2:
w = np.real(sp_fft.fft(p))
n = (M + 1) // 2
w = w[:n]
w = np.concatenate((w[n - 1:0:-1], w))
else:
p = p * np.exp(1.j * np.pi / M * np.r_[0:M])
w = np.real(sp_fft.fft(p))
n = M // 2 + 1
w = np.concatenate((w[n - 1:0:-1], w[1:n]))
w = w / max(w)
return _truncate(w, needs_trunc)
def slepian(M, width, sym=True):
"""Return a digital Slepian (DPSS) window.
Used to maximize the energy concentration in the main lobe. Also called
the digital prolate spheroidal sequence (DPSS).
.. note:: Deprecated in SciPy 1.1.
`slepian` will be removed in a future version of SciPy, it is
replaced by `dpss`, which uses the standard definition of a
digital Slepian window.
Parameters
----------
M : int
Number of points in the output window. If zero or less, an empty
array is returned.
width : float
Bandwidth
sym : bool, optional
When True (default), generates a symmetric window, for use in filter
design.
When False, generates a periodic window, for use in spectral analysis.
Returns
-------
w : ndarray
The window, with the maximum value always normalized to 1
See Also
--------
dpss
References
----------
.. [1] D. Slepian & H. O. Pollak: "Prolate spheroidal wave functions,
Fourier analysis and uncertainty-I," Bell Syst. Tech. J., vol.40,
pp.43-63, 1961. https://archive.org/details/bstj40-1-43
.. [2] H. J. Landau & H. O. Pollak: "Prolate spheroidal wave functions,
Fourier analysis and uncertainty-II," Bell Syst. Tech. J. , vol.40,
pp.65-83, 1961. https://archive.org/details/bstj40-1-65
Examples
--------
Plot the window and its frequency response:
>>> from scipy import signal
>>> from scipy.fft import fft, fftshift
>>> import matplotlib.pyplot as plt
>>> window = signal.slepian(51, width=0.3)
>>> plt.plot(window)
>>> plt.title("Slepian (DPSS) window (BW=0.3)")
>>> plt.ylabel("Amplitude")
>>> plt.xlabel("Sample")
>>> plt.figure()
>>> A = fft(window, 2048) / (len(window)/2.0)
>>> freq = np.linspace(-0.5, 0.5, len(A))
>>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max())))
>>> plt.plot(freq, response)
>>> plt.axis([-0.5, 0.5, -120, 0])
>>> plt.title("Frequency response of the Slepian window (BW=0.3)")
>>> plt.ylabel("Normalized magnitude [dB]")
>>> plt.xlabel("Normalized frequency [cycles per sample]")
"""
warnings.warn('slepian is deprecated and will be removed in a future '
'version, use dpss instead', DeprecationWarning)
if _len_guards(M):
return np.ones(M)
M, needs_trunc = _extend(M, sym)
# our width is the full bandwidth
width = width / 2
# to match the old version
width = width / 2
m = np.arange(M, dtype='d')
H = np.zeros((2, M))
H[0, 1:] = m[1:] * (M - m[1:]) / 2
H[1, :] = ((M - 1 - 2 * m) / 2)**2 * np.cos(2 * np.pi * width)
_, win = linalg.eig_banded(H, select='i', select_range=(M-1, M-1))
win = win.ravel() / win.max()
return _truncate(win, needs_trunc)
def cosine(M, sym=True):
"""Return a window with a simple cosine shape.
Parameters
----------
M : int
Number of points in the output window. If zero or less, an empty
array is returned.
sym : bool, optional
When True (default), generates a symmetric window, for use in filter
design.
When False, generates a periodic window, for use in spectral analysis.
Returns
-------
w : ndarray
The window, with the maximum value normalized to 1 (though the value 1
does not appear if `M` is even and `sym` is True).
Notes
-----
.. versionadded:: 0.13.0
Examples
--------
Plot the window and its frequency response:
>>> from scipy import signal
>>> from scipy.fft import fft, fftshift
>>> import matplotlib.pyplot as plt
>>> window = signal.cosine(51)
>>> plt.plot(window)
>>> plt.title("Cosine window")
>>> plt.ylabel("Amplitude")
>>> plt.xlabel("Sample")
>>> plt.figure()
>>> A = fft(window, 2048) / (len(window)/2.0)
>>> freq = np.linspace(-0.5, 0.5, len(A))
>>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max())))
>>> plt.plot(freq, response)
>>> plt.axis([-0.5, 0.5, -120, 0])
>>> plt.title("Frequency response of the cosine window")
>>> plt.ylabel("Normalized magnitude [dB]")
>>> plt.xlabel("Normalized frequency [cycles per sample]")
>>> plt.show()
"""
if _len_guards(M):
return np.ones(M)
M, needs_trunc = _extend(M, sym)
w = np.sin(np.pi / M * (np.arange(0, M) + .5))
return _truncate(w, needs_trunc)
def exponential(M, center=None, tau=1., sym=True):
r"""Return an exponential (or Poisson) window.
Parameters
----------
M : int
Number of points in the output window. If zero or less, an empty
array is returned.
center : float, optional
Parameter defining the center location of the window function.
The default value if not given is ``center = (M-1) / 2``. This
parameter must take its default value for symmetric windows.
tau : float, optional
Parameter defining the decay. For ``center = 0`` use
``tau = -(M-1) / ln(x)`` if ``x`` is the fraction of the window
remaining at the end.
sym : bool, optional
When True (default), generates a symmetric window, for use in filter
design.
When False, generates a periodic window, for use in spectral analysis.
Returns
-------
w : ndarray
The window, with the maximum value normalized to 1 (though the value 1
does not appear if `M` is even and `sym` is True).
Notes
-----
The Exponential window is defined as
.. math:: w(n) = e^{-|n-center| / \tau}
References
----------
S. Gade and H. Herlufsen, "Windows to FFT analysis (Part I)",
Technical Review 3, Bruel & Kjaer, 1987.
Examples
--------
Plot the symmetric window and its frequency response:
>>> from scipy import signal
>>> from scipy.fft import fft, fftshift
>>> import matplotlib.pyplot as plt
>>> M = 51
>>> tau = 3.0
>>> window = signal.exponential(M, tau=tau)
>>> plt.plot(window)
>>> plt.title("Exponential Window (tau=3.0)")
>>> plt.ylabel("Amplitude")
>>> plt.xlabel("Sample")
>>> plt.figure()
>>> A = fft(window, 2048) / (len(window)/2.0)
>>> freq = np.linspace(-0.5, 0.5, len(A))
>>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max())))
>>> plt.plot(freq, response)
>>> plt.axis([-0.5, 0.5, -35, 0])
>>> plt.title("Frequency response of the Exponential window (tau=3.0)")
>>> plt.ylabel("Normalized magnitude [dB]")
>>> plt.xlabel("Normalized frequency [cycles per sample]")
This function can also generate non-symmetric windows:
>>> tau2 = -(M-1) / np.log(0.01)
>>> window2 = signal.exponential(M, 0, tau2, False)
>>> plt.figure()
>>> plt.plot(window2)
>>> plt.ylabel("Amplitude")
>>> plt.xlabel("Sample")
"""
if sym and center is not None:
raise ValueError("If sym==True, center must be None.")
if _len_guards(M):
return np.ones(M)
M, needs_trunc = _extend(M, sym)
if center is None:
center = (M-1) / 2
n = np.arange(0, M)
w = np.exp(-np.abs(n-center) / tau)
return _truncate(w, needs_trunc)
def dpss(M, NW, Kmax=None, sym=True, norm=None, return_ratios=False):
"""
Compute the Discrete Prolate Spheroidal Sequences (DPSS).
DPSS (or Slepian sequences) are often used in multitaper power spectral
density estimation (see [1]_). The first window in the sequence can be
used to maximize the energy concentration in the main lobe, and is also
called the Slepian window.
Parameters
----------
M : int
Window length.
NW : float
Standardized half bandwidth corresponding to ``2*NW = BW/f0 = BW*N*dt``
where ``dt`` is taken as 1.
Kmax : int | None, optional
Number of DPSS windows to return (orders ``0`` through ``Kmax-1``).
If None (default), return only a single window of shape ``(M,)``
instead of an array of windows of shape ``(Kmax, M)``.
sym : bool, optional
When True (default), generates a symmetric window, for use in filter
design.
When False, generates a periodic window, for use in spectral analysis.
norm : {2, 'approximate', 'subsample'} | None, optional
If 'approximate' or 'subsample', then the windows are normalized by the
maximum, and a correction scale-factor for even-length windows
is applied either using ``M**2/(M**2+NW)`` ("approximate") or
a FFT-based subsample shift ("subsample"), see Notes for details.
If None, then "approximate" is used when ``Kmax=None`` and 2 otherwise
(which uses the l2 norm).
return_ratios : bool, optional
If True, also return the concentration ratios in addition to the
windows.
Returns
-------
v : ndarray, shape (Kmax, N) or (N,)
The DPSS windows. Will be 1D if `Kmax` is None.
r : ndarray, shape (Kmax,) or float, optional
The concentration ratios for the windows. Only returned if
`return_ratios` evaluates to True. Will be 0D if `Kmax` is None.
Notes
-----
This computation uses the tridiagonal eigenvector formulation given
in [2]_.
The default normalization for ``Kmax=None``, i.e. window-generation mode,
simply using the l-infinity norm would create a window with two unity
values, which creates slight normalization differences between even and odd
orders. The approximate correction of ``M**2/float(M**2+NW)`` for even
sample numbers is used to counteract this effect (see Examples below).
For very long signals (e.g., 1e6 elements), it can be useful to compute
windows orders of magnitude shorter and use interpolation (e.g.,
`scipy.interpolate.interp1d`) to obtain tapers of length `M`,
but this in general will not preserve orthogonality between the tapers.
.. versionadded:: 1.1
References
----------
.. [1] Percival DB, Walden WT. Spectral Analysis for Physical Applications:
Multitaper and Conventional Univariate Techniques.
Cambridge University Press; 1993.
.. [2] Slepian, D. Prolate spheroidal wave functions, Fourier analysis, and
uncertainty V: The discrete case. Bell System Technical Journal,
Volume 57 (1978), 1371430.
.. [3] Kaiser, JF, Schafer RW. On the Use of the I0-Sinh Window for
Spectrum Analysis. IEEE Transactions on Acoustics, Speech and
Signal Processing. ASSP-28 (1): 105-107; 1980.
Examples
--------
We can compare the window to `kaiser`, which was invented as an alternative
that was easier to calculate [3]_ (example adapted from
`here <https://ccrma.stanford.edu/~jos/sasp/Kaiser_DPSS_Windows_Compared.html>`_):
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.signal import windows, freqz
>>> N = 51
>>> fig, axes = plt.subplots(3, 2, figsize=(5, 7))
>>> for ai, alpha in enumerate((1, 3, 5)):
... win_dpss = windows.dpss(N, alpha)
... beta = alpha*np.pi
... win_kaiser = windows.kaiser(N, beta)
... for win, c in ((win_dpss, 'k'), (win_kaiser, 'r')):
... win /= win.sum()
... axes[ai, 0].plot(win, color=c, lw=1.)
... axes[ai, 0].set(xlim=[0, N-1], title=r'$\\alpha$ = %s' % alpha,
... ylabel='Amplitude')
... w, h = freqz(win)
... axes[ai, 1].plot(w, 20 * np.log10(np.abs(h)), color=c, lw=1.)
... axes[ai, 1].set(xlim=[0, np.pi],
... title=r'$\\beta$ = %0.2f' % beta,
... ylabel='Magnitude (dB)')
>>> for ax in axes.ravel():
... ax.grid(True)
>>> axes[2, 1].legend(['DPSS', 'Kaiser'])
>>> fig.tight_layout()
>>> plt.show()
And here are examples of the first four windows, along with their
concentration ratios:
>>> M = 512
>>> NW = 2.5
>>> win, eigvals = windows.dpss(M, NW, 4, return_ratios=True)
>>> fig, ax = plt.subplots(1)
>>> ax.plot(win.T, linewidth=1.)
>>> ax.set(xlim=[0, M-1], ylim=[-0.1, 0.1], xlabel='Samples',
... title='DPSS, M=%d, NW=%0.1f' % (M, NW))
>>> ax.legend(['win[%d] (%0.4f)' % (ii, ratio)
... for ii, ratio in enumerate(eigvals)])
>>> fig.tight_layout()
>>> plt.show()
Using a standard :math:`l_{\\infty}` norm would produce two unity values
for even `M`, but only one unity value for odd `M`. This produces uneven
window power that can be counteracted by the approximate correction
``M**2/float(M**2+NW)``, which can be selected by using
``norm='approximate'`` (which is the same as ``norm=None`` when
``Kmax=None``, as is the case here). Alternatively, the slower
``norm='subsample'`` can be used, which uses subsample shifting in the
frequency domain (FFT) to compute the correction:
>>> Ms = np.arange(1, 41)
>>> factors = (50, 20, 10, 5, 2.0001)
>>> energy = np.empty((3, len(Ms), len(factors)))
>>> for mi, M in enumerate(Ms):
... for fi, factor in enumerate(factors):
... NW = M / float(factor)
... # Corrected using empirical approximation (default)
... win = windows.dpss(M, NW)
... energy[0, mi, fi] = np.sum(win ** 2) / np.sqrt(M)
... # Corrected using subsample shifting
... win = windows.dpss(M, NW, norm='subsample')
... energy[1, mi, fi] = np.sum(win ** 2) / np.sqrt(M)
... # Uncorrected (using l-infinity norm)
... win /= win.max()
... energy[2, mi, fi] = np.sum(win ** 2) / np.sqrt(M)
>>> fig, ax = plt.subplots(1)
>>> hs = ax.plot(Ms, energy[2], '-o', markersize=4,
... markeredgecolor='none')
>>> leg = [hs[-1]]
>>> for hi, hh in enumerate(hs):
... h1 = ax.plot(Ms, energy[0, :, hi], '-o', markersize=4,
... color=hh.get_color(), markeredgecolor='none',
... alpha=0.66)
... h2 = ax.plot(Ms, energy[1, :, hi], '-o', markersize=4,
... color=hh.get_color(), markeredgecolor='none',
... alpha=0.33)
... if hi == len(hs) - 1:
... leg.insert(0, h1[0])
... leg.insert(0, h2[0])
>>> ax.set(xlabel='M (samples)', ylabel=r'Power / $\\sqrt{M}$')
>>> ax.legend(leg, ['Uncorrected', r'Corrected: $\\frac{M^2}{M^2+NW}$',
... 'Corrected (subsample)'])
>>> fig.tight_layout()
""" # noqa: E501
if _len_guards(M):
return np.ones(M)
if norm is None:
norm = 'approximate' if Kmax is None else 2
known_norms = (2, 'approximate', 'subsample')
if norm not in known_norms:
raise ValueError('norm must be one of %s, got %s'
% (known_norms, norm))
if Kmax is None:
singleton = True
Kmax = 1
else:
singleton = False
Kmax = operator.index(Kmax)
if not 0 < Kmax <= M:
raise ValueError('Kmax must be greater than 0 and less than M')
if NW >= M/2.:
raise ValueError('NW must be less than M/2.')
if NW <= 0:
raise ValueError('NW must be positive')
M, needs_trunc = _extend(M, sym)
W = float(NW) / M
nidx = np.arange(M)
# Here we want to set up an optimization problem to find a sequence
# whose energy is maximally concentrated within band [-W,W].
# Thus, the measure lambda(T,W) is the ratio between the energy within
# that band, and the total energy. This leads to the eigen-system
# (A - (l1)I)v = 0, where the eigenvector corresponding to the largest
# eigenvalue is the sequence with maximally concentrated energy. The
# collection of eigenvectors of this system are called Slepian
# sequences, or discrete prolate spheroidal sequences (DPSS). Only the
# first K, K = 2NW/dt orders of DPSS will exhibit good spectral
# concentration
# [see https://en.wikipedia.org/wiki/Spectral_concentration_problem]
# Here we set up an alternative symmetric tri-diagonal eigenvalue
# problem such that
# (B - (l2)I)v = 0, and v are our DPSS (but eigenvalues l2 != l1)
# the main diagonal = ([N-1-2*t]/2)**2 cos(2PIW), t=[0,1,2,...,N-1]
# and the first off-diagonal = t(N-t)/2, t=[1,2,...,N-1]
# [see Percival and Walden, 1993]
d = ((M - 1 - 2 * nidx) / 2.) ** 2 * np.cos(2 * np.pi * W)
e = nidx[1:] * (M - nidx[1:]) / 2.
# only calculate the highest Kmax eigenvalues
w, windows = linalg.eigh_tridiagonal(
d, e, select='i', select_range=(M - Kmax, M - 1))
w = w[::-1]
windows = windows[:, ::-1].T
# By convention (Percival and Walden, 1993 pg 379)
# * symmetric tapers (k=0,2,4,...) should have a positive average.
fix_even = (windows[::2].sum(axis=1) < 0)
for i, f in enumerate(fix_even):
if f:
windows[2 * i] *= -1
# * antisymmetric tapers should begin with a positive lobe
# (this depends on the definition of "lobe", here we'll take the first
# point above the numerical noise, which should be good enough for
# sufficiently smooth functions, and more robust than relying on an
# algorithm that uses max(abs(w)), which is susceptible to numerical
# noise problems)
thresh = max(1e-7, 1. / M)
for i, w in enumerate(windows[1::2]):
if w[w * w > thresh][0] < 0:
windows[2 * i + 1] *= -1
# Now find the eigenvalues of the original spectral concentration problem
# Use the autocorr sequence technique from Percival and Walden, 1993 pg 390
if return_ratios:
dpss_rxx = _fftautocorr(windows)
r = 4 * W * np.sinc(2 * W * nidx)
r[0] = 2 * W
ratios = np.dot(dpss_rxx, r)
if singleton:
ratios = ratios[0]
# Deal with sym and Kmax=None
if norm != 2:
windows /= windows.max()
if M % 2 == 0:
if norm == 'approximate':
correction = M**2 / float(M**2 + NW)
else:
s = sp_fft.rfft(windows[0])
shift = -(1 - 1./M) * np.arange(1, M//2 + 1)
s[1:] *= 2 * np.exp(-1j * np.pi * shift)
correction = M / s.real.sum()
windows *= correction
# else we're already l2 normed, so do nothing
if needs_trunc:
windows = windows[:, :-1]
if singleton:
windows = windows[0]
return (windows, ratios) if return_ratios else windows
def _fftautocorr(x):
"""Compute the autocorrelation of a real array and crop the result."""
N = x.shape[-1]
use_N = sp_fft.next_fast_len(2*N-1)
x_fft = sp_fft.rfft(x, use_N, axis=-1)
cxy = sp_fft.irfft(x_fft * x_fft.conj(), n=use_N)[:, :N]
# Or equivalently (but in most cases slower):
# cxy = np.array([np.convolve(xx, yy[::-1], mode='full')
# for xx, yy in zip(x, x)])[:, N-1:2*N-1]
return cxy
_win_equiv_raw = {
('barthann', 'brthan', 'bth'): (barthann, False),
('bartlett', 'bart', 'brt'): (bartlett, False),
('blackman', 'black', 'blk'): (blackman, False),
('blackmanharris', 'blackharr', 'bkh'): (blackmanharris, False),
('bohman', 'bman', 'bmn'): (bohman, False),
('boxcar', 'box', 'ones',
'rect', 'rectangular'): (boxcar, False),
('chebwin', 'cheb'): (chebwin, True),
('cosine', 'halfcosine'): (cosine, False),
('exponential', 'poisson'): (exponential, True),
('flattop', 'flat', 'flt'): (flattop, False),
('gaussian', 'gauss', 'gss'): (gaussian, True),
('general gaussian', 'general_gaussian',
'general gauss', 'general_gauss', 'ggs'): (general_gaussian, True),
('hamming', 'hamm', 'ham'): (hamming, False),
('hanning', 'hann', 'han'): (hann, False),
('kaiser', 'ksr'): (kaiser, True),
('nuttall', 'nutl', 'nut'): (nuttall, False),
('parzen', 'parz', 'par'): (parzen, False),
('slepian', 'slep', 'optimal', 'dpss', 'dss'): (slepian, True),
('triangle', 'triang', 'tri'): (triang, False),
('tukey', 'tuk'): (tukey, True),
}
# Fill dict with all valid window name strings
_win_equiv = {}
for k, v in _win_equiv_raw.items():
for key in k:
_win_equiv[key] = v[0]
# Keep track of which windows need additional parameters
_needs_param = set()
for k, v in _win_equiv_raw.items():
if v[1]:
_needs_param.update(k)
def get_window(window, Nx, fftbins=True):
"""
Return a window of a given length and type.
Parameters
----------
window : string, float, or tuple
The type of window to create. See below for more details.
Nx : int
The number of samples in the window.
fftbins : bool, optional
If True (default), create a "periodic" window, ready to use with
`ifftshift` and be multiplied by the result of an FFT (see also
:func:`~scipy.fft.fftfreq`).
If False, create a "symmetric" window, for use in filter design.
Returns
-------
get_window : ndarray
Returns a window of length `Nx` and type `window`
Notes
-----
Window types:
- `~scipy.signal.windows.boxcar`
- `~scipy.signal.windows.triang`
- `~scipy.signal.windows.blackman`
- `~scipy.signal.windows.hamming`
- `~scipy.signal.windows.hann`
- `~scipy.signal.windows.bartlett`
- `~scipy.signal.windows.flattop`
- `~scipy.signal.windows.parzen`
- `~scipy.signal.windows.bohman`
- `~scipy.signal.windows.blackmanharris`
- `~scipy.signal.windows.nuttall`
- `~scipy.signal.windows.barthann`
- `~scipy.signal.windows.kaiser` (needs beta)
- `~scipy.signal.windows.gaussian` (needs standard deviation)
- `~scipy.signal.windows.general_gaussian` (needs power, width)
- `~scipy.signal.windows.slepian` (needs width)
- `~scipy.signal.windows.dpss` (needs normalized half-bandwidth)
- `~scipy.signal.windows.chebwin` (needs attenuation)
- `~scipy.signal.windows.exponential` (needs decay scale)
- `~scipy.signal.windows.tukey` (needs taper fraction)
If the window requires no parameters, then `window` can be a string.
If the window requires parameters, then `window` must be a tuple
with the first argument the string name of the window, and the next
arguments the needed parameters.
If `window` is a floating point number, it is interpreted as the beta
parameter of the `~scipy.signal.windows.kaiser` window.
Each of the window types listed above is also the name of
a function that can be called directly to create a window of
that type.
Examples
--------
>>> from scipy import signal
>>> signal.get_window('triang', 7)
array([ 0.125, 0.375, 0.625, 0.875, 0.875, 0.625, 0.375])
>>> signal.get_window(('kaiser', 4.0), 9)
array([ 0.08848053, 0.29425961, 0.56437221, 0.82160913, 0.97885093,
0.97885093, 0.82160913, 0.56437221, 0.29425961])
>>> signal.get_window(4.0, 9)
array([ 0.08848053, 0.29425961, 0.56437221, 0.82160913, 0.97885093,
0.97885093, 0.82160913, 0.56437221, 0.29425961])
"""
sym = not fftbins
try:
beta = float(window)
except (TypeError, ValueError):
args = ()
if isinstance(window, tuple):
winstr = window[0]
if len(window) > 1:
args = window[1:]
elif isinstance(window, str):
if window in _needs_param:
raise ValueError("The '" + window + "' window needs one or "
"more parameters -- pass a tuple.")
else:
winstr = window
else:
raise ValueError("%s as window type is not supported." %
str(type(window)))
try:
winfunc = _win_equiv[winstr]
except KeyError:
raise ValueError("Unknown window type.")
params = (Nx,) + args + (sym,)
else:
winfunc = kaiser
params = (Nx, beta, sym)
return winfunc(*params)
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