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"""Tools for spectral analysis.
"""
import numpy as np
from scipy import fft as sp_fft
from . import _signaltools
from .windows import get_window
from ._spectral import _lombscargle
from ._arraytools import const_ext, even_ext, odd_ext, zero_ext
import warnings
__all__ = ['periodogram', 'welch', 'lombscargle', 'csd', 'coherence',
'spectrogram', 'stft', 'istft', 'check_COLA', 'check_NOLA']
def lombscargle(x,
y,
freqs,
precenter=False,
normalize=False):
"""
lombscargle(x, y, freqs)
Computes the Lomb-Scargle periodogram.
The Lomb-Scargle periodogram was developed by Lomb [1]_ and further
extended by Scargle [2]_ to find, and test the significance of weak
periodic signals with uneven temporal sampling.
When *normalize* is False (default) the computed periodogram
is unnormalized, it takes the value ``(A**2) * N/4`` for a harmonic
signal with amplitude A for sufficiently large N.
When *normalize* is True the computed periodogram is normalized by
the residuals of the data around a constant reference model (at zero).
Input arrays should be 1-D and will be cast to float64.
Parameters
----------
x : array_like
Sample times.
y : array_like
Measurement values.
freqs : array_like
Angular frequencies for output periodogram.
precenter : bool, optional
Pre-center measurement values by subtracting the mean.
normalize : bool, optional
Compute normalized periodogram.
Returns
-------
pgram : array_like
Lomb-Scargle periodogram.
Raises
------
ValueError
If the input arrays `x` and `y` do not have the same shape.
See Also
--------
istft: Inverse Short Time Fourier Transform
check_COLA: Check whether the Constant OverLap Add (COLA) constraint is met
welch: Power spectral density by Welch's method
spectrogram: Spectrogram by Welch's method
csd: Cross spectral density by Welch's method
Notes
-----
This subroutine calculates the periodogram using a slightly
modified algorithm due to Townsend [3]_ which allows the
periodogram to be calculated using only a single pass through
the input arrays for each frequency.
The algorithm running time scales roughly as O(x * freqs) or O(N^2)
for a large number of samples and frequencies.
References
----------
.. [1] N.R. Lomb "Least-squares frequency analysis of unequally spaced
data", Astrophysics and Space Science, vol 39, pp. 447-462, 1976
.. [2] J.D. Scargle "Studies in astronomical time series analysis. II -
Statistical aspects of spectral analysis of unevenly spaced data",
The Astrophysical Journal, vol 263, pp. 835-853, 1982
.. [3] R.H.D. Townsend, "Fast calculation of the Lomb-Scargle
periodogram using graphics processing units.", The Astrophysical
Journal Supplement Series, vol 191, pp. 247-253, 2010
Examples
--------
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> rng = np.random.default_rng()
First define some input parameters for the signal:
>>> A = 2.
>>> w0 = 1. # rad/sec
>>> nin = 150
>>> nout = 100000
Randomly generate sample times:
>>> x = rng.uniform(0, 10*np.pi, nin)
Plot a sine wave for the selected times:
>>> y = A * np.cos(w0*x)
Define the array of frequencies for which to compute the periodogram:
>>> w = np.linspace(0.01, 10, nout)
Calculate Lomb-Scargle periodogram:
>>> import scipy.signal as signal
>>> pgram = signal.lombscargle(x, y, w, normalize=True)
Now make a plot of the input data:
>>> fig, (ax_t, ax_w) = plt.subplots(2, 1, constrained_layout=True)
>>> ax_t.plot(x, y, 'b+')
>>> ax_t.set_xlabel('Time [s]')
Then plot the normalized periodogram:
>>> ax_w.plot(w, pgram)
>>> ax_w.set_xlabel('Angular frequency [rad/s]')
>>> ax_w.set_ylabel('Normalized amplitude')
>>> plt.show()
"""
x = np.ascontiguousarray(x, dtype=np.float64)
y = np.ascontiguousarray(y, dtype=np.float64)
freqs = np.ascontiguousarray(freqs, dtype=np.float64)
assert x.ndim == 1
assert y.ndim == 1
assert freqs.ndim == 1
if precenter:
pgram = _lombscargle(x, y - y.mean(), freqs)
else:
pgram = _lombscargle(x, y, freqs)
if normalize:
pgram *= 2 / np.dot(y, y)
return pgram
def periodogram(x, fs=1.0, window='boxcar', nfft=None, detrend='constant',
return_onesided=True, scaling='density', axis=-1):
"""
Estimate power spectral density using a periodogram.
Parameters
----------
x : array_like
Time series of measurement values
fs : float, optional
Sampling frequency of the `x` time series. Defaults to 1.0.
window : str or tuple or array_like, optional
Desired window to use. If `window` is a string or tuple, it is
passed to `get_window` to generate the window values, which are
DFT-even by default. See `get_window` for a list of windows and
required parameters. If `window` is array_like it will be used
directly as the window and its length must be equal to the length
of the axis over which the periodogram is computed. Defaults
to 'boxcar'.
nfft : int, optional
Length of the FFT used. If `None` the length of `x` will be
used.
detrend : str or function or `False`, optional
Specifies how to detrend each segment. If `detrend` is a
string, it is passed as the `type` argument to the `detrend`
function. If it is a function, it takes a segment and returns a
detrended segment. If `detrend` is `False`, no detrending is
done. Defaults to 'constant'.
return_onesided : bool, optional
If `True`, return a one-sided spectrum for real data. If
`False` return a two-sided spectrum. Defaults to `True`, but for
complex data, a two-sided spectrum is always returned.
scaling : { 'density', 'spectrum' }, optional
Selects between computing the power spectral density ('density')
where `Pxx` has units of V**2/Hz and computing the power
spectrum ('spectrum') where `Pxx` has units of V**2, if `x`
is measured in V and `fs` is measured in Hz. Defaults to
'density'
axis : int, optional
Axis along which the periodogram is computed; the default is
over the last axis (i.e. ``axis=-1``).
Returns
-------
f : ndarray
Array of sample frequencies.
Pxx : ndarray
Power spectral density or power spectrum of `x`.
See Also
--------
welch: Estimate power spectral density using Welch's method
lombscargle: Lomb-Scargle periodogram for unevenly sampled data
Notes
-----
.. versionadded:: 0.12.0
Examples
--------
>>> import numpy as np
>>> from scipy import signal
>>> import matplotlib.pyplot as plt
>>> rng = np.random.default_rng()
Generate a test signal, a 2 Vrms sine wave at 1234 Hz, corrupted by
0.001 V**2/Hz of white noise sampled at 10 kHz.
>>> fs = 10e3
>>> N = 1e5
>>> amp = 2*np.sqrt(2)
>>> freq = 1234.0
>>> noise_power = 0.001 * fs / 2
>>> time = np.arange(N) / fs
>>> x = amp*np.sin(2*np.pi*freq*time)
>>> x += rng.normal(scale=np.sqrt(noise_power), size=time.shape)
Compute and plot the power spectral density.
>>> f, Pxx_den = signal.periodogram(x, fs)
>>> plt.semilogy(f, Pxx_den)
>>> plt.ylim([1e-7, 1e2])
>>> plt.xlabel('frequency [Hz]')
>>> plt.ylabel('PSD [V**2/Hz]')
>>> plt.show()
If we average the last half of the spectral density, to exclude the
peak, we can recover the noise power on the signal.
>>> np.mean(Pxx_den[25000:])
0.000985320699252543
Now compute and plot the power spectrum.
>>> f, Pxx_spec = signal.periodogram(x, fs, 'flattop', scaling='spectrum')
>>> plt.figure()
>>> plt.semilogy(f, np.sqrt(Pxx_spec))
>>> plt.ylim([1e-4, 1e1])
>>> plt.xlabel('frequency [Hz]')
>>> plt.ylabel('Linear spectrum [V RMS]')
>>> plt.show()
The peak height in the power spectrum is an estimate of the RMS
amplitude.
>>> np.sqrt(Pxx_spec.max())
2.0077340678640727
"""
x = np.asarray(x)
if x.size == 0:
return np.empty(x.shape), np.empty(x.shape)
if window is None:
window = 'boxcar'
if nfft is None:
nperseg = x.shape[axis]
elif nfft == x.shape[axis]:
nperseg = nfft
elif nfft > x.shape[axis]:
nperseg = x.shape[axis]
elif nfft < x.shape[axis]:
s = [np.s_[:]]*len(x.shape)
s[axis] = np.s_[:nfft]
x = x[tuple(s)]
nperseg = nfft
nfft = None
if hasattr(window, 'size'):
if window.size != nperseg:
raise ValueError('the size of the window must be the same size '
'of the input on the specified axis')
return welch(x, fs=fs, window=window, nperseg=nperseg, noverlap=0,
nfft=nfft, detrend=detrend, return_onesided=return_onesided,
scaling=scaling, axis=axis)
def welch(x, fs=1.0, window='hann', nperseg=None, noverlap=None, nfft=None,
detrend='constant', return_onesided=True, scaling='density',
axis=-1, average='mean'):
r"""
Estimate power spectral density using Welch's method.
Welch's method [1]_ computes an estimate of the power spectral
density by dividing the data into overlapping segments, computing a
modified periodogram for each segment and averaging the
periodograms.
Parameters
----------
x : array_like
Time series of measurement values
fs : float, optional
Sampling frequency of the `x` time series. Defaults to 1.0.
window : str or tuple or array_like, optional
Desired window to use. If `window` is a string or tuple, it is
passed to `get_window` to generate the window values, which are
DFT-even by default. See `get_window` for a list of windows and
required parameters. If `window` is array_like it will be used
directly as the window and its length must be nperseg. Defaults
to a Hann window.
nperseg : int, optional
Length of each segment. Defaults to None, but if window is str or
tuple, is set to 256, and if window is array_like, is set to the
length of the window.
noverlap : int, optional
Number of points to overlap between segments. If `None`,
``noverlap = nperseg // 2``. Defaults to `None`.
nfft : int, optional
Length of the FFT used, if a zero padded FFT is desired. If
`None`, the FFT length is `nperseg`. Defaults to `None`.
detrend : str or function or `False`, optional
Specifies how to detrend each segment. If `detrend` is a
string, it is passed as the `type` argument to the `detrend`
function. If it is a function, it takes a segment and returns a
detrended segment. If `detrend` is `False`, no detrending is
done. Defaults to 'constant'.
return_onesided : bool, optional
If `True`, return a one-sided spectrum for real data. If
`False` return a two-sided spectrum. Defaults to `True`, but for
complex data, a two-sided spectrum is always returned.
scaling : { 'density', 'spectrum' }, optional
Selects between computing the power spectral density ('density')
where `Pxx` has units of V**2/Hz and computing the power
spectrum ('spectrum') where `Pxx` has units of V**2, if `x`
is measured in V and `fs` is measured in Hz. Defaults to
'density'
axis : int, optional
Axis along which the periodogram is computed; the default is
over the last axis (i.e. ``axis=-1``).
average : { 'mean', 'median' }, optional
Method to use when averaging periodograms. Defaults to 'mean'.
.. versionadded:: 1.2.0
Returns
-------
f : ndarray
Array of sample frequencies.
Pxx : ndarray
Power spectral density or power spectrum of x.
See Also
--------
periodogram: Simple, optionally modified periodogram
lombscargle: Lomb-Scargle periodogram for unevenly sampled data
Notes
-----
An appropriate amount of overlap will depend on the choice of window
and on your requirements. For the default Hann window an overlap of
50% is a reasonable trade off between accurately estimating the
signal power, while not over counting any of the data. Narrower
windows may require a larger overlap.
If `noverlap` is 0, this method is equivalent to Bartlett's method
[2]_.
.. versionadded:: 0.12.0
References
----------
.. [1] P. Welch, "The use of the fast Fourier transform for the
estimation of power spectra: A method based on time averaging
over short, modified periodograms", IEEE Trans. Audio
Electroacoust. vol. 15, pp. 70-73, 1967.
.. [2] M.S. Bartlett, "Periodogram Analysis and Continuous Spectra",
Biometrika, vol. 37, pp. 1-16, 1950.
Examples
--------
>>> import numpy as np
>>> from scipy import signal
>>> import matplotlib.pyplot as plt
>>> rng = np.random.default_rng()
Generate a test signal, a 2 Vrms sine wave at 1234 Hz, corrupted by
0.001 V**2/Hz of white noise sampled at 10 kHz.
>>> fs = 10e3
>>> N = 1e5
>>> amp = 2*np.sqrt(2)
>>> freq = 1234.0
>>> noise_power = 0.001 * fs / 2
>>> time = np.arange(N) / fs
>>> x = amp*np.sin(2*np.pi*freq*time)
>>> x += rng.normal(scale=np.sqrt(noise_power), size=time.shape)
Compute and plot the power spectral density.
>>> f, Pxx_den = signal.welch(x, fs, nperseg=1024)
>>> plt.semilogy(f, Pxx_den)
>>> plt.ylim([0.5e-3, 1])
>>> plt.xlabel('frequency [Hz]')
>>> plt.ylabel('PSD [V**2/Hz]')
>>> plt.show()
If we average the last half of the spectral density, to exclude the
peak, we can recover the noise power on the signal.
>>> np.mean(Pxx_den[256:])
0.0009924865443739191
Now compute and plot the power spectrum.
>>> f, Pxx_spec = signal.welch(x, fs, 'flattop', 1024, scaling='spectrum')
>>> plt.figure()
>>> plt.semilogy(f, np.sqrt(Pxx_spec))
>>> plt.xlabel('frequency [Hz]')
>>> plt.ylabel('Linear spectrum [V RMS]')
>>> plt.show()
The peak height in the power spectrum is an estimate of the RMS
amplitude.
>>> np.sqrt(Pxx_spec.max())
2.0077340678640727
If we now introduce a discontinuity in the signal, by increasing the
amplitude of a small portion of the signal by 50, we can see the
corruption of the mean average power spectral density, but using a
median average better estimates the normal behaviour.
>>> x[int(N//2):int(N//2)+10] *= 50.
>>> f, Pxx_den = signal.welch(x, fs, nperseg=1024)
>>> f_med, Pxx_den_med = signal.welch(x, fs, nperseg=1024, average='median')
>>> plt.semilogy(f, Pxx_den, label='mean')
>>> plt.semilogy(f_med, Pxx_den_med, label='median')
>>> plt.ylim([0.5e-3, 1])
>>> plt.xlabel('frequency [Hz]')
>>> plt.ylabel('PSD [V**2/Hz]')
>>> plt.legend()
>>> plt.show()
"""
freqs, Pxx = csd(x, x, fs=fs, window=window, nperseg=nperseg,
noverlap=noverlap, nfft=nfft, detrend=detrend,
return_onesided=return_onesided, scaling=scaling,
axis=axis, average=average)
return freqs, Pxx.real
def csd(x, y, fs=1.0, window='hann', nperseg=None, noverlap=None, nfft=None,
detrend='constant', return_onesided=True, scaling='density',
axis=-1, average='mean'):
r"""
Estimate the cross power spectral density, Pxy, using Welch's method.
Parameters
----------
x : array_like
Time series of measurement values
y : array_like
Time series of measurement values
fs : float, optional
Sampling frequency of the `x` and `y` time series. Defaults
to 1.0.
window : str or tuple or array_like, optional
Desired window to use. If `window` is a string or tuple, it is
passed to `get_window` to generate the window values, which are
DFT-even by default. See `get_window` for a list of windows and
required parameters. If `window` is array_like it will be used
directly as the window and its length must be nperseg. Defaults
to a Hann window.
nperseg : int, optional
Length of each segment. Defaults to None, but if window is str or
tuple, is set to 256, and if window is array_like, is set to the
length of the window.
noverlap: int, optional
Number of points to overlap between segments. If `None`,
``noverlap = nperseg // 2``. Defaults to `None`.
nfft : int, optional
Length of the FFT used, if a zero padded FFT is desired. If
`None`, the FFT length is `nperseg`. Defaults to `None`.
detrend : str or function or `False`, optional
Specifies how to detrend each segment. If `detrend` is a
string, it is passed as the `type` argument to the `detrend`
function. If it is a function, it takes a segment and returns a
detrended segment. If `detrend` is `False`, no detrending is
done. Defaults to 'constant'.
return_onesided : bool, optional
If `True`, return a one-sided spectrum for real data. If
`False` return a two-sided spectrum. Defaults to `True`, but for
complex data, a two-sided spectrum is always returned.
scaling : { 'density', 'spectrum' }, optional
Selects between computing the cross spectral density ('density')
where `Pxy` has units of V**2/Hz and computing the cross spectrum
('spectrum') where `Pxy` has units of V**2, if `x` and `y` are
measured in V and `fs` is measured in Hz. Defaults to 'density'
axis : int, optional
Axis along which the CSD is computed for both inputs; the
default is over the last axis (i.e. ``axis=-1``).
average : { 'mean', 'median' }, optional
Method to use when averaging periodograms. If the spectrum is
complex, the average is computed separately for the real and
imaginary parts. Defaults to 'mean'.
.. versionadded:: 1.2.0
Returns
-------
f : ndarray
Array of sample frequencies.
Pxy : ndarray
Cross spectral density or cross power spectrum of x,y.
See Also
--------
periodogram: Simple, optionally modified periodogram
lombscargle: Lomb-Scargle periodogram for unevenly sampled data
welch: Power spectral density by Welch's method. [Equivalent to
csd(x,x)]
coherence: Magnitude squared coherence by Welch's method.
Notes
-----
By convention, Pxy is computed with the conjugate FFT of X
multiplied by the FFT of Y.
If the input series differ in length, the shorter series will be
zero-padded to match.
An appropriate amount of overlap will depend on the choice of window
and on your requirements. For the default Hann window an overlap of
50% is a reasonable trade off between accurately estimating the
signal power, while not over counting any of the data. Narrower
windows may require a larger overlap.
.. versionadded:: 0.16.0
References
----------
.. [1] P. Welch, "The use of the fast Fourier transform for the
estimation of power spectra: A method based on time averaging
over short, modified periodograms", IEEE Trans. Audio
Electroacoust. vol. 15, pp. 70-73, 1967.
.. [2] Rabiner, Lawrence R., and B. Gold. "Theory and Application of
Digital Signal Processing" Prentice-Hall, pp. 414-419, 1975
Examples
--------
>>> import numpy as np
>>> from scipy import signal
>>> import matplotlib.pyplot as plt
>>> rng = np.random.default_rng()
Generate two test signals with some common features.
>>> fs = 10e3
>>> N = 1e5
>>> amp = 20
>>> freq = 1234.0
>>> noise_power = 0.001 * fs / 2
>>> time = np.arange(N) / fs
>>> b, a = signal.butter(2, 0.25, 'low')
>>> x = rng.normal(scale=np.sqrt(noise_power), size=time.shape)
>>> y = signal.lfilter(b, a, x)
>>> x += amp*np.sin(2*np.pi*freq*time)
>>> y += rng.normal(scale=0.1*np.sqrt(noise_power), size=time.shape)
Compute and plot the magnitude of the cross spectral density.
>>> f, Pxy = signal.csd(x, y, fs, nperseg=1024)
>>> plt.semilogy(f, np.abs(Pxy))
>>> plt.xlabel('frequency [Hz]')
>>> plt.ylabel('CSD [V**2/Hz]')
>>> plt.show()
"""
freqs, _, Pxy = _spectral_helper(x, y, fs, window, nperseg, noverlap, nfft,
detrend, return_onesided, scaling, axis,
mode='psd')
# Average over windows.
if len(Pxy.shape) >= 2 and Pxy.size > 0:
if Pxy.shape[-1] > 1:
if average == 'median':
# np.median must be passed real arrays for the desired result
bias = _median_bias(Pxy.shape[-1])
if np.iscomplexobj(Pxy):
Pxy = (np.median(np.real(Pxy), axis=-1)
+ 1j * np.median(np.imag(Pxy), axis=-1))
else:
Pxy = np.median(Pxy, axis=-1)
Pxy /= bias
elif average == 'mean':
Pxy = Pxy.mean(axis=-1)
else:
raise ValueError('average must be "median" or "mean", got %s'
% (average,))
else:
Pxy = np.reshape(Pxy, Pxy.shape[:-1])
return freqs, Pxy
def spectrogram(x, fs=1.0, window=('tukey', .25), nperseg=None, noverlap=None,
nfft=None, detrend='constant', return_onesided=True,
scaling='density', axis=-1, mode='psd'):
"""Compute a spectrogram with consecutive Fourier transforms.
Spectrograms can be used as a way of visualizing the change of a
nonstationary signal's frequency content over time.
Parameters
----------
x : array_like
Time series of measurement values
fs : float, optional
Sampling frequency of the `x` time series. Defaults to 1.0.
window : str or tuple or array_like, optional
Desired window to use. If `window` is a string or tuple, it is
passed to `get_window` to generate the window values, which are
DFT-even by default. See `get_window` for a list of windows and
required parameters. If `window` is array_like it will be used
directly as the window and its length must be nperseg.
Defaults to a Tukey window with shape parameter of 0.25.
nperseg : int, optional
Length of each segment. Defaults to None, but if window is str or
tuple, is set to 256, and if window is array_like, is set to the
length of the window.
noverlap : int, optional
Number of points to overlap between segments. If `None`,
``noverlap = nperseg // 8``. Defaults to `None`.
nfft : int, optional
Length of the FFT used, if a zero padded FFT is desired. If
`None`, the FFT length is `nperseg`. Defaults to `None`.
detrend : str or function or `False`, optional
Specifies how to detrend each segment. If `detrend` is a
string, it is passed as the `type` argument to the `detrend`
function. If it is a function, it takes a segment and returns a
detrended segment. If `detrend` is `False`, no detrending is
done. Defaults to 'constant'.
return_onesided : bool, optional
If `True`, return a one-sided spectrum for real data. If
`False` return a two-sided spectrum. Defaults to `True`, but for
complex data, a two-sided spectrum is always returned.
scaling : { 'density', 'spectrum' }, optional
Selects between computing the power spectral density ('density')
where `Sxx` has units of V**2/Hz and computing the power
spectrum ('spectrum') where `Sxx` has units of V**2, if `x`
is measured in V and `fs` is measured in Hz. Defaults to
'density'.
axis : int, optional
Axis along which the spectrogram is computed; the default is over
the last axis (i.e. ``axis=-1``).
mode : str, optional
Defines what kind of return values are expected. Options are
['psd', 'complex', 'magnitude', 'angle', 'phase']. 'complex' is
equivalent to the output of `stft` with no padding or boundary
extension. 'magnitude' returns the absolute magnitude of the
STFT. 'angle' and 'phase' return the complex angle of the STFT,
with and without unwrapping, respectively.
Returns
-------
f : ndarray
Array of sample frequencies.
t : ndarray
Array of segment times.
Sxx : ndarray
Spectrogram of x. By default, the last axis of Sxx corresponds
to the segment times.
See Also
--------
periodogram: Simple, optionally modified periodogram
lombscargle: Lomb-Scargle periodogram for unevenly sampled data
welch: Power spectral density by Welch's method.
csd: Cross spectral density by Welch's method.
Notes
-----
An appropriate amount of overlap will depend on the choice of window
and on your requirements. In contrast to welch's method, where the
entire data stream is averaged over, one may wish to use a smaller
overlap (or perhaps none at all) when computing a spectrogram, to
maintain some statistical independence between individual segments.
It is for this reason that the default window is a Tukey window with
1/8th of a window's length overlap at each end.
.. versionadded:: 0.16.0
References
----------
.. [1] Oppenheim, Alan V., Ronald W. Schafer, John R. Buck
"Discrete-Time Signal Processing", Prentice Hall, 1999.
Examples
--------
>>> import numpy as np
>>> from scipy import signal
>>> from scipy.fft import fftshift
>>> import matplotlib.pyplot as plt
>>> rng = np.random.default_rng()
Generate a test signal, a 2 Vrms sine wave whose frequency is slowly
modulated around 3kHz, corrupted by white noise of exponentially
decreasing magnitude sampled at 10 kHz.
>>> fs = 10e3
>>> N = 1e5
>>> amp = 2 * np.sqrt(2)
>>> noise_power = 0.01 * fs / 2
>>> time = np.arange(N) / float(fs)
>>> mod = 500*np.cos(2*np.pi*0.25*time)
>>> carrier = amp * np.sin(2*np.pi*3e3*time + mod)
>>> noise = rng.normal(scale=np.sqrt(noise_power), size=time.shape)
>>> noise *= np.exp(-time/5)
>>> x = carrier + noise
Compute and plot the spectrogram.
>>> f, t, Sxx = signal.spectrogram(x, fs)
>>> plt.pcolormesh(t, f, Sxx, shading='gouraud')
>>> plt.ylabel('Frequency [Hz]')
>>> plt.xlabel('Time [sec]')
>>> plt.show()
Note, if using output that is not one sided, then use the following:
>>> f, t, Sxx = signal.spectrogram(x, fs, return_onesided=False)
>>> plt.pcolormesh(t, fftshift(f), fftshift(Sxx, axes=0), shading='gouraud')
>>> plt.ylabel('Frequency [Hz]')
>>> plt.xlabel('Time [sec]')
>>> plt.show()
"""
modelist = ['psd', 'complex', 'magnitude', 'angle', 'phase']
if mode not in modelist:
raise ValueError('unknown value for mode {}, must be one of {}'
.format(mode, modelist))
# need to set default for nperseg before setting default for noverlap below
window, nperseg = _triage_segments(window, nperseg,
input_length=x.shape[axis])
# Less overlap than welch, so samples are more statisically independent
if noverlap is None:
noverlap = nperseg // 8
if mode == 'psd':
freqs, time, Sxx = _spectral_helper(x, x, fs, window, nperseg,
noverlap, nfft, detrend,
return_onesided, scaling, axis,
mode='psd')
else:
freqs, time, Sxx = _spectral_helper(x, x, fs, window, nperseg,
noverlap, nfft, detrend,
return_onesided, scaling, axis,
mode='stft')
if mode == 'magnitude':
Sxx = np.abs(Sxx)
elif mode in ['angle', 'phase']:
Sxx = np.angle(Sxx)
if mode == 'phase':
# Sxx has one additional dimension for time strides
if axis < 0:
axis -= 1
Sxx = np.unwrap(Sxx, axis=axis)
# mode =='complex' is same as `stft`, doesn't need modification
return freqs, time, Sxx
def check_COLA(window, nperseg, noverlap, tol=1e-10):
r"""Check whether the Constant OverLap Add (COLA) constraint is met.
Parameters
----------
window : str or tuple or array_like
Desired window to use. If `window` is a string or tuple, it is
passed to `get_window` to generate the window values, which are
DFT-even by default. See `get_window` for a list of windows and
required parameters. If `window` is array_like it will be used
directly as the window and its length must be nperseg.
nperseg : int
Length of each segment.
noverlap : int
Number of points to overlap between segments.
tol : float, optional
The allowed variance of a bin's weighted sum from the median bin
sum.
Returns
-------
verdict : bool
`True` if chosen combination satisfies COLA within `tol`,
`False` otherwise
See Also
--------
check_NOLA: Check whether the Nonzero Overlap Add (NOLA) constraint is met
stft: Short Time Fourier Transform
istft: Inverse Short Time Fourier Transform
Notes
-----
In order to enable inversion of an STFT via the inverse STFT in
`istft`, it is sufficient that the signal windowing obeys the constraint of
"Constant OverLap Add" (COLA). This ensures that every point in the input
data is equally weighted, thereby avoiding aliasing and allowing full
reconstruction.
Some examples of windows that satisfy COLA:
- Rectangular window at overlap of 0, 1/2, 2/3, 3/4, ...
- Bartlett window at overlap of 1/2, 3/4, 5/6, ...
- Hann window at 1/2, 2/3, 3/4, ...
- Any Blackman family window at 2/3 overlap
- Any window with ``noverlap = nperseg-1``
A very comprehensive list of other windows may be found in [2]_,
wherein the COLA condition is satisfied when the "Amplitude
Flatness" is unity.
.. versionadded:: 0.19.0
References
----------
.. [1] Julius O. Smith III, "Spectral Audio Signal Processing", W3K
Publishing, 2011,ISBN 978-0-9745607-3-1.
.. [2] G. Heinzel, A. Ruediger and R. Schilling, "Spectrum and
spectral density estimation by the Discrete Fourier transform
(DFT), including a comprehensive list of window functions and
some new at-top windows", 2002,
http://hdl.handle.net/11858/00-001M-0000-0013-557A-5
Examples
--------
>>> from scipy import signal
Confirm COLA condition for rectangular window of 75% (3/4) overlap:
>>> signal.check_COLA(signal.windows.boxcar(100), 100, 75)
True
COLA is not true for 25% (1/4) overlap, though:
>>> signal.check_COLA(signal.windows.boxcar(100), 100, 25)
False
"Symmetrical" Hann window (for filter design) is not COLA:
>>> signal.check_COLA(signal.windows.hann(120, sym=True), 120, 60)
False
"Periodic" or "DFT-even" Hann window (for FFT analysis) is COLA for
overlap of 1/2, 2/3, 3/4, etc.:
>>> signal.check_COLA(signal.windows.hann(120, sym=False), 120, 60)
True
>>> signal.check_COLA(signal.windows.hann(120, sym=False), 120, 80)
True
>>> signal.check_COLA(signal.windows.hann(120, sym=False), 120, 90)
True
"""
nperseg = int(nperseg)
if nperseg < 1:
raise ValueError('nperseg must be a positive integer')
if noverlap >= nperseg:
raise ValueError('noverlap must be less than nperseg.')
noverlap = int(noverlap)
if isinstance(window, str) or type(window) is tuple:
win = get_window(window, nperseg)
else:
win = np.asarray(window)
if len(win.shape) != 1:
raise ValueError('window must be 1-D')
if win.shape[0] != nperseg:
raise ValueError('window must have length of nperseg')
step = nperseg - noverlap
binsums = sum(win[ii*step:(ii+1)*step] for ii in range(nperseg//step))
if nperseg % step != 0:
binsums[:nperseg % step] += win[-(nperseg % step):]
deviation = binsums - np.median(binsums)
return np.max(np.abs(deviation)) < tol
def check_NOLA(window, nperseg, noverlap, tol=1e-10):
r"""Check whether the Nonzero Overlap Add (NOLA) constraint is met.
Parameters
----------
window : str or tuple or array_like
Desired window to use. If `window` is a string or tuple, it is
passed to `get_window` to generate the window values, which are
DFT-even by default. See `get_window` for a list of windows and
required parameters. If `window` is array_like it will be used
directly as the window and its length must be nperseg.
nperseg : int
Length of each segment.
noverlap : int
Number of points to overlap between segments.
tol : float, optional
The allowed variance of a bin's weighted sum from the median bin
sum.
Returns
-------
verdict : bool
`True` if chosen combination satisfies the NOLA constraint within
`tol`, `False` otherwise
See Also
--------
check_COLA: Check whether the Constant OverLap Add (COLA) constraint is met
stft: Short Time Fourier Transform
istft: Inverse Short Time Fourier Transform
Notes
-----
In order to enable inversion of an STFT via the inverse STFT in
`istft`, the signal windowing must obey the constraint of "nonzero
overlap add" (NOLA):
.. math:: \sum_{t}w^{2}[n-tH] \ne 0
for all :math:`n`, where :math:`w` is the window function, :math:`t` is the
frame index, and :math:`H` is the hop size (:math:`H` = `nperseg` -
`noverlap`).
This ensures that the normalization factors in the denominator of the
overlap-add inversion equation are not zero. Only very pathological windows
will fail the NOLA constraint.
.. versionadded:: 1.2.0
References
----------
.. [1] Julius O. Smith III, "Spectral Audio Signal Processing", W3K
Publishing, 2011,ISBN 978-0-9745607-3-1.
.. [2] G. Heinzel, A. Ruediger and R. Schilling, "Spectrum and
spectral density estimation by the Discrete Fourier transform
(DFT), including a comprehensive list of window functions and
some new at-top windows", 2002,
http://hdl.handle.net/11858/00-001M-0000-0013-557A-5
Examples
--------
>>> import numpy as np
>>> from scipy import signal
Confirm NOLA condition for rectangular window of 75% (3/4) overlap:
>>> signal.check_NOLA(signal.windows.boxcar(100), 100, 75)
True
NOLA is also true for 25% (1/4) overlap:
>>> signal.check_NOLA(signal.windows.boxcar(100), 100, 25)
True
"Symmetrical" Hann window (for filter design) is also NOLA:
>>> signal.check_NOLA(signal.windows.hann(120, sym=True), 120, 60)
True
As long as there is overlap, it takes quite a pathological window to fail
NOLA:
>>> w = np.ones(64, dtype="float")
>>> w[::2] = 0
>>> signal.check_NOLA(w, 64, 32)
False
If there is not enough overlap, a window with zeros at the ends will not
work: