package scipy

Get an attribute of this module as a Py.Object.t. This is useful to pass a Python function to another function.

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.boxcar(100), 100, 75) True

COLA is not true for 25% (1/4) overlap, though:

>>> signal.check_COLA(signal.boxcar(100), 100, 25) False

'Symmetrical' Hann window (for filter design) is not COLA:

>>> signal.check_COLA(signal.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.hann(120, sym=False), 120, 60) True

>>> signal.check_COLA(signal.hann(120, sym=False), 120, 80) True

>>> signal.check_COLA(signal.hann(120, sym=False), 120, 90) True

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_

w^

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 -------- >>> from scipy import signal

Confirm NOLA condition for rectangular window of 75% (3/4) overlap:

>>> signal.check_NOLA(signal.boxcar(100), 100, 75) True

NOLA is also true for 25% (1/4) overlap:

>>> signal.check_NOLA(signal.boxcar(100), 100, 25) True

'Symmetrical' Hann window (for filter design) is also NOLA:

>>> signal.check_NOLA(signal.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:

>>> signal.check_NOLA(signal.hann(64), 64, 0) False >>> signal.check_NOLA(signal.hann(64), 64, 1) False >>> signal.check_NOLA(signal.hann(64), 64, 2) True

Estimate the magnitude squared coherence estimate, Cxy, of discrete-time signals X and Y using Welch's method.

``Cxy = abs(Pxy)**2/(Pxx*Pyy)``, where `Pxx` and `Pyy` are power spectral density estimates of X and Y, and `Pxy` is the cross spectral density estimate of X and Y.

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'. axis : int, optional Axis along which the coherence is computed for both inputs; the default is over the last axis (i.e. ``axis=-1``).

Returns ------- f : ndarray Array of sample frequencies. Cxy : ndarray Magnitude squared coherence of x and y.

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. 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 Stoica, Petre, and Randolph Moses, 'Spectral Analysis of Signals' Prentice Hall, 2005

Examples -------- >>> from scipy import signal >>> import matplotlib.pyplot as plt

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 = np.random.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 += np.random.normal(scale=0.1*np.sqrt(noise_power), size=time.shape)

Compute and plot the coherence.

>>> f, Cxy = signal.coherence(x, y, fs, nperseg=1024) >>> plt.semilogy(f, Cxy) >>> plt.xlabel('frequency Hz') >>> plt.ylabel('Coherence') >>> plt.show()

Constant extension at the boundaries of an array

Generate a new ndarray that is a constant extension of `x` along an axis.

The extension repeats the values at the first and last element of the axis.

Parameters ---------- x : ndarray The array to be extended. n : int The number of elements by which to extend `x` at each end of the axis. axis : int, optional The axis along which to extend `x`. Default is -1.

Examples -------- >>> from scipy.signal._arraytools import const_ext >>> a = np.array([1, 2, 3, 4, 5], [0, 1, 4, 9, 16]) >>> const_ext(a, 2) array([ 1, 1, 1, 2, 3, 4, 5, 5, 5], [ 0, 0, 0, 1, 4, 9, 16, 16, 16])

Constant extension continues with the same values as the endpoints of the array:

>>> t = np.linspace(0, 1.5, 100) >>> a = 0.9 * np.sin(2 * np.pi * t**2) >>> b = const_ext(a, 40) >>> import matplotlib.pyplot as plt >>> plt.plot(arange(-40, 140), b, 'b', lw=1, label='constant extension') >>> plt.plot(arange(100), a, 'r', lw=2, label='original') >>> plt.legend(loc='best') >>> plt.show()

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. 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 -------- >>> from scipy import signal >>> import matplotlib.pyplot as plt

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 = np.random.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 += np.random.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()

Even extension at the boundaries of an array

Generate a new ndarray by making an even extension of `x` along an axis.

Parameters ---------- x : ndarray The array to be extended. n : int The number of elements by which to extend `x` at each end of the axis. axis : int, optional The axis along which to extend `x`. Default is -1.

Examples -------- >>> from scipy.signal._arraytools import even_ext >>> a = np.array([1, 2, 3, 4, 5], [0, 1, 4, 9, 16]) >>> even_ext(a, 2) array([ 3, 2, 1, 2, 3, 4, 5, 4, 3], [ 4, 1, 0, 1, 4, 9, 16, 9, 4])

Even extension is a 'mirror image' at the boundaries of the original array:

>>> t = np.linspace(0, 1.5, 100) >>> a = 0.9 * np.sin(2 * np.pi * t**2) >>> b = even_ext(a, 40) >>> import matplotlib.pyplot as plt >>> plt.plot(arange(-40, 140), b, 'b', lw=1, label='even extension') >>> plt.plot(arange(100), a, 'r', lw=2, label='original') >>> plt.legend(loc='best') >>> plt.show()

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)

Perform the inverse Short Time Fourier transform (iSTFT).

Parameters ---------- Zxx : array_like STFT of the signal to be reconstructed. If a purely real array is passed, it will be cast to a complex data type. fs : float, optional Sampling frequency of the 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. Must match the window used to generate the STFT for faithful inversion. nperseg : int, optional Number of data points corresponding to each STFT segment. This parameter must be specified if the number of data points per segment is odd, or if the STFT was padded via ``nfft > nperseg``. If `None`, the value depends on the shape of `Zxx` and `input_onesided`. If `input_onesided` is `True`, ``nperseg=2*(Zxx.shapefreq_axis - 1)``. Otherwise, ``nperseg=Zxx.shapefreq_axis``. Defaults to `None`. noverlap : int, optional Number of points to overlap between segments. If `None`, half of the segment length. Defaults to `None`. When specified, the COLA constraint must be met (see Notes below), and should match the parameter used to generate the STFT. Defaults to `None`. nfft : int, optional Number of FFT points corresponding to each STFT segment. This parameter must be specified if the STFT was padded via ``nfft > nperseg``. If `None`, the default values are the same as for `nperseg`, detailed above, with one exception: if `input_onesided` is True and ``nperseg==2*Zxx.shapefreq_axis - 1``, `nfft` also takes on that value. This case allows the proper inversion of an odd-length unpadded STFT using ``nfft=None``. Defaults to `None`. input_onesided : bool, optional If `True`, interpret the input array as one-sided FFTs, such as is returned by `stft` with ``return_onesided=True`` and `numpy.fft.rfft`. If `False`, interpret the input as a a two-sided FFT. Defaults to `True`. boundary : bool, optional Specifies whether the input signal was extended at its boundaries by supplying a non-`None` ``boundary`` argument to `stft`. Defaults to `True`. time_axis : int, optional Where the time segments of the STFT is located; the default is the last axis (i.e. ``axis=-1``). freq_axis : int, optional Where the frequency axis of the STFT is located; the default is the penultimate axis (i.e. ``axis=-2``).

Returns ------- t : ndarray Array of output data times. x : ndarray iSTFT of `Zxx`.

See Also -------- stft: Short Time Fourier Transform check_COLA: Check whether the Constant OverLap Add (COLA) constraint is met check_NOLA: Check whether the Nonzero Overlap Add (NOLA) constraint is met

Notes ----- In order to enable inversion of an STFT via the inverse STFT with `istft`, the signal windowing must obey the constraint of 'nonzero overlap add' (NOLA):

.. math:: \sum_

w^

n-tH \ne 0

This ensures that the normalization factors that appear in the denominator of the overlap-add reconstruction equation

.. math:: xn=\frac\sum_{tx_

nwn-tH

}

\sum_{tw^

n-tH

}

are not zero. The NOLA constraint can be checked with the `check_NOLA` function.

An STFT which has been modified (via masking or otherwise) is not guaranteed to correspond to a exactly realizible signal. This function implements the iSTFT via the least-squares estimation algorithm detailed in 2_, which produces a signal that minimizes the mean squared error between the STFT of the returned signal and the modified STFT.

.. versionadded:: 0.19.0

References ---------- .. 1 Oppenheim, Alan V., Ronald W. Schafer, John R. Buck 'Discrete-Time Signal Processing', Prentice Hall, 1999. .. 2 Daniel W. Griffin, Jae S. Lim 'Signal Estimation from Modified Short-Time Fourier Transform', IEEE 1984, 10.1109/TASSP.1984.1164317

Examples -------- >>> from scipy import signal >>> import matplotlib.pyplot as plt

Generate a test signal, a 2 Vrms sine wave at 50Hz corrupted by 0.001 V**2/Hz of white noise sampled at 1024 Hz.

>>> fs = 1024 >>> N = 10*fs >>> nperseg = 512 >>> amp = 2 * np.sqrt(2) >>> noise_power = 0.001 * fs / 2 >>> time = np.arange(N) / float(fs) >>> carrier = amp * np.sin(2*np.pi*50*time) >>> noise = np.random.normal(scale=np.sqrt(noise_power), ... size=time.shape) >>> x = carrier + noise

Compute the STFT, and plot its magnitude

>>> f, t, Zxx = signal.stft(x, fs=fs, nperseg=nperseg) >>> plt.figure() >>> plt.pcolormesh(t, f, np.abs(Zxx), vmin=0, vmax=amp) >>> plt.ylim(f[1], f[-1]) >>> plt.title('STFT Magnitude') >>> plt.ylabel('Frequency Hz') >>> plt.xlabel('Time sec') >>> plt.yscale('log') >>> plt.show()

Zero the components that are 10% or less of the carrier magnitude, then convert back to a time series via inverse STFT

>>> Zxx = np.where(np.abs(Zxx) >= amp/10, Zxx, 0) >>> _, xrec = signal.istft(Zxx, fs)

Compare the cleaned signal with the original and true carrier signals.

>>> plt.figure() >>> plt.plot(time, x, time, xrec, time, carrier) >>> plt.xlim(2, 2.1) >>> plt.xlabel('Time sec') >>> plt.ylabel('Signal') >>> plt.legend('Carrier + Noise', 'Filtered via STFT', 'True Carrier') >>> plt.show()

Note that the cleaned signal does not start as abruptly as the original, since some of the coefficients of the transient were also removed:

>>> plt.figure() >>> plt.plot(time, x, time, xrec, time, carrier) >>> plt.xlim(0, 0.1) >>> plt.xlabel('Time sec') >>> plt.ylabel('Signal') >>> plt.legend('Carrier + Noise', 'Filtered via STFT', 'True Carrier') >>> plt.show()

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 one-dimensional 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 amplitudes 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.

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

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

Examples -------- >>> import matplotlib.pyplot as plt

First define some input parameters for the signal:

>>> A = 2. >>> w = 1. >>> phi = 0.5 * np.pi >>> nin = 1000 >>> nout = 100000 >>> frac_points = 0.9 # Fraction of points to select

Randomly select a fraction of an array with timesteps:

>>> r = np.random.rand(nin) >>> x = np.linspace(0.01, 10*np.pi, nin) >>> x = xr >= frac_points

Plot a sine wave for the selected times:

>>> y = A * np.sin(w*x+phi)

Define the array of frequencies for which to compute the periodogram:

>>> f = np.linspace(0.01, 10, nout)

Calculate Lomb-Scargle periodogram:

>>> import scipy.signal as signal >>> pgram = signal.lombscargle(x, y, f, normalize=True)

Now make a plot of the input data:

>>> plt.subplot(2, 1, 1) >>> plt.plot(x, y, 'b+')

Then plot the normalized periodogram:

>>> plt.subplot(2, 1, 2) >>> plt.plot(f, pgram) >>> plt.show()

Odd extension at the boundaries of an array

Generate a new ndarray by making an odd extension of `x` along an axis.

Parameters ---------- x : ndarray The array to be extended. n : int The number of elements by which to extend `x` at each end of the axis. axis : int, optional The axis along which to extend `x`. Default is -1.

Examples -------- >>> from scipy.signal._arraytools import odd_ext >>> a = np.array([1, 2, 3, 4, 5], [0, 1, 4, 9, 16]) >>> odd_ext(a, 2) array([-1, 0, 1, 2, 3, 4, 5, 6, 7], [-4, -1, 0, 1, 4, 9, 16, 23, 28])

Odd extension is a '180 degree rotation' at the endpoints of the original array:

>>> t = np.linspace(0, 1.5, 100) >>> a = 0.9 * np.sin(2 * np.pi * t**2) >>> b = odd_ext(a, 40) >>> import matplotlib.pyplot as plt >>> plt.plot(arange(-40, 140), b, 'b', lw=1, label='odd extension') >>> plt.plot(arange(100), a, 'r', lw=2, label='original') >>> plt.legend(loc='best') >>> plt.show()

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 nperseg. 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`.

Notes ----- .. versionadded:: 0.12.0

See Also -------- welch: Estimate power spectral density using Welch's method lombscargle: Lomb-Scargle periodogram for unevenly sampled data

Examples -------- >>> from scipy import signal >>> import matplotlib.pyplot as plt >>> np.random.seed(1234)

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 += np.random.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_den25000:) 0.00099728892368242854

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

============================================== Discrete Fourier transforms (:mod:`scipy.fft`) ==============================================

.. currentmodule:: scipy.fft

Fast Fourier Transforms (FFTs) ==============================

.. autosummary:: :toctree: generated/

fft - Fast (discrete) Fourier Transform (FFT) ifft - Inverse FFT fft2 - Two dimensional FFT ifft2 - Two dimensional inverse FFT fftn - n-dimensional FFT ifftn - n-dimensional inverse FFT rfft - FFT of strictly real-valued sequence irfft - Inverse of rfft rfft2 - Two dimensional FFT of real sequence irfft2 - Inverse of rfft2 rfftn - n-dimensional FFT of real sequence irfftn - Inverse of rfftn hfft - FFT of a Hermitian sequence (real spectrum) ihfft - Inverse of hfft hfft2 - Two dimensional FFT of a Hermitian sequence ihfft2 - Inverse of hfft2 hfftn - n-dimensional FFT of a Hermitian sequence ihfftn - Inverse of hfftn

Discrete Sin and Cosine Transforms (DST and DCT) ================================================ .. autosummary:: :toctree: generated/

dct - Discrete cosine transform idct - Inverse discrete cosine transform dctn - n-dimensional Discrete cosine transform idctn - n-dimensional Inverse discrete cosine transform dst - Discrete sine transform idst - Inverse discrete sine transform dstn - n-dimensional Discrete sine transform idstn - n-dimensional Inverse discrete sine transform

Helper functions ================

.. autosummary:: :toctree: generated/

fftshift - Shift the zero-frequency component to the center of the spectrum ifftshift - The inverse of `fftshift` fftfreq - Return the Discrete Fourier Transform sample frequencies rfftfreq - DFT sample frequencies (for usage with rfft, irfft) next_fast_len - Find the optimal length to zero-pad an FFT for speed set_workers - Context manager to set default number of workers get_workers - Get the current default number of workers

Backend control ===============

.. autosummary:: :toctree: generated/

set_backend - Context manager to set the backend within a fixed scope skip_backend - Context manager to skip a backend within a fixed scope set_global_backend - Sets the global fft backend register_backend - Register a backend for permanent use

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 -------- >>> from scipy import signal >>> from scipy.fft import fftshift >>> import matplotlib.pyplot as plt

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 = np.random.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) >>> 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)) >>> plt.ylabel('Frequency Hz') >>> plt.xlabel('Time sec') >>> plt.show()

Compute the Short Time Fourier Transform (STFT).

STFTs can be used as a way of quantifying the change of a nonstationary signal's frequency and phase 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 Hann window. nperseg : int, optional Length of each segment. Defaults to 256. noverlap : int, optional Number of points to overlap between segments. If `None`, ``noverlap = nperseg // 2``. Defaults to `None`. When specified, the COLA constraint must be met (see Notes below). 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 `False`. 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. boundary : str or None, optional Specifies whether the input signal is extended at both ends, and how to generate the new values, in order to center the first windowed segment on the first input point. This has the benefit of enabling reconstruction of the first input point when the employed window function starts at zero. Valid options are ``'even', 'odd', 'constant', 'zeros', None``. Defaults to 'zeros', for zero padding extension. I.e. ``1, 2, 3, 4`` is extended to ``0, 1, 2, 3, 4, 0`` for ``nperseg=3``. padded : bool, optional Specifies whether the input signal is zero-padded at the end to make the signal fit exactly into an integer number of window segments, so that all of the signal is included in the output. Defaults to `True`. Padding occurs after boundary extension, if `boundary` is not `None`, and `padded` is `True`, as is the default. axis : int, optional Axis along which the STFT is computed; the default is over the last axis (i.e. ``axis=-1``).

Returns ------- f : ndarray Array of sample frequencies. t : ndarray Array of segment times. Zxx : ndarray STFT of `x`. By default, the last axis of `Zxx` corresponds to the segment times.

See Also -------- istft: Inverse Short Time Fourier Transform check_COLA: Check whether the Constant OverLap Add (COLA) constraint is met check_NOLA: Check whether the Nonzero Overlap Add (NOLA) 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. lombscargle: Lomb-Scargle periodogram for unevenly sampled data

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), and the input signal must have complete windowing coverage (i.e. ``(x.shapeaxis - nperseg) % (nperseg-noverlap) == 0``). The `padded` argument may be used to accomplish this.

Given a time-domain signal :math:`xn`, a window :math:`wn`, and a hop size :math:`H` = `nperseg - noverlap`, the windowed frame at time index :math:`t` is given by

.. math:: x_

n=xnwn-tH

The overlap-add (OLA) reconstruction equation is given by

.. math:: xn=\frac\sum_{tx_

nwn-tH

}

\sum_{tw^

n-tH

}

The NOLA constraint ensures that every normalization term that appears in the denomimator of the OLA reconstruction equation is nonzero. Whether a choice of `window`, `nperseg`, and `noverlap` satisfy this constraint can be tested with `check_NOLA`.

.. versionadded:: 0.19.0

References ---------- .. 1 Oppenheim, Alan V., Ronald W. Schafer, John R. Buck 'Discrete-Time Signal Processing', Prentice Hall, 1999. .. 2 Daniel W. Griffin, Jae S. Lim 'Signal Estimation from Modified Short-Time Fourier Transform', IEEE 1984, 10.1109/TASSP.1984.1164317

Examples -------- >>> from scipy import signal >>> import matplotlib.pyplot as plt

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 = np.random.normal(scale=np.sqrt(noise_power), ... size=time.shape) >>> noise *= np.exp(-time/5) >>> x = carrier + noise

Compute and plot the STFT's magnitude.

>>> f, t, Zxx = signal.stft(x, fs, nperseg=1000) >>> plt.pcolormesh(t, f, np.abs(Zxx), vmin=0, vmax=amp) >>> plt.title('STFT Magnitude') >>> plt.ylabel('Frequency Hz') >>> plt.xlabel('Time sec') >>> plt.show()

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 -------- >>> from scipy import signal >>> import matplotlib.pyplot as plt >>> np.random.seed(1234)

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 += np.random.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_den256:) 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.

>>> xint(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()

Zero padding at the boundaries of an array

Generate a new ndarray that is a zero padded extension of `x` along an axis.

Parameters ---------- x : ndarray The array to be extended. n : int The number of elements by which to extend `x` at each end of the axis. axis : int, optional The axis along which to extend `x`. Default is -1.

Examples -------- >>> from scipy.signal._arraytools import zero_ext >>> a = np.array([1, 2, 3, 4, 5], [0, 1, 4, 9, 16]) >>> zero_ext(a, 2) array([ 0, 0, 1, 2, 3, 4, 5, 0, 0], [ 0, 0, 0, 1, 4, 9, 16, 0, 0])