""" Spectral analysis functions and utilities. Some of the functions defined here were ported directly from CuSignal under terms of the MIT license, under the following notice Copyright (c) 2019-2023 NVIDIA CORPORATION & AFFILIATES. All rights reserved. Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions: The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software. THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. """ import warnings import cupy from cupyx.scipy.signal.windows._windows import get_window from cupyx.scipy.signal._spectral_impl import ( _lombscargle, _spectral_helper, _median_bias, _triage_segments) 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 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 """ x = cupy.asarray(x, dtype=cupy.float64) y = cupy.asarray(y, dtype=cupy.float64) freqs = cupy.asarray(freqs, dtype=cupy.float64) pgram = cupy.empty(freqs.shape[0], dtype=cupy.float64) assert x.ndim == 1 assert y.ndim == 1 assert freqs.ndim == 1 # Check input sizes if x.shape[0] != y.shape[0]: raise ValueError("Input arrays do not have the same size.") y_dot = cupy.zeros(1, dtype=cupy.float64) if normalize: cupy.dot(y, y, out=y_dot) if precenter: y_in = y - y.mean() else: y_in = y _lombscargle(x, y_in, freqs, pgram, y_dot) 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 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`. See Also -------- welch: Estimate power spectral density using Welch's method lombscargle: Lomb-Scargle periodogram for unevenly sampled data """ x = cupy.asarray(x) if x.size == 0: return cupy.empty(x.shape), cupy.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]: # cupy.s_ not implemented s = [cupy.s_[:]] * len(x.shape) s[axis] = cupy.s_[:nfft] x = cupy.asarray(x[tuple(s)]) nperseg = nfft nfft = None 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'. 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]_. 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. """ 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. Defaults to 'mean'. 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. """ x = cupy.asarray(x) y = cupy.asarray(y) 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": Pxy = cupy.median(Pxy, axis=-1) / _median_bias(Pxy.shape[-1]) elif average == "mean": Pxy = Pxy.mean(axis=-1) else: raise ValueError( 'average must be "median" or "mean", got %s' % (average,) ) else: Pxy = cupy.reshape(Pxy, Pxy.shape[:-1]) return freqs, Pxy 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. See [1]_ for more information. 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 """ 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 = cupy.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 - cupy.median(binsums) return cupy.max(cupy.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. See [1]_, [2]_ for more information. 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 """ 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') if noverlap < 0: raise ValueError('noverlap must be a nonnegative integer') noverlap = int(noverlap) if isinstance(window, str) or type(window) is tuple: win = get_window(window, nperseg) else: win = cupy.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] ** 2 for ii in range(nperseg//step)) if nperseg % step != 0: binsums[:nperseg % step] += win[-(nperseg % step):]**2 return cupy.min(binsums) > tol def stft( x, fs=1.0, window="hann", nperseg=256, noverlap=None, nfft=None, detrend=False, return_onesided=True, boundary="zeros", padded=True, axis=-1, scaling='spectrum' ): r""" 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``). scaling: {'spectrum', 'psd'} The default 'spectrum' scaling allows each frequency line of `Zxx` to be interpreted as a magnitude spectrum. The 'psd' option scales each line to a power spectral density - it allows to calculate the signal's energy by numerically integrating over ``abs(Zxx)**2``. 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 -------- 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.shape[axis] - nperseg) % (nperseg-noverlap) == 0``). The `padded` argument may be used to accomplish this. Given a time-domain signal :math:`x[n]`, a window :math:`w[n]`, and a hop size :math:`H` = `nperseg - noverlap`, the windowed frame at time index :math:`t` is given by .. math:: x_{t}[n]=x[n]w[n-tH] The overlap-add (OLA) reconstruction equation is given by .. math:: x[n]=\frac{\sum_{t}x_{t}[n]w[n-tH]}{\sum_{t}w^{2}[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`. See [1]_, [2]_ for more information. 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 -------- >>> import cupy >>> import cupyx.scipy.signal import stft >>> 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 * cupy.sqrt(2) >>> noise_power = 0.01 * fs / 2 >>> time = cupy.arange(N) / float(fs) >>> mod = 500*cupy.cos(2*cupy.pi*0.25*time) >>> carrier = amp * cupy.sin(2*cupy.pi*3e3*time + mod) >>> noise = cupy.random.normal(scale=cupy.sqrt(noise_power), ... size=time.shape) >>> noise *= cupy.exp(-time/5) >>> x = carrier + noise Compute and plot the STFT's magnitude. >>> f, t, Zxx = stft(x, fs, nperseg=1000) >>> plt.pcolormesh(cupy.asnumpy(t), cupy.asnumpy(f), ... cupy.asnumpy(cupy.abs(Zxx)), vmin=0, vmax=amp) >>> plt.title('STFT Magnitude') >>> plt.ylabel('Frequency [Hz]') >>> plt.xlabel('Time [sec]') >>> plt.show() """ if scaling == 'psd': scaling = 'density' elif scaling != 'spectrum': raise ValueError(f"Parameter {scaling=} not in ['spectrum', 'psd']!") freqs, time, Zxx = _spectral_helper( x, x, fs, window, nperseg, noverlap, nfft, detrend, return_onesided, scaling=scaling, axis=axis, mode="stft", boundary=boundary, padded=padded, ) return freqs, time, Zxx def istft( Zxx, fs=1.0, window="hann", nperseg=None, noverlap=None, nfft=None, input_onesided=True, boundary=True, time_axis=-1, freq_axis=-2, scaling='spectrum' ): r""" 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.shape[freq_axis] - 1)``. Otherwise, ``nperseg=Zxx.shape[freq_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.shape[freq_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 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``). scaling: {'spectrum', 'psd'} The default 'spectrum' scaling allows each frequency line of `Zxx` to be interpreted as a magnitude spectrum. The 'psd' option scales each line to a power spectral density - it allows to calculate the signal's energy by numerically integrating over ``abs(Zxx)**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_{t}w^{2}[n-tH] \ne 0 This ensures that the normalization factors that appear in the denominator of the overlap-add reconstruction equation .. math:: x[n]=\frac{\sum_{t}x_{t}[n]w[n-tH]}{\sum_{t}w^{2}[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. See [1]_, [2]_ for more information. 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 -------- >>> import cupy >>> from cupyx.scipy.signal import stft, istft >>> 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 = cupy.arange(N) / float(fs) >>> carrier = amp * cupy.sin(2*cupy.pi*50*time) >>> noise = cupy.random.normal(scale=cupy.sqrt(noise_power), ... size=time.shape) >>> x = carrier + noise Compute the STFT, and plot its magnitude >>> f, t, Zxx = cusignal.stft(x, fs=fs, nperseg=nperseg) >>> f = cupy.asnumpy(f) >>> t = cupy.asnumpy(t) >>> Zxx = cupy.asnumpy(Zxx) >>> plt.figure() >>> plt.pcolormesh(t, f, np.abs(Zxx), vmin=0, vmax=amp, shading='gouraud') >>> 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 = cupy.where(cupy.abs(Zxx) >= amp/10, Zxx, 0) >>> _, xrec = cusignal.istft(Zxx, fs) >>> xrec = cupy.asnumpy(xrec) >>> x = cupy.asnumpy(x) >>> time = cupy.asnumpy(time) >>> carrier = cupy.asnumpy(carrier) 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() """ # Make sure input is an ndarray of appropriate complex dtype Zxx = cupy.asarray(Zxx) + 0j freq_axis = int(freq_axis) time_axis = int(time_axis) if Zxx.ndim < 2: raise ValueError("Input stft must be at least 2d!") if freq_axis == time_axis: raise ValueError("Must specify differing time and frequency axes!") nseg = Zxx.shape[time_axis] if input_onesided: # Assume even segment length n_default = 2 * (Zxx.shape[freq_axis] - 1) else: n_default = Zxx.shape[freq_axis] # Check windowing parameters if nperseg is None: nperseg = n_default else: nperseg = int(nperseg) if nperseg < 1: raise ValueError("nperseg must be a positive integer") if nfft is None: if (input_onesided) and (nperseg == n_default + 1): # Odd nperseg, no FFT padding nfft = nperseg else: nfft = n_default elif nfft < nperseg: raise ValueError("nfft must be greater than or equal to nperseg.") else: nfft = int(nfft) if noverlap is None: noverlap = nperseg // 2 else: noverlap = int(noverlap) if noverlap >= nperseg: raise ValueError("noverlap must be less than nperseg.") nstep = nperseg - noverlap # Rearrange axes if necessary if time_axis != Zxx.ndim - 1 or freq_axis != Zxx.ndim - 2: # Turn negative indices to positive for the call to transpose if freq_axis < 0: freq_axis = Zxx.ndim + freq_axis if time_axis < 0: time_axis = Zxx.ndim + time_axis zouter = list(range(Zxx.ndim)) for ax in sorted([time_axis, freq_axis], reverse=True): zouter.pop(ax) Zxx = cupy.transpose(Zxx, zouter + [freq_axis, time_axis]) # Get window as array if isinstance(window, str) or type(window) is tuple: win = get_window(window, nperseg) else: win = cupy.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 {0}".format(nperseg)) ifunc = cupy.fft.irfft if input_onesided else cupy.fft.ifft xsubs = ifunc(Zxx, axis=-2, n=nfft)[..., :nperseg, :] # Initialize output and normalization arrays outputlength = nperseg + (nseg - 1) * nstep x = cupy.zeros(list(Zxx.shape[:-2]) + [outputlength], dtype=xsubs.dtype) norm = cupy.zeros(outputlength, dtype=xsubs.dtype) if cupy.result_type(win, xsubs) != xsubs.dtype: win = win.astype(xsubs.dtype) if scaling == 'spectrum': xsubs *= win.sum() elif scaling == 'psd': xsubs *= cupy.sqrt(fs * cupy.sum(win**2)) else: raise ValueError(f"Parameter {scaling=} not in ['spectrum', 'psd']!") for ii in range(nseg): # Window the ifft x[..., ii * nstep:ii * nstep + nperseg] += xsubs[..., ii] * win norm[..., ii * nstep:ii * nstep + nperseg] += win**2 # Remove extension points if boundary: x = x[..., nperseg // 2: -(nperseg // 2)] norm = norm[..., nperseg // 2: -(nperseg // 2)] # Divide out normalization where non-tiny if cupy.sum(norm > 1e-10) != len(norm): warnings.warn("NOLA condition failed, STFT may not be invertible") x /= cupy.where(norm > 1e-10, norm, 1.0) if input_onesided: x = x.real # Put axes back if x.ndim > 1: if time_axis != Zxx.ndim - 1: if freq_axis < time_axis: time_axis -= 1 x = cupy.rollaxis(x, -1, time_axis) time = cupy.arange(x.shape[0]) / float(fs) return time, x def spectrogram( x, fs=1.0, window=("tukey", 0.25), 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. See [1]_ for more information. References ---------- .. [1] Oppenheim, Alan V., Ronald W. Schafer, John R. Buck "Discrete-Time Signal Processing", Prentice Hall, 1999. Examples -------- >>> import cupy >>> from cupyx.scipy.signal import spectrogram >>> 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 * cupy.sqrt(2) >>> noise_power = 0.01 * fs / 2 >>> time = cupy.arange(N) / float(fs) >>> mod = 500*cupy.cos(2*cupy.pi*0.25*time) >>> carrier = amp * cupy.sin(2*cupy.pi*3e3*time + mod) >>> noise = cupy.random.normal( ... scale=cupy.sqrt(noise_power), size=time.shape) >>> noise *= cupy.exp(-time/5) >>> x = carrier + noise Compute and plot the spectrogram. >>> f, t, Sxx = spectrogram(x, fs) >>> plt.pcolormesh(cupy.asnumpy(t), cupy.asnumpy(f), cupy.asnumpy(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 = spectrogram(x, fs, return_onesided=False) >>> plt.pcolormesh(cupy.asnumpy(t), cupy.fft.fftshift(f), \ cupy.fft.fftshift(Sxx, axes=0)) >>> 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 = cupy.abs(Sxx) elif mode in ["angle", "phase"]: Sxx = cupy.angle(Sxx) if mode == "phase": # Sxx has one additional dimension for time strides if axis < 0: axis -= 1 Sxx = cupy.unwrap(Sxx, axis=axis) # mode =='complex' is same as `stft`, doesn't need modification return freqs, time, Sxx def coherence( x, y, fs=1.0, window="hann", nperseg=None, noverlap=None, nfft=None, detrend="constant", axis=-1, ): r""" 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. See [1]_ and [2]_ for more information. 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 -------- >>> import cupy as cp >>> from cupyx.scipy.signal import butter, lfilter, coherence >>> 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 = cupy.arange(N) / fs >>> b, a = butter(2, 0.25, 'low') >>> x = cupy.random.normal( ... scale=cupy.sqrt(noise_power), size=time.shape) >>> y = lfilter(b, a, x) >>> x += amp * cupy.sin(2*cupy.pi*freq*time) >>> y += cupy.random.normal( ... scale=0.1*cupy.sqrt(noise_power), size=time.shape) Compute and plot the coherence. >>> f, Cxy = coherence(x, y, fs, nperseg=1024) >>> plt.semilogy(cupy.asnumpy(f), cupy.asnumpy(Cxy)) >>> plt.xlabel('frequency [Hz]') >>> plt.ylabel('Coherence') >>> plt.show() """ freqs, Pxx = welch( x, fs=fs, window=window, nperseg=nperseg, noverlap=noverlap, nfft=nfft, detrend=detrend, axis=axis, ) _, Pyy = welch( y, fs=fs, window=window, nperseg=nperseg, noverlap=noverlap, nfft=nfft, detrend=detrend, axis=axis, ) _, Pxy = csd( x, y, fs=fs, window=window, nperseg=nperseg, noverlap=noverlap, nfft=nfft, detrend=detrend, axis=axis, ) Cxy = cupy.abs(Pxy) ** 2 / Pxx / Pyy return freqs, Cxy def vectorstrength(events, period): """ Determine the vector strength of the events corresponding to the given period. The vector strength is a measure of phase synchrony, how well the timing of the events is synchronized to a single period of a periodic signal. If multiple periods are used, calculate the vector strength of each. This is called the "resonating vector strength". Parameters ---------- events : 1D array_like An array of time points containing the timing of the events. period : float or array_like The period of the signal that the events should synchronize to. The period is in the same units as `events`. It can also be an array of periods, in which case the outputs are arrays of the same length. Returns ------- strength : float or 1D array The strength of the synchronization. 1.0 is perfect synchronization and 0.0 is no synchronization. If `period` is an array, this is also an array with each element containing the vector strength at the corresponding period. phase : float or array The phase that the events are most strongly synchronized to in radians. If `period` is an array, this is also an array with each element containing the phase for the corresponding period. Notes ----- See [1]_, [2]_ and [3]_ for more information. References ---------- .. [1] van Hemmen, JL, Longtin, A, and Vollmayr, AN. Testing resonating vector strength: Auditory system, electric fish, and noise. Chaos 21, 047508 (2011). .. [2] van Hemmen, JL. Vector strength after Goldberg, Brown, and von Mises: biological and mathematical perspectives. Biol Cybern. 2013 Aug;107(4):385-96. .. [3] van Hemmen, JL and Vollmayr, AN. Resonating vector strength: what happens when we vary the "probing" frequency while keeping the spike times fixed. Biol Cybern. 2013 Aug;107(4):491-94. """ events = cupy.asarray(events) period = cupy.asarray(period) if events.ndim > 1: raise ValueError("events cannot have dimensions more than 1") if period.ndim > 1: raise ValueError("period cannot have dimensions more than 1") # we need to know later if period was originally a scalar scalarperiod = not period.ndim events = cupy.atleast_2d(events) period = cupy.atleast_2d(period) if (period <= 0).any(): raise ValueError("periods must be positive") # this converts the times to vectors vectors = cupy.exp(cupy.dot(2j * cupy.pi / period.T, events)) # the vector strength is just the magnitude of the mean of the vectors # the vector phase is the angle of the mean of the vectors vectormean = cupy.mean(vectors, axis=1) strength = cupy.abs(vectormean) phase = cupy.angle(vectormean) # if the original period was a scalar, return scalars if scalarperiod: strength = strength[0] phase = phase[0] return strength, phase