"""Tools for spectral analysis. """ import numpy as np import numpy.typing as npt from scipy import fft as sp_fft from . import _signaltools from .windows import get_window from ._arraytools import const_ext, even_ext, odd_ext, zero_ext import warnings from typing import Literal __all__ = ['periodogram', 'welch', 'lombscargle', 'csd', 'coherence', 'spectrogram', 'stft', 'istft', 'check_COLA', 'check_NOLA'] def lombscargle( x: npt.ArrayLike, y: npt.ArrayLike, freqs: npt.ArrayLike, precenter: bool = False, normalize: bool | Literal["power", "normalize", "amplitude"] = False, *, weights: npt.NDArray | None = None, floating_mean: bool = False, ) -> npt.NDArray: """ Compute the generalized 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. The algorithm used here is based on a weighted least-squares fit of the form ``y(ω) = a*cos(ω*x) + b*sin(ω*x) + c``, where the fit is calculated for each frequency independently. This algorithm was developed by Zechmeister and Kürster which improves the Lomb-Scargle periodogram by enabling the weighting of individual samples and calculating an unknown y offset (also called a "floating-mean" model) [3]_. For more details, and practical considerations, see the excellent reference on the Lomb-Scargle periodogram [4]_. When *normalize* is False (or "power") (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. Where N is the length of x or y. When *normalize* is True (or "normalize") the computed periodogram is normalized by the residuals of the data around a constant reference model (at zero). When *normalize* is "amplitude" the computed periodogram is the complex representation of the amplitude and phase. Input arrays should be 1-D of a real floating data type, which are converted into float64 arrays before processing. Parameters ---------- x : array_like Sample times. y : array_like Measurement values. Values are assumed to have a baseline of ``y = 0``. If there is a possibility of a y offset, it is recommended to set `floating_mean` to True. freqs : array_like Angular frequencies (e.g., having unit rad/s=2π/s for `x` having unit s) for output periodogram. Frequencies are normally >= 0, as any peak at ``-freq`` will also exist at ``+freq``. precenter : bool, optional Pre-center measurement values by subtracting the mean, if True. This is a legacy parameter and unnecessary if `floating_mean` is True. normalize : bool | str, optional Compute normalized or complex (amplitude + phase) periodogram. Valid options are: ``False``/``"power"``, ``True``/``"normalize"``, or ``"amplitude"``. weights : array_like, optional Weights for each sample. Weights must be nonnegative. floating_mean : bool, optional Determines a y offset for each frequency independently, if True. Else the y offset is assumed to be `0`. Returns ------- pgram : array_like Lomb-Scargle periodogram. Raises ------ ValueError If any of the input arrays x, y, freqs, or weights are not 1D, or if any are zero length. Or, if the input arrays x, y, and weights do not have the same shape as each other. ValueError If any weight is < 0, or the sum of the weights is <= 0. ValueError If the normalize parameter is not one of the allowed options. See Also -------- periodogram: Power spectral density using a periodogram welch: Power spectral density by Welch's method csd: Cross spectral density by Welch's method Notes ----- The algorithm used will not automatically account for any unknown y offset, unless floating_mean is True. Therefore, for most use cases, if there is a possibility of a y offset, it is recommended to set floating_mean to True. If precenter is True, it performs the operation ``y -= y.mean()``. However, precenter is a legacy parameter, and unnecessary when floating_mean is True. Furthermore, the mean removed by precenter does not account for sample weights, nor will it correct for any bias due to consistently missing observations at peaks and/or troughs. When the normalize parameter is "amplitude", for any frequency in freqs that is below ``(2*pi)/(x.max() - x.min())``, the predicted amplitude will tend towards infinity. The concept of a "Nyquist frequency" limit (see Nyquist-Shannon sampling theorem) is not generally applicable to unevenly sampled data. Therefore, with unevenly sampled data, valid frequencies in freqs can often be much higher than expected. References ---------- .. [1] N.R. Lomb "Least-squares frequency analysis of unequally spaced data", Astrophysics and Space Science, vol 39, pp. 447-462, 1976 :doi:`10.1007/bf00648343` .. [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 :doi:`10.1086/160554` .. [3] M. Zechmeister and M. Kürster, "The generalised Lomb-Scargle periodogram. A new formalism for the floating-mean and Keplerian periodograms," Astronomy and Astrophysics, vol. 496, pp. 577-584, 2009 :doi:`10.1051/0004-6361:200811296` .. [4] J.T. VanderPlas, "Understanding the Lomb-Scargle Periodogram," The Astrophysical Journal Supplement Series, vol. 236, no. 1, p. 16, May 2018 :doi:`10.3847/1538-4365/aab766` Examples -------- >>> import numpy as np >>> rng = np.random.default_rng() First define some input parameters for the signal: >>> A = 2. # amplitude >>> c = 2. # offset >>> w0 = 1. # rad/sec >>> nin = 150 >>> nout = 1002 Randomly generate sample times: >>> x = rng.uniform(0, 10*np.pi, nin) Plot a sine wave for the selected times: >>> y = A * np.cos(w0*x) + c Define the array of frequencies for which to compute the periodogram: >>> w = np.linspace(0.25, 10, nout) Calculate Lomb-Scargle periodogram for each of the normalize options: >>> from scipy.signal import lombscargle >>> pgram_power = lombscargle(x, y, w, normalize=False) >>> pgram_norm = lombscargle(x, y, w, normalize=True) >>> pgram_amp = lombscargle(x, y, w, normalize='amplitude') ... >>> pgram_power_f = lombscargle(x, y, w, normalize=False, floating_mean=True) >>> pgram_norm_f = lombscargle(x, y, w, normalize=True, floating_mean=True) >>> pgram_amp_f = lombscargle(x, y, w, normalize='amplitude', floating_mean=True) Now make a plot of the input data: >>> import matplotlib.pyplot as plt >>> fig, (ax_t, ax_p, ax_n, ax_a) = plt.subplots(4, 1, figsize=(5, 6)) >>> ax_t.plot(x, y, 'b+') >>> ax_t.set_xlabel('Time [s]') >>> ax_t.set_ylabel('Amplitude') Then plot the periodogram for each of the normalize options, as well as with and without floating_mean=True: >>> ax_p.plot(w, pgram_power, label='default') >>> ax_p.plot(w, pgram_power_f, label='floating_mean=True') >>> ax_p.set_xlabel('Angular frequency [rad/s]') >>> ax_p.set_ylabel('Power') >>> ax_p.legend(prop={'size': 7}) ... >>> ax_n.plot(w, pgram_norm, label='default') >>> ax_n.plot(w, pgram_norm_f, label='floating_mean=True') >>> ax_n.set_xlabel('Angular frequency [rad/s]') >>> ax_n.set_ylabel('Normalized') >>> ax_n.legend(prop={'size': 7}) ... >>> ax_a.plot(w, np.abs(pgram_amp), label='default') >>> ax_a.plot(w, np.abs(pgram_amp_f), label='floating_mean=True') >>> ax_a.set_xlabel('Angular frequency [rad/s]') >>> ax_a.set_ylabel('Amplitude') >>> ax_a.legend(prop={'size': 7}) ... >>> plt.tight_layout() >>> plt.show() """ # if no weights are provided, assume all data points are equally important if weights is None: weights = np.ones_like(y, dtype=np.float64) else: # if provided, make sure weights is an array and cast to float64 weights = np.asarray(weights, dtype=np.float64) # make sure other inputs are arrays and cast to float64 # done before validation, in case they were not arrays x = np.asarray(x, dtype=np.float64) y = np.asarray(y, dtype=np.float64) freqs = np.asarray(freqs, dtype=np.float64) # validate input shapes if not (x.ndim == 1 and x.size > 0 and x.shape == y.shape == weights.shape): raise ValueError("Parameters x, y, weights must be 1-D arrays of " "equal non-zero length!") if not (freqs.ndim == 1 and freqs.size > 0): raise ValueError("Parameter freqs must be a 1-D array of non-zero length!") # validate weights if not (np.all(weights >= 0) and np.sum(weights) > 0): raise ValueError("Parameter weights must have only non-negative entries " "which sum to a positive value!") # validate normalize parameter if isinstance(normalize, bool): # if bool, convert to str literal normalize = "normalize" if normalize else "power" if normalize not in ["power", "normalize", "amplitude"]: raise ValueError( "Normalize must be: False (or 'power'), True (or 'normalize'), " "or 'amplitude'." ) # weight vector must sum to 1 weights *= 1.0 / weights.sum() # if requested, perform precenter if precenter: y -= y.mean() # transform arrays # row vector freqs = freqs.reshape(1, -1) # column vectors x = x.reshape(-1, 1) y = y.reshape(-1, 1) weights = weights.reshape(-1, 1) # store frequent intermediates weights_y = weights * y freqst = freqs * x coswt = np.cos(freqst) sinwt = np.sin(freqst) Y = np.dot(weights.T, y) # Eq. 7 CC = np.dot(weights.T, coswt * coswt) # Eq. 13 SS = 1.0 - CC # trig identity: S^2 = 1 - C^2 Eq.14 CS = np.dot(weights.T, coswt * sinwt) # Eq. 15 if floating_mean: C = np.dot(weights.T, coswt) # Eq. 8 S = np.dot(weights.T, sinwt) # Eq. 9 CC -= C * C # Eq. 13 SS -= S * S # Eq. 14 CS -= C * S # Eq. 15 # calculate tau (phase offset to eliminate CS variable) tau = 0.5 * np.arctan2(2.0 * CS, CC - SS) # Eq. 19 freqst_tau = freqst - tau # coswt and sinwt are now offset by tau, which eliminates CS coswt_tau = np.cos(freqst_tau) sinwt_tau = np.sin(freqst_tau) YC = np.dot(weights_y.T, coswt_tau) # Eq. 11 YS = np.dot(weights_y.T, sinwt_tau) # Eq. 12 CC = np.dot(weights.T, coswt_tau * coswt_tau) # Eq. 13, CC range is [0.5, 1.0] SS = 1.0 - CC # trig identity: S^2 = 1 - C^2 Eq. 14, SS range is [0.0, 0.5] if floating_mean: C = np.dot(weights.T, coswt_tau) # Eq. 8 S = np.dot(weights.T, sinwt_tau) # Eq. 9 YC -= Y * C # Eq. 11 YS -= Y * S # Eq. 12 CC -= C * C # Eq. 13, CC range is now [0.0, 1.0] SS -= S * S # Eq. 14, SS range is now [0.0, 0.5] # to prevent division by zero errors with a and b, as well as correcting for # numerical precision errors that lead to CC or SS being approximately -0.0, # make sure CC and SS are both > 0 epsneg = np.finfo(dtype=y.dtype).epsneg CC[CC < epsneg] = epsneg SS[SS < epsneg] = epsneg # calculate a and b # where: y(w) = a*cos(w) + b*sin(w) + c a = YC / CC # Eq. A.4 and 6, eliminating CS b = YS / SS # Eq. A.4 and 6, eliminating CS # c = Y - a * C - b * S # store final value as power in A^2 (i.e., (y units)^2) pgram = 2.0 * (a * YC + b * YS) # squeeze back to a vector pgram = np.squeeze(pgram) if normalize == "power": # (default) # return the legacy power units ((A**2) * N/4) pgram *= float(x.shape[0]) / 4.0 elif normalize == "normalize": # return the normalized power (power at current frequency wrt the entire signal) # range will be [0, 1] YY = np.dot(weights_y.T, y) # Eq. 10 if floating_mean: YY -= Y * Y # Eq. 10 pgram *= 0.5 / np.squeeze(YY) # Eq. 20 else: # normalize == "amplitude": # return the complex representation of the best-fit amplitude and phase # squeeze back to vectors a = np.squeeze(a) b = np.squeeze(b) tau = np.squeeze(tau) # calculate the complex representation, and correct for tau rotation pgram = (a + 1j * b) * np.exp(1j * tau) return pgram def periodogram(x, fs=1.0, window='boxcar', nfft=None, detrend='constant', return_onesided=True, scaling='density', axis=-1): """ Estimate power spectral density using a periodogram. Parameters ---------- x : array_like Time series of measurement values fs : float, optional Sampling frequency of the `x` time series. Defaults to 1.0. window : str or tuple or array_like, optional Desired window to use. If `window` is a string or tuple, it is passed to `get_window` to generate the window values, which are DFT-even by default. See `get_window` for a list of windows and required parameters. If `window` is array_like it will be used directly as the window and its length must be equal to the length of the axis over which the periodogram is computed. Defaults to 'boxcar'. nfft : int, optional Length of the FFT used. If `None` the length of `x` will be used. detrend : str or function or `False`, optional Specifies how to detrend each segment. If `detrend` is a string, it is passed as the `type` argument to the `detrend` function. If it is a function, it takes a segment and returns a detrended segment. If `detrend` is `False`, no detrending is done. Defaults to 'constant'. return_onesided : bool, optional If `True`, return a one-sided spectrum for real data. If `False` return a two-sided spectrum. Defaults to `True`, but for complex data, a two-sided spectrum is always returned. scaling : { 'density', 'spectrum' }, optional Selects between computing the power spectral density ('density') where `Pxx` has units of V**2/Hz and computing the squared magnitude spectrum ('spectrum') where `Pxx` has units of V**2, if `x` is measured in V and `fs` is measured in Hz. Defaults to 'density' axis : int, optional Axis along which the periodogram is computed; the default is over the last axis (i.e. ``axis=-1``). Returns ------- f : ndarray Array of sample frequencies. Pxx : ndarray Power spectral density or power spectrum of `x`. See Also -------- welch: Estimate power spectral density using Welch's method lombscargle: Lomb-Scargle periodogram for unevenly sampled data Notes ----- Consult the :ref:`tutorial_SpectralAnalysis` section of the :ref:`user_guide` for a discussion of the scalings of the power spectral density and the magnitude (squared) spectrum. .. versionadded:: 0.12.0 Examples -------- >>> import numpy as np >>> from scipy import signal >>> import matplotlib.pyplot as plt >>> rng = np.random.default_rng() Generate a test signal, a 2 Vrms sine wave at 1234 Hz, corrupted by 0.001 V**2/Hz of white noise sampled at 10 kHz. >>> fs = 10e3 >>> N = 1e5 >>> amp = 2*np.sqrt(2) >>> freq = 1234.0 >>> noise_power = 0.001 * fs / 2 >>> time = np.arange(N) / fs >>> x = amp*np.sin(2*np.pi*freq*time) >>> x += rng.normal(scale=np.sqrt(noise_power), size=time.shape) Compute and plot the power spectral density. >>> f, Pxx_den = signal.periodogram(x, fs) >>> plt.semilogy(f, Pxx_den) >>> plt.ylim([1e-7, 1e2]) >>> plt.xlabel('frequency [Hz]') >>> plt.ylabel('PSD [V**2/Hz]') >>> plt.show() If we average the last half of the spectral density, to exclude the peak, we can recover the noise power on the signal. >>> np.mean(Pxx_den[25000:]) 0.000985320699252543 Now compute and plot the power spectrum. >>> f, Pxx_spec = signal.periodogram(x, fs, 'flattop', scaling='spectrum') >>> plt.figure() >>> plt.semilogy(f, np.sqrt(Pxx_spec)) >>> plt.ylim([1e-4, 1e1]) >>> plt.xlabel('frequency [Hz]') >>> plt.ylabel('Linear spectrum [V RMS]') >>> plt.show() The peak height in the power spectrum is an estimate of the RMS amplitude. >>> np.sqrt(Pxx_spec.max()) 2.0077340678640727 """ x = np.asarray(x) if x.size == 0: return np.empty(x.shape), np.empty(x.shape) if window is None: window = 'boxcar' if nfft is None: nperseg = x.shape[axis] elif nfft == x.shape[axis]: nperseg = nfft elif nfft > x.shape[axis]: nperseg = x.shape[axis] elif nfft < x.shape[axis]: s = [np.s_[:]]*len(x.shape) s[axis] = np.s_[:nfft] x = x[tuple(s)] nperseg = nfft nfft = None if hasattr(window, 'size'): if window.size != nperseg: raise ValueError('the size of the window must be the same size ' 'of the input on the specified axis') return welch(x, fs=fs, window=window, nperseg=nperseg, noverlap=0, nfft=nfft, detrend=detrend, return_onesided=return_onesided, scaling=scaling, axis=axis) def welch(x, fs=1.0, window='hann', nperseg=None, noverlap=None, nfft=None, detrend='constant', return_onesided=True, scaling='density', axis=-1, average='mean'): r""" Estimate power spectral density using Welch's method. Welch's method [1]_ computes an estimate of the power spectral density by dividing the data into overlapping segments, computing a modified periodogram for each segment and averaging the periodograms. Parameters ---------- x : array_like Time series of measurement values fs : float, optional Sampling frequency of the `x` time series. Defaults to 1.0. window : str or tuple or array_like, optional Desired window to use. If `window` is a string or tuple, it is passed to `get_window` to generate the window values, which are DFT-even by default. See `get_window` for a list of windows and required parameters. If `window` is array_like it will be used directly as the window and its length must be nperseg. Defaults to a Hann window. nperseg : int, optional Length of each segment. Defaults to None, but if window is str or tuple, is set to 256, and if window is array_like, is set to the length of the window. noverlap : int, optional Number of points to overlap between segments. If `None`, ``noverlap = nperseg // 2``. Defaults to `None`. nfft : int, optional Length of the FFT used, if a zero padded FFT is desired. If `None`, the FFT length is `nperseg`. Defaults to `None`. detrend : str or function or `False`, optional Specifies how to detrend each segment. If `detrend` is a string, it is passed as the `type` argument to the `detrend` function. If it is a function, it takes a segment and returns a detrended segment. If `detrend` is `False`, no detrending is done. Defaults to 'constant'. return_onesided : bool, optional If `True`, return a one-sided spectrum for real data. If `False` return a two-sided spectrum. Defaults to `True`, but for complex data, a two-sided spectrum is always returned. scaling : { 'density', 'spectrum' }, optional Selects between computing the power spectral density ('density') where `Pxx` has units of V**2/Hz and computing the squared magnitude 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]_. Consult the :ref:`tutorial_SpectralAnalysis` section of the :ref:`user_guide` for a discussion of the scalings of the power spectral density and the (squared) magnitude spectrum. .. versionadded:: 0.12.0 References ---------- .. [1] P. Welch, "The use of the fast Fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms", IEEE Trans. Audio Electroacoust. vol. 15, pp. 70-73, 1967. .. [2] M.S. Bartlett, "Periodogram Analysis and Continuous Spectra", Biometrika, vol. 37, pp. 1-16, 1950. Examples -------- >>> import numpy as np >>> from scipy import signal >>> import matplotlib.pyplot as plt >>> rng = np.random.default_rng() Generate a test signal, a 2 Vrms sine wave at 1234 Hz, corrupted by 0.001 V**2/Hz of white noise sampled at 10 kHz. >>> fs = 10e3 >>> N = 1e5 >>> amp = 2*np.sqrt(2) >>> freq = 1234.0 >>> noise_power = 0.001 * fs / 2 >>> time = np.arange(N) / fs >>> x = amp*np.sin(2*np.pi*freq*time) >>> x += rng.normal(scale=np.sqrt(noise_power), size=time.shape) Compute and plot the power spectral density. >>> f, Pxx_den = signal.welch(x, fs, nperseg=1024) >>> plt.semilogy(f, Pxx_den) >>> plt.ylim([0.5e-3, 1]) >>> plt.xlabel('frequency [Hz]') >>> plt.ylabel('PSD [V**2/Hz]') >>> plt.show() If we average the last half of the spectral density, to exclude the peak, we can recover the noise power on the signal. >>> np.mean(Pxx_den[256:]) 0.0009924865443739191 Now compute and plot the power spectrum. >>> f, Pxx_spec = signal.welch(x, fs, 'flattop', 1024, scaling='spectrum') >>> plt.figure() >>> plt.semilogy(f, np.sqrt(Pxx_spec)) >>> plt.xlabel('frequency [Hz]') >>> plt.ylabel('Linear spectrum [V RMS]') >>> plt.show() The peak height in the power spectrum is an estimate of the RMS amplitude. >>> np.sqrt(Pxx_spec.max()) 2.0077340678640727 If we now introduce a discontinuity in the signal, by increasing the amplitude of a small portion of the signal by 50, we can see the corruption of the mean average power spectral density, but using a median average better estimates the normal behaviour. >>> x[int(N//2):int(N//2)+10] *= 50. >>> f, Pxx_den = signal.welch(x, fs, nperseg=1024) >>> f_med, Pxx_den_med = signal.welch(x, fs, nperseg=1024, average='median') >>> plt.semilogy(f, Pxx_den, label='mean') >>> plt.semilogy(f_med, Pxx_den_med, label='median') >>> plt.ylim([0.5e-3, 1]) >>> plt.xlabel('frequency [Hz]') >>> plt.ylabel('PSD [V**2/Hz]') >>> plt.legend() >>> plt.show() """ freqs, Pxx = csd(x, x, fs=fs, window=window, nperseg=nperseg, noverlap=noverlap, nfft=nfft, detrend=detrend, return_onesided=return_onesided, scaling=scaling, axis=axis, average=average) return freqs, Pxx.real def csd(x, y, fs=1.0, window='hann', nperseg=None, noverlap=None, nfft=None, detrend='constant', return_onesided=True, scaling='density', axis=-1, average='mean'): r""" Estimate the cross power spectral density, Pxy, using Welch's method. Parameters ---------- x : array_like Time series of measurement values y : array_like Time series of measurement values fs : float, optional Sampling frequency of the `x` and `y` time series. Defaults to 1.0. window : str or tuple or array_like, optional Desired window to use. If `window` is a string or tuple, it is passed to `get_window` to generate the window values, which are DFT-even by default. See `get_window` for a list of windows and required parameters. If `window` is array_like it will be used directly as the window and its length must be nperseg. Defaults to a Hann window. nperseg : int, optional Length of each segment. Defaults to None, but if window is str or tuple, is set to 256, and if window is array_like, is set to the length of the window. noverlap: int, optional Number of points to overlap between segments. If `None`, ``noverlap = nperseg // 2``. Defaults to `None`. nfft : int, optional Length of the FFT used, if a zero padded FFT is desired. If `None`, the FFT length is `nperseg`. Defaults to `None`. detrend : str or function or `False`, optional Specifies how to detrend each segment. If `detrend` is a string, it is passed as the `type` argument to the `detrend` function. If it is a function, it takes a segment and returns a detrended segment. If `detrend` is `False`, no detrending is done. Defaults to 'constant'. return_onesided : bool, optional If `True`, return a one-sided spectrum for real data. If `False` return a two-sided spectrum. Defaults to `True`, but for complex data, a two-sided spectrum is always returned. scaling : { 'density', 'spectrum' }, optional Selects between computing the cross spectral density ('density') where `Pxy` has units of V**2/Hz and computing the cross spectrum ('spectrum') where `Pxy` has units of V**2, if `x` and `y` are measured in V and `fs` is measured in Hz. Defaults to 'density' axis : int, optional Axis along which the CSD is computed for both inputs; the default is over the last axis (i.e. ``axis=-1``). average : { 'mean', 'median' }, optional Method to use when averaging periodograms. If the spectrum is complex, the average is computed separately for the real and imaginary parts. Defaults to 'mean'. .. versionadded:: 1.2.0 Returns ------- f : ndarray Array of sample frequencies. Pxy : ndarray Cross spectral density or cross power spectrum of x,y. See Also -------- periodogram: Simple, optionally modified periodogram lombscargle: Lomb-Scargle periodogram for unevenly sampled data welch: Power spectral density by Welch's method. [Equivalent to csd(x,x)] coherence: Magnitude squared coherence by Welch's method. Notes ----- By convention, Pxy is computed with the conjugate FFT of X multiplied by the FFT of Y. If the input series differ in length, the shorter series will be zero-padded to match. An appropriate amount of overlap will depend on the choice of window and on your requirements. For the default Hann window an overlap of 50% is a reasonable trade off between accurately estimating the signal power, while not over counting any of the data. Narrower windows may require a larger overlap. Consult the :ref:`tutorial_SpectralAnalysis` section of the :ref:`user_guide` for a discussion of the scalings of a spectral density and an (amplitude) spectrum. .. versionadded:: 0.16.0 References ---------- .. [1] P. Welch, "The use of the fast Fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms", IEEE Trans. Audio Electroacoust. vol. 15, pp. 70-73, 1967. .. [2] Rabiner, Lawrence R., and B. Gold. "Theory and Application of Digital Signal Processing" Prentice-Hall, pp. 414-419, 1975 Examples -------- >>> import numpy as np >>> from scipy import signal >>> import matplotlib.pyplot as plt >>> rng = np.random.default_rng() Generate two test signals with some common features. >>> fs = 10e3 >>> N = 1e5 >>> amp = 20 >>> freq = 1234.0 >>> noise_power = 0.001 * fs / 2 >>> time = np.arange(N) / fs >>> b, a = signal.butter(2, 0.25, 'low') >>> x = rng.normal(scale=np.sqrt(noise_power), size=time.shape) >>> y = signal.lfilter(b, a, x) >>> x += amp*np.sin(2*np.pi*freq*time) >>> y += rng.normal(scale=0.1*np.sqrt(noise_power), size=time.shape) Compute and plot the magnitude of the cross spectral density. >>> f, Pxy = signal.csd(x, y, fs, nperseg=1024) >>> plt.semilogy(f, np.abs(Pxy)) >>> plt.xlabel('frequency [Hz]') >>> plt.ylabel('CSD [V**2/Hz]') >>> plt.show() """ freqs, _, Pxy = _spectral_helper(x, y, fs, window, nperseg, noverlap, nfft, detrend, return_onesided, scaling, axis, mode='psd') # Average over windows. if len(Pxy.shape) >= 2 and Pxy.size > 0: if Pxy.shape[-1] > 1: if average == 'median': # np.median must be passed real arrays for the desired result bias = _median_bias(Pxy.shape[-1]) if np.iscomplexobj(Pxy): Pxy = (np.median(np.real(Pxy), axis=-1) + 1j * np.median(np.imag(Pxy), axis=-1)) else: Pxy = np.median(Pxy, axis=-1) Pxy /= bias elif average == 'mean': Pxy = Pxy.mean(axis=-1) else: raise ValueError(f'average must be "median" or "mean", got {average}') else: Pxy = np.reshape(Pxy, Pxy.shape[:-1]) return freqs, Pxy def spectrogram(x, fs=1.0, window=('tukey', .25), nperseg=None, noverlap=None, nfft=None, detrend='constant', return_onesided=True, scaling='density', axis=-1, mode='psd'): """Compute a spectrogram with consecutive Fourier transforms (legacy function). Spectrograms can be used as a way of visualizing the change of a nonstationary signal's frequency content over time. .. legacy:: function :class:`ShortTimeFFT` is a newer STFT / ISTFT implementation with more features also including a :meth:`~ShortTimeFFT.spectrogram` method. A :ref:`comparison ` between the implementations can be found in the :ref:`tutorial_stft` section of the :ref:`user_guide`. 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. ShortTimeFFT: Newer STFT/ISTFT implementation providing more features, which also includes a :meth:`~ShortTimeFFT.spectrogram` method. Notes ----- An appropriate amount of overlap will depend on the choice of window and on your requirements. In contrast to welch's method, where the entire data stream is averaged over, one may wish to use a smaller overlap (or perhaps none at all) when computing a spectrogram, to maintain some statistical independence between individual segments. It is for this reason that the default window is a Tukey window with 1/8th of a window's length overlap at each end. .. versionadded:: 0.16.0 References ---------- .. [1] Oppenheim, Alan V., Ronald W. Schafer, John R. Buck "Discrete-Time Signal Processing", Prentice Hall, 1999. Examples -------- >>> import numpy as np >>> from scipy import signal >>> from scipy.fft import fftshift >>> import matplotlib.pyplot as plt >>> rng = np.random.default_rng() Generate a test signal, a 2 Vrms sine wave whose frequency is slowly modulated around 3kHz, corrupted by white noise of exponentially decreasing magnitude sampled at 10 kHz. >>> fs = 10e3 >>> N = 1e5 >>> amp = 2 * np.sqrt(2) >>> noise_power = 0.01 * fs / 2 >>> time = np.arange(N) / float(fs) >>> mod = 500*np.cos(2*np.pi*0.25*time) >>> carrier = amp * np.sin(2*np.pi*3e3*time + mod) >>> noise = rng.normal(scale=np.sqrt(noise_power), size=time.shape) >>> noise *= np.exp(-time/5) >>> x = carrier + noise Compute and plot the spectrogram. >>> f, t, Sxx = signal.spectrogram(x, fs) >>> plt.pcolormesh(t, f, Sxx, shading='gouraud') >>> plt.ylabel('Frequency [Hz]') >>> plt.xlabel('Time [sec]') >>> plt.show() Note, if using output that is not one sided, then use the following: >>> f, t, Sxx = signal.spectrogram(x, fs, return_onesided=False) >>> plt.pcolormesh(t, fftshift(f), fftshift(Sxx, axes=0), shading='gouraud') >>> plt.ylabel('Frequency [Hz]') >>> plt.xlabel('Time [sec]') >>> plt.show() """ modelist = ['psd', 'complex', 'magnitude', 'angle', 'phase'] if mode not in modelist: raise ValueError(f'unknown value for mode {mode}, must be one of {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 statistically independent if noverlap is None: noverlap = nperseg // 8 if mode == 'psd': freqs, time, Sxx = _spectral_helper(x, x, fs, window, nperseg, noverlap, nfft, detrend, return_onesided, scaling, axis, mode='psd') else: freqs, time, Sxx = _spectral_helper(x, x, fs, window, nperseg, noverlap, nfft, detrend, return_onesided, scaling, axis, mode='stft') if mode == 'magnitude': Sxx = np.abs(Sxx) elif mode in ['angle', 'phase']: Sxx = np.angle(Sxx) if mode == 'phase': # Sxx has one additional dimension for time strides if axis < 0: axis -= 1 Sxx = np.unwrap(Sxx, axis=axis) # mode =='complex' is same as `stft`, doesn't need modification return freqs, time, Sxx def check_COLA(window, nperseg, noverlap, tol=1e-10): r"""Check whether the Constant OverLap Add (COLA) constraint is met. Parameters ---------- window : str or tuple or array_like Desired window to use. If `window` is a string or tuple, it is passed to `get_window` to generate the window values, which are DFT-even by default. See `get_window` for a list of windows and required parameters. If `window` is array_like it will be used directly as the window and its length must be nperseg. nperseg : int Length of each segment. noverlap : int Number of points to overlap between segments. tol : float, optional The allowed variance of a bin's weighted sum from the median bin sum. Returns ------- verdict : bool `True` if chosen combination satisfies COLA within `tol`, `False` otherwise See Also -------- check_NOLA: Check whether the Nonzero Overlap Add (NOLA) constraint is met stft: Short Time Fourier Transform istft: Inverse Short Time Fourier Transform Notes ----- In order to enable inversion of an STFT via the inverse STFT in `istft`, it is sufficient that the signal windowing obeys the constraint of "Constant OverLap Add" (COLA). This ensures that every point in the input data is equally weighted, thereby avoiding aliasing and allowing full reconstruction. Some examples of windows that satisfy COLA: - Rectangular window at overlap of 0, 1/2, 2/3, 3/4, ... - Bartlett window at overlap of 1/2, 3/4, 5/6, ... - Hann window at 1/2, 2/3, 3/4, ... - Any Blackman family window at 2/3 overlap - Any window with ``noverlap = nperseg-1`` A very comprehensive list of other windows may be found in [2]_, wherein the COLA condition is satisfied when the "Amplitude Flatness" is unity. .. versionadded:: 0.19.0 References ---------- .. [1] Julius O. Smith III, "Spectral Audio Signal Processing", W3K Publishing, 2011,ISBN 978-0-9745607-3-1. .. [2] G. Heinzel, A. Ruediger and R. Schilling, "Spectrum and spectral density estimation by the Discrete Fourier transform (DFT), including a comprehensive list of window functions and some new at-top windows", 2002, http://hdl.handle.net/11858/00-001M-0000-0013-557A-5 Examples -------- >>> from scipy import signal Confirm COLA condition for rectangular window of 75% (3/4) overlap: >>> signal.check_COLA(signal.windows.boxcar(100), 100, 75) True COLA is not true for 25% (1/4) overlap, though: >>> signal.check_COLA(signal.windows.boxcar(100), 100, 25) False "Symmetrical" Hann window (for filter design) is not COLA: >>> signal.check_COLA(signal.windows.hann(120, sym=True), 120, 60) False "Periodic" or "DFT-even" Hann window (for FFT analysis) is COLA for overlap of 1/2, 2/3, 3/4, etc.: >>> signal.check_COLA(signal.windows.hann(120, sym=False), 120, 60) True >>> signal.check_COLA(signal.windows.hann(120, sym=False), 120, 80) True >>> signal.check_COLA(signal.windows.hann(120, sym=False), 120, 90) True """ nperseg = int(nperseg) if nperseg < 1: raise ValueError('nperseg must be a positive integer') if noverlap >= nperseg: raise ValueError('noverlap must be less than nperseg.') noverlap = int(noverlap) if isinstance(window, str) or type(window) is tuple: win = get_window(window, nperseg) else: win = np.asarray(window) if len(win.shape) != 1: raise ValueError('window must be 1-D') if win.shape[0] != nperseg: raise ValueError('window must have length of nperseg') step = nperseg - noverlap binsums = sum(win[ii*step:(ii+1)*step] for ii in range(nperseg//step)) if nperseg % step != 0: binsums[:nperseg % step] += win[-(nperseg % step):] deviation = binsums - np.median(binsums) return np.max(np.abs(deviation)) < tol def check_NOLA(window, nperseg, noverlap, tol=1e-10): r"""Check whether the Nonzero Overlap Add (NOLA) constraint is met. Parameters ---------- window : str or tuple or array_like Desired window to use. If `window` is a string or tuple, it is passed to `get_window` to generate the window values, which are DFT-even by default. See `get_window` for a list of windows and required parameters. If `window` is array_like it will be used directly as the window and its length must be nperseg. nperseg : int Length of each segment. noverlap : int Number of points to overlap between segments. tol : float, optional The allowed variance of a bin's weighted sum from the median bin sum. Returns ------- verdict : bool `True` if chosen combination satisfies the NOLA constraint within `tol`, `False` otherwise See Also -------- check_COLA: Check whether the Constant OverLap Add (COLA) constraint is met stft: Short Time Fourier Transform istft: Inverse Short Time Fourier Transform Notes ----- In order to enable inversion of an STFT via the inverse STFT in `istft`, the signal windowing must obey the constraint of "nonzero overlap add" (NOLA): .. math:: \sum_{t}w^{2}[n-tH] \ne 0 for all :math:`n`, where :math:`w` is the window function, :math:`t` is the frame index, and :math:`H` is the hop size (:math:`H` = `nperseg` - `noverlap`). This ensures that the normalization factors in the denominator of the overlap-add inversion equation are not zero. Only very pathological windows will fail the NOLA constraint. .. versionadded:: 1.2.0 References ---------- .. [1] Julius O. Smith III, "Spectral Audio Signal Processing", W3K Publishing, 2011,ISBN 978-0-9745607-3-1. .. [2] G. Heinzel, A. Ruediger and R. Schilling, "Spectrum and spectral density estimation by the Discrete Fourier transform (DFT), including a comprehensive list of window functions and some new at-top windows", 2002, http://hdl.handle.net/11858/00-001M-0000-0013-557A-5 Examples -------- >>> import numpy as np >>> from scipy import signal Confirm NOLA condition for rectangular window of 75% (3/4) overlap: >>> signal.check_NOLA(signal.windows.boxcar(100), 100, 75) True NOLA is also true for 25% (1/4) overlap: >>> signal.check_NOLA(signal.windows.boxcar(100), 100, 25) True "Symmetrical" Hann window (for filter design) is also NOLA: >>> signal.check_NOLA(signal.windows.hann(120, sym=True), 120, 60) True As long as there is overlap, it takes quite a pathological window to fail NOLA: >>> w = np.ones(64, dtype="float") >>> w[::2] = 0 >>> signal.check_NOLA(w, 64, 32) False If there is not enough overlap, a window with zeros at the ends will not work: >>> signal.check_NOLA(signal.windows.hann(64), 64, 0) False >>> signal.check_NOLA(signal.windows.hann(64), 64, 1) False >>> signal.check_NOLA(signal.windows.hann(64), 64, 2) True """ nperseg = int(nperseg) if nperseg < 1: raise ValueError('nperseg must be a positive integer') if noverlap >= nperseg: raise ValueError('noverlap must be less than nperseg') 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 = np.asarray(window) if len(win.shape) != 1: raise ValueError('window must be 1-D') if win.shape[0] != nperseg: raise ValueError('window must have length of nperseg') step = nperseg - noverlap binsums = sum(win[ii*step:(ii+1)*step]**2 for ii in range(nperseg//step)) if nperseg % step != 0: binsums[:nperseg % step] += win[-(nperseg % step):]**2 return np.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 (legacy function). STFTs can be used as a way of quantifying the change of a nonstationary signal's frequency and phase content over time. .. legacy:: function `ShortTimeFFT` is a newer STFT / ISTFT implementation with more features. A :ref:`comparison ` between the implementations can be found in the :ref:`tutorial_stft` section of the :ref:`user_guide`. 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``. .. versionadded:: 1.9.0 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 ShortTimeFFT: Newer STFT/ISTFT implementation providing more features. 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.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 denominator 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 -------- >>> import numpy as np >>> from scipy import signal >>> import matplotlib.pyplot as plt >>> rng = np.random.default_rng() Generate a test signal, a 2 Vrms sine wave whose frequency is slowly modulated around 3kHz, corrupted by white noise of exponentially decreasing magnitude sampled at 10 kHz. >>> fs = 10e3 >>> N = 1e5 >>> amp = 2 * np.sqrt(2) >>> noise_power = 0.01 * fs / 2 >>> time = np.arange(N) / float(fs) >>> mod = 500*np.cos(2*np.pi*0.25*time) >>> carrier = amp * np.sin(2*np.pi*3e3*time + mod) >>> noise = rng.normal(scale=np.sqrt(noise_power), ... size=time.shape) >>> noise *= np.exp(-time/5) >>> x = carrier + noise Compute and plot the STFT's magnitude. >>> f, t, Zxx = signal.stft(x, fs, nperseg=1000) >>> plt.pcolormesh(t, f, np.abs(Zxx), vmin=0, vmax=amp, shading='gouraud') >>> plt.title('STFT Magnitude') >>> plt.ylabel('Frequency [Hz]') >>> plt.xlabel('Time [sec]') >>> plt.show() Compare the energy of the signal `x` with the energy of its STFT: >>> E_x = sum(x**2) / fs # Energy of x >>> # Calculate a two-sided STFT with PSD scaling: >>> f, t, Zxx = signal.stft(x, fs, nperseg=1000, return_onesided=False, ... scaling='psd') >>> # Integrate numerically over abs(Zxx)**2: >>> df, dt = f[1] - f[0], t[1] - t[0] >>> E_Zxx = sum(np.sum(Zxx.real**2 + Zxx.imag**2, axis=0) * df) * dt >>> # The energy is the same, but the numerical errors are quite large: >>> np.isclose(E_x, E_Zxx, rtol=1e-2) True """ 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 (legacy function). .. legacy:: function `ShortTimeFFT` is a newer STFT / ISTFT implementation with more features. A :ref:`comparison ` between the implementations can be found in the :ref:`tutorial_stft` section of the :ref:`user_guide`. 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 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 ShortTimeFFT: Newer STFT/ISTFT implementation providing more features. 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. .. 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 -------- >>> import numpy as np >>> from scipy import signal >>> import matplotlib.pyplot as plt >>> rng = np.random.default_rng() Generate a test signal, a 2 Vrms sine wave at 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 = rng.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, 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 = 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() """ # Make sure input is an ndarray of appropriate complex dtype Zxx = np.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 = np.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 = np.asarray(window) if len(win.shape) != 1: raise ValueError('window must be 1-D') if win.shape[0] != nperseg: raise ValueError(f'window must have length of {nperseg}') ifunc = sp_fft.irfft if input_onesided else sp_fft.ifft xsubs = ifunc(Zxx, axis=-2, n=nfft)[..., :nperseg, :] # Initialize output and normalization arrays outputlength = nperseg + (nseg-1)*nstep x = np.zeros(list(Zxx.shape[:-2])+[outputlength], dtype=xsubs.dtype) norm = np.zeros(outputlength, dtype=xsubs.dtype) if np.result_type(win, xsubs) != xsubs.dtype: win = win.astype(xsubs.dtype) if scaling == 'spectrum': xsubs *= win.sum() elif scaling == 'psd': xsubs *= np.sqrt(fs * sum(win**2)) else: raise ValueError(f"Parameter {scaling=} not in ['spectrum', 'psd']!") # Construct the output from the ifft segments # This loop could perhaps be vectorized/strided somehow... 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 np.sum(norm > 1e-10) != len(norm): warnings.warn( "NOLA condition failed, STFT may not be invertible." + (" Possibly due to missing boundary" if not boundary else ""), stacklevel=2 ) x /= np.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 = np.moveaxis(x, -1, time_axis) time = np.arange(x.shape[0])/float(fs) return time, x 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. .. 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 -------- >>> import numpy as np >>> from scipy import signal >>> import matplotlib.pyplot as plt >>> rng = np.random.default_rng() Generate two test signals with some common features. >>> fs = 10e3 >>> N = 1e5 >>> amp = 20 >>> freq = 1234.0 >>> noise_power = 0.001 * fs / 2 >>> time = np.arange(N) / fs >>> b, a = signal.butter(2, 0.25, 'low') >>> x = rng.normal(scale=np.sqrt(noise_power), size=time.shape) >>> y = signal.lfilter(b, a, x) >>> x += amp*np.sin(2*np.pi*freq*time) >>> y += rng.normal(scale=0.1*np.sqrt(noise_power), size=time.shape) Compute and plot the coherence. >>> f, Cxy = signal.coherence(x, y, fs, nperseg=1024) >>> plt.semilogy(f, 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 = np.abs(Pxy)**2 / Pxx / Pyy return freqs, Cxy def _spectral_helper(x, y, fs=1.0, window='hann', nperseg=None, noverlap=None, nfft=None, detrend='constant', return_onesided=True, scaling='density', axis=-1, mode='psd', boundary=None, padded=False): """Calculate various forms of windowed FFTs for PSD, CSD, etc. This is a helper function that implements the commonality between the stft, psd, csd, and spectrogram functions. It is not designed to be called externally. The windows are not averaged over; the result from each window is returned. Parameters ---------- x : array_like Array or sequence containing the data to be analyzed. y : array_like Array or sequence containing the data to be analyzed. If this is the same object in memory as `x` (i.e. ``_spectral_helper(x, x, ...)``), the extra computations are spared. 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. 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 FFTs are computed; the default is over the last axis (i.e. ``axis=-1``). mode: str {'psd', 'stft'}, optional Defines what kind of return values are expected. Defaults to 'psd'. 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 `None`. 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 `False`. Padding occurs after boundary extension, if `boundary` is not `None`, and `padded` is `True`. Returns ------- freqs : ndarray Array of sample frequencies. t : ndarray Array of times corresponding to each data segment result : ndarray Array of output data, contents dependent on *mode* kwarg. Notes ----- Adapted from matplotlib.mlab .. versionadded:: 0.16.0 """ if mode not in ['psd', 'stft']: raise ValueError(f"Unknown value for mode {mode}, must be one of: " "{'psd', 'stft'}") boundary_funcs = {'even': even_ext, 'odd': odd_ext, 'constant': const_ext, 'zeros': zero_ext, None: None} if boundary not in boundary_funcs: raise ValueError(f"Unknown boundary option '{boundary}', " f"must be one of: {list(boundary_funcs.keys())}") # If x and y are the same object we can save ourselves some computation. same_data = y is x if not same_data and mode != 'psd': raise ValueError("x and y must be equal if mode is 'stft'") axis = int(axis) # Ensure we have np.arrays, get outdtype x = np.asarray(x) if not same_data: y = np.asarray(y) outdtype = np.result_type(x, y, np.complex64) else: outdtype = np.result_type(x, np.complex64) if not same_data: # Check if we can broadcast the outer axes together xouter = list(x.shape) youter = list(y.shape) xouter.pop(axis) youter.pop(axis) try: outershape = np.broadcast(np.empty(xouter), np.empty(youter)).shape except ValueError as e: raise ValueError('x and y cannot be broadcast together.') from e if same_data: if x.size == 0: return np.empty(x.shape), np.empty(x.shape), np.empty(x.shape) else: if x.size == 0 or y.size == 0: outshape = outershape + (min([x.shape[axis], y.shape[axis]]),) emptyout = np.moveaxis(np.empty(outshape), -1, axis) return emptyout, emptyout, emptyout if x.ndim > 1: if axis != -1: x = np.moveaxis(x, axis, -1) if not same_data and y.ndim > 1: y = np.moveaxis(y, axis, -1) # Check if x and y are the same length, zero-pad if necessary if not same_data: if x.shape[-1] != y.shape[-1]: if x.shape[-1] < y.shape[-1]: pad_shape = list(x.shape) pad_shape[-1] = y.shape[-1] - x.shape[-1] x = np.concatenate((x, np.zeros(pad_shape)), -1) else: pad_shape = list(y.shape) pad_shape[-1] = x.shape[-1] - y.shape[-1] y = np.concatenate((y, np.zeros(pad_shape)), -1) if nperseg is not None: # if specified by user nperseg = int(nperseg) if nperseg < 1: raise ValueError('nperseg must be a positive integer') # parse window; if array like, then set nperseg = win.shape win, nperseg = _triage_segments(window, nperseg, input_length=x.shape[-1]) if nfft is None: nfft = nperseg 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 # Padding occurs after boundary extension, so that the extended signal ends # in zeros, instead of introducing an impulse at the end. # I.e. if x = [..., 3, 2] # extend then pad -> [..., 3, 2, 2, 3, 0, 0, 0] # pad then extend -> [..., 3, 2, 0, 0, 0, 2, 3] if boundary is not None: ext_func = boundary_funcs[boundary] x = ext_func(x, nperseg//2, axis=-1) if not same_data: y = ext_func(y, nperseg//2, axis=-1) if padded: # Pad to integer number of windowed segments # I.e. make x.shape[-1] = nperseg + (nseg-1)*nstep, with integer nseg nadd = (-(x.shape[-1]-nperseg) % nstep) % nperseg zeros_shape = list(x.shape[:-1]) + [nadd] x = np.concatenate((x, np.zeros(zeros_shape)), axis=-1) if not same_data: zeros_shape = list(y.shape[:-1]) + [nadd] y = np.concatenate((y, np.zeros(zeros_shape)), axis=-1) # Handle detrending and window functions if not detrend: def detrend_func(d): return d elif not hasattr(detrend, '__call__'): def detrend_func(d): return _signaltools.detrend(d, type=detrend, axis=-1) elif axis != -1: # Wrap this function so that it receives a shape that it could # reasonably expect to receive. def detrend_func(d): d = np.moveaxis(d, -1, axis) d = detrend(d) return np.moveaxis(d, axis, -1) else: detrend_func = detrend if np.result_type(win, np.complex64) != outdtype: win = win.astype(outdtype) if scaling == 'density': scale = 1.0 / (fs * (win*win).sum()) elif scaling == 'spectrum': scale = 1.0 / win.sum()**2 else: raise ValueError(f'Unknown scaling: {scaling!r}') if mode == 'stft': scale = np.sqrt(scale) if return_onesided: if np.iscomplexobj(x): sides = 'twosided' warnings.warn('Input data is complex, switching to return_onesided=False', stacklevel=3) else: sides = 'onesided' if not same_data: if np.iscomplexobj(y): sides = 'twosided' warnings.warn('Input data is complex, switching to ' 'return_onesided=False', stacklevel=3) else: sides = 'twosided' if sides == 'twosided': freqs = sp_fft.fftfreq(nfft, 1/fs) elif sides == 'onesided': freqs = sp_fft.rfftfreq(nfft, 1/fs) # Perform the windowed FFTs result = _fft_helper(x, win, detrend_func, nperseg, noverlap, nfft, sides) if not same_data: # All the same operations on the y data result_y = _fft_helper(y, win, detrend_func, nperseg, noverlap, nfft, sides) result = np.conjugate(result) * result_y elif mode == 'psd': result = np.conjugate(result) * result result *= scale if sides == 'onesided' and mode == 'psd': if nfft % 2: result[..., 1:] *= 2 else: # Last point is unpaired Nyquist freq point, don't double result[..., 1:-1] *= 2 time = np.arange(nperseg/2, x.shape[-1] - nperseg/2 + 1, nperseg - noverlap)/float(fs) if boundary is not None: time -= (nperseg/2) / fs result = result.astype(outdtype) # All imaginary parts are zero anyways if same_data and mode != 'stft': result = result.real # Output is going to have new last axis for time/window index, so a # negative axis index shifts down one if axis < 0: axis -= 1 # Roll frequency axis back to axis where the data came from result = np.moveaxis(result, -1, axis) return freqs, time, result def _fft_helper(x, win, detrend_func, nperseg, noverlap, nfft, sides): """ Calculate windowed FFT, for internal use by `scipy.signal._spectral_helper`. This is a helper function that does the main FFT calculation for `_spectral helper`. All input validation is performed there, and the data axis is assumed to be the last axis of x. It is not designed to be called externally. The windows are not averaged over; the result from each window is returned. Returns ------- result : ndarray Array of FFT data Notes ----- Adapted from matplotlib.mlab .. versionadded:: 0.16.0 """ # Created sliding window view of array if nperseg == 1 and noverlap == 0: result = x[..., np.newaxis] else: step = nperseg - noverlap result = np.lib.stride_tricks.sliding_window_view( x, window_shape=nperseg, axis=-1, writeable=True ) result = result[..., 0::step, :] # Detrend each data segment individually result = detrend_func(result) # Apply window by multiplication result = win * result # Perform the fft. Acts on last axis by default. Zero-pads automatically if sides == 'twosided': func = sp_fft.fft else: result = result.real func = sp_fft.rfft result = func(result, n=nfft) return result def _triage_segments(window, nperseg, input_length): """ Parses window and nperseg arguments for spectrogram and _spectral_helper. This is a helper function, not meant to be called externally. Parameters ---------- window : string, tuple, or ndarray If window is specified by a string or tuple and nperseg is not specified, nperseg is set to the default of 256 and returns a window of that length. If instead the window is array_like and nperseg is not specified, then nperseg is set to the length of the window. A ValueError is raised if the user supplies both an array_like window and a value for nperseg but nperseg does not equal the length of the window. nperseg : int Length of each segment input_length: int Length of input signal, i.e. x.shape[-1]. Used to test for errors. Returns ------- win : ndarray window. If function was called with string or tuple than this will hold the actual array used as a window. nperseg : int Length of each segment. If window is str or tuple, nperseg is set to 256. If window is array_like, nperseg is set to the length of the window. """ # parse window; if array like, then set nperseg = win.shape if isinstance(window, str) or isinstance(window, tuple): # if nperseg not specified if nperseg is None: nperseg = 256 # then change to default if nperseg > input_length: warnings.warn(f'nperseg = {nperseg:d} is greater than input length ' f' = {input_length:d}, using nperseg = {input_length:d}', stacklevel=3) nperseg = input_length win = get_window(window, nperseg) else: win = np.asarray(window) if len(win.shape) != 1: raise ValueError('window must be 1-D') if input_length < win.shape[-1]: raise ValueError('window is longer than input signal') if nperseg is None: nperseg = win.shape[0] elif nperseg is not None: if nperseg != win.shape[0]: raise ValueError("value specified for nperseg is different" " from length of window") return win, nperseg def _median_bias(n): """ Returns the bias of the median of a set of periodograms relative to the mean. See Appendix B from [1]_ for details. Parameters ---------- n : int Numbers of periodograms being averaged. Returns ------- bias : float Calculated bias. References ---------- .. [1] B. Allen, W.G. Anderson, P.R. Brady, D.A. Brown, J.D.E. Creighton. "FINDCHIRP: an algorithm for detection of gravitational waves from inspiraling compact binaries", Physical Review D 85, 2012, :arxiv:`gr-qc/0509116` """ ii_2 = 2 * np.arange(1., (n-1) // 2 + 1) return 1 + np.sum(1. / (ii_2 + 1) - 1. / ii_2)