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periodogramwelchlombscarglecsd	coherencespectrogramstftistft
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check_NOLAFc                 C   s   t j| t jd} t j|t jd}t j|t jd}| jdks>J |jdksLJ |jdksZJ |rtt| ||  |}nt| ||}|r|dt || 9 }|S )a  
    lombscargle(x, y, freqs)

    Computes the Lomb-Scargle periodogram.

    The Lomb-Scargle periodogram was developed by Lomb [1]_ and further
    extended by Scargle [2]_ to find, and test the significance of weak
    periodic signals with uneven temporal sampling.

    When *normalize* is False (default) the computed periodogram
    is unnormalized, it takes the value ``(A**2) * N/4`` for a harmonic
    signal with amplitude A for sufficiently large N.

    When *normalize* is True the computed periodogram is normalized by
    the residuals of the data around a constant reference model (at zero).

    Input arrays should be 1-D and will be cast to float64.

    Parameters
    ----------
    x : array_like
        Sample times.
    y : array_like
        Measurement values.
    freqs : array_like
        Angular frequencies for output periodogram.
    precenter : bool, optional
        Pre-center measurement values by subtracting the mean.
    normalize : bool, optional
        Compute normalized periodogram.

    Returns
    -------
    pgram : array_like
        Lomb-Scargle periodogram.

    Raises
    ------
    ValueError
        If the input arrays `x` and `y` do not have the same shape.

    See Also
    --------
    istft: Inverse Short Time Fourier Transform
    check_COLA: Check whether the Constant OverLap Add (COLA) constraint is met
    welch: Power spectral density by Welch's method
    spectrogram: Spectrogram by Welch's method
    csd: Cross spectral density by Welch's method

    Notes
    -----
    This subroutine calculates the periodogram using a slightly
    modified algorithm due to Townsend [3]_ which allows the
    periodogram to be calculated using only a single pass through
    the input arrays for each frequency.

    The algorithm running time scales roughly as O(x * freqs) or O(N^2)
    for a large number of samples and frequencies.

    References
    ----------
    .. [1] N.R. Lomb "Least-squares frequency analysis of unequally spaced
           data", Astrophysics and Space Science, vol 39, pp. 447-462, 1976

    .. [2] J.D. Scargle "Studies in astronomical time series analysis. II -
           Statistical aspects of spectral analysis of unevenly spaced data",
           The Astrophysical Journal, vol 263, pp. 835-853, 1982

    .. [3] R.H.D. Townsend, "Fast calculation of the Lomb-Scargle
           periodogram using graphics processing units.", The Astrophysical
           Journal Supplement Series, vol 191, pp. 247-253, 2010

    Examples
    --------
    >>> import numpy as np
    >>> import matplotlib.pyplot as plt
    >>> rng = np.random.default_rng()

    First define some input parameters for the signal:

    >>> A = 2.
    >>> w0 = 1.  # rad/sec
    >>> nin = 150
    >>> nout = 100000

    Randomly generate sample times:

    >>> x = rng.uniform(0, 10*np.pi, nin)

    Plot a sine wave for the selected times:

    >>> y = A * np.cos(w0*x)

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

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

    Calculate Lomb-Scargle periodogram:

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

    Now make a plot of the input data:

    >>> fig, (ax_t, ax_w) = plt.subplots(2, 1, constrained_layout=True)
    >>> ax_t.plot(x, y, 'b+')
    >>> ax_t.set_xlabel('Time [s]')

    Then plot the normalized periodogram:

    >>> ax_w.plot(w, pgram)
    >>> ax_w.set_xlabel('Angular frequency [rad/s]')
    >>> ax_w.set_ylabel('Normalized amplitude')
    >>> plt.show()

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    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

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    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()

    )
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    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()

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dkr|	dk r|	d8 }	tj||	d}|||fS )a  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 <tutorial_stft_legacy_stft>` 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()

    )r=   complex	magnitudeanglephasezunknown value for mode z, must be one of input_lengthN   r=   r>   r   rJ   )rK   rL   rL   r   r   rA   )r7   _triage_segmentsr3   rB   r   absrK   unwrap)r   r(   r)   r*   r+   r,   r-   r.   r/   r0   r?   Zmodelistr   timeZSxxr    r    r!   r   r  s:     
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a  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

    r   "nperseg must be a positive integer#noverlap must be less than nperseg.window must be 1-Dr   "window must have length of npersegc                 3   s&   | ]}|  |d     V  qdS )r   Nr    .0iistepwinr    r!   	<genexpr>      zcheck_COLA.<locals>.<genexpr>N)intr7   
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 d 7  < t||kS )aH  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

    r   rU   z"noverlap must be less than npersegr   z&noverlap must be a nonnegative integerrW   rX   c                 3   s*   | ]"}|  |d     d V  qdS )r   r   Nr    rY   r\   r    r!   r_     r`   zcheck_NOLA.<locals>.<genexpr>Nr   )ra   r7   rb   rc   rd   r5   r   r   r1   r4   r3   re   rf   min)r)   r*   r+   rh   ri   r    r\   r!   r     s(    b
 *r      zerosspectrumc                 C   sZ   |dkrd}n|dkr&t d|dt| | |||||||||
d||	d\}}}|||fS )a  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 <tutorial_stft_legacy_stft>` 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 denomimator of the OLA reconstruction equation is nonzero. Whether a
    choice of `window`, `nperseg`, and `noverlap` satisfy this constraint can
    be tested with `check_NOLA`.


    .. versionadded:: 0.19.0

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

    Examples
    --------
    >>> 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

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 t | }||fS )aQ  Perform the inverse Short Time Fourier transform (legacy function).

    .. legacy:: function

        `ShortTimeFFT` is a newer STFT / ISTFT implementation with more
        features. A :ref:`comparison <tutorial_stft_legacy_stft>` 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()

    y                r   zInput stft must be at least 2d!z/Must specify differing time and frequency axes!r   NrU   .nfft must be greater than or equal to nperseg.rV   r   T)reverserW   zwindow must have length of rs   )r0   n.r   rm   r=   rn   ro   rT   z2NOLA condition failed, STFT may not be invertible.z! Possibly due to missing boundary 
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 | }|	|fS )a  
    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()

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 }!| |#|!fS )"a  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.
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        `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
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    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
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    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.
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        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.
    Nrk   z
nperseg = r   z! is greater than input length  = z, using nperseg = r   rx   r   rW   r&   z"window is longer than input signalr   z>value specified for nperseg is different from length of window)rb   rc   r5   r   r   r   r   r1   r4   r3   r7   )r)   r*   rN   r^   r    r    r!   rP     s0    "
rP   c                 C   s:   dt d| d d d  }dt d|d  d|   S )aG  
    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`
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