content: """Tools for spectral analysis.
  """
  
  from __future__ import division, print_function, absolute_import
  
  import numpy as np
  from scipy import fftpack
  from . import signaltools
  from .windows import get_window
  from ._spectral import _lombscargle
  from ._arraytools import const_ext, even_ext, odd_ext, zero_ext
  import warnings
  
  from scipy._lib.six import string_types
  
  __all__ = ['periodogram', 'welch', 'lombscargle', 'csd', 'coherence',
             'spectrogram', 'stft', 'istft', 'check_COLA']
  
  
  def lombscargle(x,
                  y,
                  freqs,
                  precenter=False,
                  normalize=False):
      """
      lombscargle(x, y, freqs)
  
      Computes the Lomb-Scargle periodogram.
      
      The Lomb-Scargle periodogram was developed by Lomb [1]_ and further
      extended by Scargle [2]_ to find, and test the significance of weak
      periodic signals with uneven temporal sampling.
  
      When *normalize* is False (default) the computed periodogram
      is unnormalized, it takes the value ``(A**2) * N/4`` for a harmonic
      signal with amplitude A for sufficiently large N.
  
      When *normalize* is True the computed periodogram is is normalized by
      the residuals of the data around a constant reference model (at zero).
  
      Input arrays should be one-dimensional and will be cast to float64.
  
      Parameters
      ----------
      x : array_like
          Sample times.
      y : array_like
          Measurement values.
      freqs : array_like
          Angular frequencies for output periodogram.
      precenter : bool, optional
          Pre-center amplitudes by subtracting the mean.
      normalize : bool, optional
          Compute normalized periodogram.
  
      Returns
      -------
      pgram : array_like
          Lomb-Scargle periodogram.
  
      Raises
      ------
      ValueError
          If the input arrays `x` and `y` do not have the same shape.
  
      Notes
      -----
      This subroutine calculates the periodogram using a slightly
      modified algorithm due to Townsend [3]_ which allows the
      periodogram to be calculated using only a single pass through
      the input arrays for each frequency.
  
      The algorithm running time scales roughly as O(x * freqs) or O(N^2)
      for a large number of samples and frequencies.
  
      References
      ----------
      .. [1] N.R. Lomb "Least-squares frequency analysis of unequally spaced
             data", Astrophysics and Space Science, vol 39, pp. 447-462, 1976
  
      .. [2] J.D. Scargle "Studies in astronomical time series analysis. II - 
             Statistical aspects of spectral analysis of unevenly spaced data",
             The Astrophysical Journal, vol 263, pp. 835-853, 1982
  
      .. [3] R.H.D. Townsend, "Fast calculation of the Lomb-Scargle
             periodogram using graphics processing units.", The Astrophysical
             Journal Supplement Series, vol 191, pp. 247-253, 2010
  
      Examples
      --------
      >>> import scipy.signal
      >>> import matplotlib.pyplot as plt
  
      First define some input parameters for the signal:
  
      >>> A = 2.
      >>> w = 1.
      >>> phi = 0.5 * np.pi
      >>> nin = 1000
      >>> nout = 100000
      >>> frac_points = 0.9 # Fraction of points to select
       
      Randomly select a fraction of an array with timesteps:
  
      >>> r = np.random.rand(nin)
      >>> x = np.linspace(0.01, 10*np.pi, nin)
      >>> x = x[r >= frac_points]
       
      Plot a sine wave for the selected times:
  
      >>> y = A * np.sin(w*x+phi)
  
      Define the array of frequencies for which to compute the periodogram:
      
      >>> f = np.linspace(0.01, 10, nout)
       
      Calculate Lomb-Scargle periodogram:
  
      >>> import scipy.signal as signal
      >>> pgram = signal.lombscargle(x, y, f, normalize=True)
  
      Now make a plot of the input data:
  
      >>> plt.subplot(2, 1, 1)
      >>> plt.plot(x, y, 'b+')
  
      Then plot the normalized periodogram:
  
      >>> plt.subplot(2, 1, 2)
      >>> plt.plot(f, pgram)
      >>> plt.show()
  
      """
  
      x = np.asarray(x, dtype=np.float64)
      y = np.asarray(y, dtype=np.float64)
      freqs = np.asarray(freqs, dtype=np.float64)
  
      assert x.ndim == 1
      assert y.ndim == 1
      assert freqs.ndim == 1
  
      if precenter:
          pgram = _lombscargle(x, y - y.mean(), freqs)
      else:
          pgram = _lombscargle(x, y, freqs)
  
      if normalize:
          pgram *= 2 / np.dot(y, y)
  
      return pgram
  
  
  def periodogram(x, fs=1.0, window='boxcar', nfft=None, detrend='constant',
                  return_onesided=True, scaling='density', axis=-1):
      """
      Estimate power spectral density using a periodogram.
  
      Parameters
      ----------
      x : array_like
          Time series of measurement values
      fs : float, optional
          Sampling frequency of the `x` time series. Defaults to 1.0.
      window : str or tuple or array_like, optional
          Desired window to use. If `window` is a string or tuple, it is
          passed to `get_window` to generate the window values, which are
          DFT-even by default. See `get_window` for a list of windows and
          required parameters. If `window` is array_like it will be used
          directly as the window and its length must be nperseg. Defaults
          to 'boxcar'.
      nfft : int, optional
          Length of the FFT used. If `None` the length of `x` will be
          used.
      detrend : str or function or `False`, optional
          Specifies how to detrend each segment. If `detrend` is a
          string, it is passed as the `type` argument to the `detrend`
          function. If it is a function, it takes a segment and returns a
          detrended segment. If `detrend` is `False`, no detrending is
          done. Defaults to 'constant'.
      return_onesided : bool, optional
          If `True`, return a one-sided spectrum for real data. If
          `False` return a two-sided spectrum. Note that for complex
          data, a two-sided spectrum is always returned.
      scaling : { 'density', 'spectrum' }, optional
          Selects between computing the power spectral density ('density')
          where `Pxx` has units of V**2/Hz and computing the power
          spectrum ('spectrum') where `Pxx` has units of V**2, if `x`
          is measured in V and `fs` is measured in Hz. Defaults to
          'density'
      axis : int, optional
          Axis along which the periodogram is computed; the default is
          over the last axis (i.e. ``axis=-1``).
  
      Returns
      -------
      f : ndarray
          Array of sample frequencies.
      Pxx : ndarray
          Power spectral density or power spectrum of `x`.
  
      Notes
      -----
      .. versionadded:: 0.12.0
  
      See Also
      --------
      welch: Estimate power spectral density using Welch's method
      lombscargle: Lomb-Scargle periodogram for unevenly sampled data
  
      Examples
      --------
      >>> from scipy import signal
      >>> import matplotlib.pyplot as plt
      >>> np.random.seed(1234)
  
      Generate a test signal, a 2 Vrms sine wave at 1234 Hz, corrupted by
      0.001 V**2/Hz of white noise sampled at 10 kHz.
  
      >>> fs = 10e3
      >>> N = 1e5
      >>> amp = 2*np.sqrt(2)
      >>> freq = 1234.0
      >>> noise_power = 0.001 * fs / 2
      >>> time = np.arange(N) / fs
      >>> x = amp*np.sin(2*np.pi*freq*time)
      >>> x += np.random.normal(scale=np.sqrt(noise_power), size=time.shape)
  
      Compute and plot the power spectral density.
  
      >>> f, Pxx_den = signal.periodogram(x, fs)
      >>> plt.semilogy(f, Pxx_den)
      >>> plt.ylim([1e-7, 1e2])
      >>> plt.xlabel('frequency [Hz]')
      >>> plt.ylabel('PSD [V**2/Hz]')
      >>> plt.show()
  
      If we average the last half of the spectral density, to exclude the
      peak, we can recover the noise power on the signal.
  
      >>> np.mean(Pxx_den[25000:])
      0.00099728892368242854
  
      Now compute and plot the power spectrum.
  
      >>> f, Pxx_spec = signal.periodogram(x, fs, 'flattop', scaling='spectrum')
      >>> plt.figure()
      >>> plt.semilogy(f, np.sqrt(Pxx_spec))
      >>> plt.ylim([1e-4, 1e1])
      >>> plt.xlabel('frequency [Hz]')
      >>> plt.ylabel('Linear spectrum [V RMS]')
      >>> plt.show()
  
      The peak height in the power spectrum is an estimate of the RMS
      amplitude.
  
      >>> np.sqrt(Pxx_spec.max())
      2.0077340678640727
  
      """
      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[s]
          nperseg = nfft
          nfft = None
  
      return welch(x, fs, window, nperseg, 0, nfft, detrend, return_onesided,
                   scaling, axis)
  
  
  def welch(x, fs=1.0, window='hann', nperseg=None, noverlap=None, nfft=None,
            detrend='constant', return_onesided=True, scaling='density',
            axis=-1):
      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. Note that for complex
          data, a two-sided spectrum is always returned.
      scaling : { 'density', 'spectrum' }, optional
          Selects between computing the power spectral density ('density')
          where `Pxx` has units of V**2/Hz and computing the power
          spectrum ('spectrum') where `Pxx` has units of V**2, if `x`
          is measured in V and `fs` is measured in Hz. Defaults to
          'density'
      axis : int, optional
          Axis along which the periodogram is computed; the default is
          over the last axis (i.e. ``axis=-1``).
  
      Returns
      -------
      f : ndarray
          Array of sample frequencies.
      Pxx : ndarray
          Power spectral density or power spectrum of x.
  
      See Also
      --------
      periodogram: Simple, optionally modified periodogram
      lombscargle: Lomb-Scargle periodogram for unevenly sampled data
  
      Notes
      -----
      An appropriate amount of overlap will depend on the choice of window
      and on your requirements. For the default Hann window an overlap of
      50% is a reasonable trade off between accurately estimating the
      signal power, while not over counting any of the data. Narrower
      windows may require a larger overlap.
  
      If `noverlap` is 0, this method is equivalent to Bartlett's method
      [2]_.
  
      .. versionadded:: 0.12.0
  
      References
      ----------
      .. [1] P. Welch, "The use of the fast Fourier transform for the
             estimation of power spectra: A method based on time averaging
             over short, modified periodograms", IEEE Trans. Audio
             Electroacoust. vol. 15, pp. 70-73, 1967.
      .. [2] M.S. Bartlett, "Periodogram Analysis and Continuous Spectra",
             Biometrika, vol. 37, pp. 1-16, 1950.
  
      Examples
      --------
      >>> from scipy import signal
      >>> import matplotlib.pyplot as plt
      >>> np.random.seed(1234)
  
      Generate a test signal, a 2 Vrms sine wave at 1234 Hz, corrupted by
      0.001 V**2/Hz of white noise sampled at 10 kHz.
  
      >>> fs = 10e3
      >>> N = 1e5
      >>> amp = 2*np.sqrt(2)
      >>> freq = 1234.0
      >>> noise_power = 0.001 * fs / 2
      >>> time = np.arange(N) / fs
      >>> x = amp*np.sin(2*np.pi*freq*time)
      >>> x += np.random.normal(scale=np.sqrt(noise_power), size=time.shape)
  
      Compute and plot the power spectral density.
  
      >>> f, Pxx_den = signal.welch(x, fs, nperseg=1024)
      >>> plt.semilogy(f, Pxx_den)
      >>> plt.ylim([0.5e-3, 1])
      >>> plt.xlabel('frequency [Hz]')
      >>> plt.ylabel('PSD [V**2/Hz]')
      >>> plt.show()
  
      If we average the last half of the spectral density, to exclude the
      peak, we can recover the noise power on the signal.
  
      >>> np.mean(Pxx_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
  
      """
  
      freqs, Pxx = csd(x, x, fs, window, nperseg, noverlap, nfft, detrend,
                       return_onesided, scaling, axis)
  
      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):
      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. Note that 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``).
  
      Returns
      -------
      f : ndarray
          Array of sample frequencies.
      Pxy : ndarray
          Cross spectral density or cross power spectrum of x,y.
  
      See Also
      --------
      periodogram: Simple, optionally modified periodogram
      lombscargle: Lomb-Scargle periodogram for unevenly sampled data
      welch: Power spectral density by Welch's method. [Equivalent to
             csd(x,x)]
      coherence: Magnitude squared coherence by Welch's method.
  
      Notes
      --------
      By convention, Pxy is computed with the conjugate FFT of X
      multiplied by the FFT of Y.
  
      If the input series differ in length, the shorter series will be
      zero-padded to match.
  
      An appropriate amount of overlap will depend on the choice of window
      and on your requirements. For the default Hann window an overlap of
      50% is a reasonable trade off between accurately estimating the
      signal power, while not over counting any of the data. Narrower
      windows may require a larger overlap.
  
      .. versionadded:: 0.16.0
  
      References
      ----------
      .. [1] P. Welch, "The use of the fast Fourier transform for the
             estimation of power spectra: A method based on time averaging
             over short, modified periodograms", IEEE Trans. Audio
             Electroacoust. vol. 15, pp. 70-73, 1967.
      .. [2] Rabiner, Lawrence R., and B. Gold. "Theory and Application of
             Digital Signal Processing" Prentice-Hall, pp. 414-419, 1975
  
      Examples
      --------
      >>> from scipy import signal
      >>> import matplotlib.pyplot as plt
  
      Generate two test signals with some common features.
  
      >>> fs = 10e3
      >>> N = 1e5
      >>> amp = 20
      >>> freq = 1234.0
      >>> noise_power = 0.001 * fs / 2
      >>> time = np.arange(N) / fs
      >>> b, a = signal.butter(2, 0.25, 'low')
      >>> x = np.random.normal(scale=np.sqrt(noise_power), size=time.shape)
      >>> y = signal.lfilter(b, a, x)
      >>> x += amp*np.sin(2*np.pi*freq*time)
      >>> y += np.random.normal(scale=0.1*np.sqrt(noise_power), size=time.shape)
  
      Compute and plot the magnitude of the cross spectral density.
  
      >>> f, Pxy = signal.csd(x, y, fs, nperseg=1024)
      >>> plt.semilogy(f, np.abs(Pxy))
      >>> plt.xlabel('frequency [Hz]')
      >>> plt.ylabel('CSD [V**2/Hz]')
      >>> plt.show()
      """
  
      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:
              Pxy = Pxy.mean(axis=-1)
          else:
              Pxy = np.reshape(Pxy, Pxy.shape[:-1])
  
      return freqs, Pxy
  
  
  def spectrogram(x, fs=1.0, window=('tukey',.25), nperseg=None, noverlap=None,
                  nfft=None, detrend='constant', return_onesided=True,
                  scaling='density', axis=-1, mode='psd'):
      """
      Compute a spectrogram with consecutive Fourier transforms.
  
      Spectrograms can be used as a way of visualizing the change of a
      nonstationary signal's frequency content over time.
  
      Parameters
      ----------
      x : array_like
          Time series of measurement values
      fs : float, optional
          Sampling frequency of the `x` time series. Defaults to 1.0.
      window : str or tuple or array_like, optional
          Desired window to use. If `window` is a string or tuple, it is
          passed to `get_window` to generate the window values, which are
          DFT-even by default. See `get_window` for a list of windows and
          required parameters. If `window` is array_like it will be used
          directly as the window and its length must be nperseg.
          Defaults to a Tukey window with shape parameter of 0.25.
      nperseg : int, optional
          Length of each segment. Defaults to None, but if window is str or
          tuple, is set to 256, and if window is array_like, is set to the
          length of the window.
      noverlap : int, optional
          Number of points to overlap between segments. If `None`,
          ``noverlap = nperseg // 8``. Defaults to `None`.
      nfft : int, optional
          Length of the FFT used, if a zero padded FFT is desired. If
          `None`, the FFT length is `nperseg`. Defaults to `None`.
      detrend : str or function or `False`, optional
          Specifies how to detrend each segment. If `detrend` is a
          string, it is passed as the `type` argument to the `detrend`
          function. If it is a function, it takes a segment and returns a
          detrended segment. If `detrend` is `False`, no detrending is
          done. Defaults to 'constant'.
      return_onesided : bool, optional
          If `True`, return a one-sided spectrum for real data. If
          `False` return a two-sided spectrum. Note that for complex
          data, a two-sided spectrum is always returned.
      scaling : { 'density', 'spectrum' }, optional
          Selects between computing the power spectral density ('density')
          where `Sxx` has units of V**2/Hz and computing the power
          spectrum ('spectrum') where `Sxx` has units of V**2, if `x`
          is measured in V and `fs` is measured in Hz. Defaults to
          'density'.
      axis : int, optional
          Axis along which the spectrogram is computed; the default is over
          the last axis (i.e. ``axis=-1``).
      mode : str, optional
          Defines what kind of return values are expected. Options are
          ['psd', 'complex', 'magnitude', 'angle', 'phase']. 'complex' is
          equivalent to the output of `stft` with no padding or boundary
          extension. 'magnitude' returns the absolute magnitude of the
          STFT. 'angle' and 'phase' return the complex angle of the STFT,
          with and without unwrapping, respectively.
  
      Returns
      -------
      f : ndarray
          Array of sample frequencies.
      t : ndarray
          Array of segment times.
      Sxx : ndarray
          Spectrogram of x. By default, the last axis of Sxx corresponds
          to the segment times.
  
      See Also
      --------
      periodogram: Simple, optionally modified periodogram
      lombscargle: Lomb-Scargle periodogram for unevenly sampled data
      welch: Power spectral density by Welch's method.
      csd: Cross spectral density by Welch's method.
  
      Notes
      -----
      An appropriate amount of overlap will depend on the choice of window
      and on your requirements. In contrast to welch's method, where the
      entire data stream is averaged over, one may wish to use a smaller
      overlap (or perhaps none at all) when computing a spectrogram, to
      maintain some statistical independence between individual segments.
      It is for this reason that the default window is a Tukey window with
      1/8th of a window's length overlap at each end.
  
      .. versionadded:: 0.16.0
  
      References
      ----------
      .. [1] Oppenheim, Alan V., Ronald W. Schafer, John R. Buck
             "Discrete-Time Signal Processing", Prentice Hall, 1999.
  
      Examples
      --------
      >>> from scipy import signal
      >>> import matplotlib.pyplot as plt
  
      Generate a test signal, a 2 Vrms sine wave whose frequency is slowly
      modulated around 3kHz, corrupted by white noise of exponentially
      decreasing magnitude sampled at 10 kHz.
  
      >>> fs = 10e3
      >>> N = 1e5
      >>> amp = 2 * np.sqrt(2)
      >>> noise_power = 0.01 * fs / 2
      >>> time = np.arange(N) / float(fs)
      >>> mod = 500*np.cos(2*np.pi*0.25*time)
      >>> carrier = amp * np.sin(2*np.pi*3e3*time + mod)
      >>> noise = np.random.normal(scale=np.sqrt(noise_power), size=time.shape)
      >>> noise *= np.exp(-time/5)
      >>> x = carrier + noise
  
      Compute and plot the spectrogram.
  
      >>> f, t, Sxx = signal.spectrogram(x, fs)
      >>> plt.pcolormesh(t, f, Sxx)
      >>> plt.ylabel('Frequency [Hz]')
      >>> plt.xlabel('Time [sec]')
      >>> plt.show()
      """
      modelist = ['psd', 'complex', 'magnitude', 'angle', 'phase']
      if mode not in modelist:
          raise ValueError('unknown value for mode {}, must be one of {}'
                           .format(mode, modelist))
  
      # need to set default for nperseg before setting default for noverlap below
      window, nperseg = _triage_segments(window, nperseg,
                                         input_length=x.shape[axis])
  
      # Less overlap than welch, so samples are more statisically independent
      if noverlap is None:
          noverlap = nperseg // 8
  
      if mode == 'psd':
          freqs, time, Sxx = _spectral_helper(x, x, fs, window, nperseg,
                                              noverlap, nfft, detrend,
                                              return_onesided, scaling, axis,
                                              mode='psd')
  
      else:
          freqs, time, Sxx = _spectral_helper(x, x, fs, window, nperseg,
                                              noverlap, nfft, detrend,
                                              return_onesided, scaling, axis,
                                              mode='stft')
  
          if mode == 'magnitude':
              Sxx = np.abs(Sxx)
          elif mode in ['angle', 'phase']:
              Sxx = np.angle(Sxx)
              if mode == 'phase':
                  # Sxx has one additional dimension for time strides
                  if axis < 0:
                      axis -= 1
                  Sxx = np.unwrap(Sxx, axis=axis)
  
          # mode =='complex' is same as `stft`, doesn't need modification
  
      return freqs, time, Sxx
  
  
  def check_COLA(window, nperseg, noverlap, tol=1e-10):
      r"""
      Check whether the Constant OverLap Add (COLA) constraint is met
  
      Parameters
      ----------
      window : str or tuple or array_like
          Desired window to use. If `window` is a string or tuple, it is
          passed to `get_window` to generate the window values, which are
          DFT-even by default. See `get_window` for a list of windows and
          required parameters. If `window` is array_like it will be used
          directly as the window and its length must be nperseg.
      nperseg : int
          Length of each segment.
      noverlap : int
          Number of points to overlap between segments.
      tol : float, optional
          The allowed variance of a bin's weighted sum from the median bin
          sum.
  
      Returns
      -------
      verdict : bool
          `True` if chosen combination satisfies COLA within `tol`,
          `False` otherwise
  
      See Also
      --------
      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 "Constant
      OverLap Add" (COLA). This ensures that every point in the input data
      is equally weighted, thereby avoiding aliasing and allowing full
      reconstruction.
  
      Some examples of windows that satisfy COLA:
          - Rectangular window at overlap of 0, 1/2, 2/3, 3/4, ...
          - Bartlett window at overlap of 1/2, 3/4, 5/6, ...
          - Hann window at 1/2, 2/3, 3/4, ...
          - Any Blackman family window at 2/3 overlap
          - Any window with ``noverlap = nperseg-1``
  
      A very comprehensive list of other windows may be found in [2]_,
      wherein the COLA condition is satisfied when the "Amplitude
      Flatness" is unity.
  
      .. versionadded:: 0.19.0
  
      References
      ----------
      .. [1] Julius O. Smith III, "Spectral Audio Signal Processing", W3K
             Publishing, 2011,ISBN 978-0-9745607-3-1.
      .. [2] G. Heinzel, A. Ruediger and R. Schilling, "Spectrum and
             spectral density estimation by the Discrete Fourier transform
             (DFT), including a comprehensive list of window functions and
             some new at-top windows", 2002,
             http://hdl.handle.net/11858/00-001M-0000-0013-557A-5
  
      Examples
      --------
      >>> from scipy import signal
  
      Confirm COLA condition for rectangular window of 75% (3/4) overlap:
  
      >>> signal.check_COLA(signal.boxcar(100), 100, 75)
      True
  
      COLA is not true for 25% (1/4) overlap, though:
  
      >>> signal.check_COLA(signal.boxcar(100), 100, 25)
      False
  
      "Symmetrical" Hann window (for filter design) is not COLA:
  
      >>> signal.check_COLA(signal.hann(120, sym=True), 120, 60)
      False
  
      "Periodic" or "DFT-even" Hann window (for FFT analysis) is COLA for
      overlap of 1/2, 2/3, 3/4, etc.:
  
      >>> signal.check_COLA(signal.hann(120, sym=False), 120, 60)
      True
  
      >>> signal.check_COLA(signal.hann(120, sym=False), 120, 80)
      True
  
      >>> signal.check_COLA(signal.hann(120, sym=False), 120, 90)
      True
  
      """
  
      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, string_types) 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 stft(x, fs=1.0, window='hann', nperseg=256, noverlap=None, nfft=None,
           detrend=False, return_onesided=True, boundary='zeros', padded=True,
           axis=-1):
      r"""
      Compute the Short Time Fourier Transform (STFT).
  
      STFTs can be used as a way of quantifying the change of a
      nonstationary signal's frequency and phase content over time.
  
      Parameters
      ----------
      x : array_like
          Time series of measurement values
      fs : float, optional
          Sampling frequency of the `x` time series. Defaults to 1.0.
      window : str or tuple or array_like, optional
          Desired window to use. If `window` is a string or tuple, it is
          passed to `get_window` to generate the window values, which are
          DFT-even by default. See `get_window` for a list of windows and
          required parameters. If `window` is array_like it will be used
          directly as the window and its length must be nperseg. Defaults
          to a Hann window.
      nperseg : int, optional
          Length of each segment. Defaults to 256.
      noverlap : int, optional
          Number of points to overlap between segments. If `None`,
          ``noverlap = nperseg // 2``. Defaults to `None`. When
          specified, the COLA constraint must be met (see Notes below).
      nfft : int, optional
          Length of the FFT used, if a zero padded FFT is desired. If
          `None`, the FFT length is `nperseg`. Defaults to `None`.
      detrend : str or function or `False`, optional
          Specifies how to detrend each segment. If `detrend` is a
          string, it is passed as the `type` argument to the `detrend`
          function. If it is a function, it takes a segment and returns a
          detrended segment. If `detrend` is `False`, no detrending is
          done. Defaults to `False`.
      return_onesided : bool, optional
          If `True`, return a one-sided spectrum for real data. If
          `False` return a two-sided spectrum. Note that for complex
          data, a two-sided spectrum is always returned. Defaults to
          `True`.
      boundary : str or None, optional
          Specifies whether the input signal is extended at both ends, and
          how to generate the new values, in order to center the first
          windowed segment on the first input point. This has the benefit
          of enabling reconstruction of the first input point when the
          employed window function starts at zero. Valid options are
          ``['even', 'odd', 'constant', 'zeros', None]``. Defaults to
          'zeros', for zero padding extension. I.e. ``[1, 2, 3, 4]`` is
          extended to ``[0, 1, 2, 3, 4, 0]`` for ``nperseg=3``.
      padded : bool, optional
          Specifies whether the input signal is zero-padded at the end to
          make the signal fit exactly into an integer number of window
          segments, so that all of the signal is included in the output.
          Defaults to `True`. Padding occurs after boundary extension, if
          `boundary` is not `None`, and `padded` is `True`, as is the
          default.
      axis : int, optional
          Axis along which the STFT is computed; the default is over the
          last axis (i.e. ``axis=-1``).
  
      Returns
      -------
      f : ndarray
          Array of sample frequencies.
      t : ndarray
          Array of segment times.
      Zxx : ndarray
          STFT of `x`. By default, the last axis of `Zxx` corresponds
          to the segment times.
  
      See Also
      --------
      istft: Inverse Short Time Fourier Transform
      check_COLA: Check whether the Constant OverLap Add (COLA) constraint
                  is met
      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 "Constant
      OverLap Add" (COLA), 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.
  
      The COLA constraint ensures that every point in the input data is
      equally weighted, thereby avoiding aliasing and allowing full
      reconstruction. Whether a choice of `window`, `nperseg`, and
      `noverlap` satisfy this constraint can be tested with
      `check_COLA`.
  
      .. 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. Limdt "Signal Estimation from
             Modified Short Fourier Transform", IEEE 1984,
             10.1109/TASSP.1984.1164317
  
      Examples
      --------
      >>> from scipy import signal
      >>> import matplotlib.pyplot as plt
  
      Generate a test signal, a 2 Vrms sine wave whose frequency is slowly
      modulated around 3kHz, corrupted by white noise of exponentially
      decreasing magnitude sampled at 10 kHz.
  
      >>> fs = 10e3
      >>> N = 1e5
      >>> amp = 2 * np.sqrt(2)
      >>> noise_power = 0.01 * fs / 2
      >>> time = np.arange(N) / float(fs)
      >>> mod = 500*np.cos(2*np.pi*0.25*time)
      >>> carrier = amp * np.sin(2*np.pi*3e3*time + mod)
      >>> noise = np.random.normal(scale=np.sqrt(noise_power),
      ...                          size=time.shape)
      >>> noise *= np.exp(-time/5)
      >>> x = carrier + noise
  
      Compute and plot the STFT's magnitude.
  
      >>> f, t, Zxx = signal.stft(x, fs, nperseg=1000)
      >>> plt.pcolormesh(t, f, np.abs(Zxx), vmin=0, vmax=amp)
      >>> plt.title('STFT Magnitude')
      >>> plt.ylabel('Frequency [Hz]')
      >>> plt.xlabel('Time [sec]')
      >>> plt.show()
      """
  
      freqs, time, Zxx = _spectral_helper(x, x, fs, window, nperseg, noverlap,
                                          nfft, detrend, return_onesided,
                                          scaling='spectrum', 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):
      r"""
      Perform the inverse Short Time Fourier transform (iSTFT).
  
      Parameters
      ----------
      Zxx : array_like
          STFT of the signal to be reconstructed. If a purely real array
          is passed, it will be cast to a complex data type.
      fs : float, optional
          Sampling frequency of the time series. Defaults to 1.0.
      window : str or tuple or array_like, optional
          Desired window to use. If `window` is a string or tuple, it is
          passed to `get_window` to generate the window values, which are
          DFT-even by default. See `get_window` for a list of windows and
          required parameters. If `window` is array_like it will be used
          directly as the window and its length must be nperseg. Defaults
          to a Hann window. Must match the window used to generate the
          STFT for faithful inversion.
      nperseg : int, optional
          Number of data points corresponding to each STFT segment. This
          parameter must be specified if the number of data points per
          segment is odd, or if the STFT was padded via ``nfft >
          nperseg``. If `None`, the value depends on the shape of
          `Zxx` and `input_onesided`. If `input_onesided` is True,
          ``nperseg=2*(Zxx.shape[freq_axis] - 1)``. Otherwise,
          ``nperseg=Zxx.shape[freq_axis]``. Defaults to `None`.
      noverlap : int, optional
          Number of points to overlap between segments. If `None`, half
          of the segment length. Defaults to `None`. When specified, the
          COLA constraint must be met (see Notes below), and should match
          the parameter used to generate the STFT. Defaults to `None`.
      nfft : int, optional
          Number of FFT points corresponding to each STFT segment. This
          parameter must be specified if the STFT was padded via ``nfft >
          nperseg``. If `None`, the default values are the same as for
          `nperseg`, detailed above, with one exception: if
          `input_onesided` is True and
          ``nperseg==2*Zxx.shape[freq_axis] - 1``, `nfft` also takes on
          that value. This case allows the proper inversion of an
          odd-length unpadded STFT using ``nfft=None``. Defaults to
          `None`.
      input_onesided : bool, optional
          If `True`, interpret the input array as one-sided FFTs, such
          as is returned by `stft` with ``return_onesided=True`` and
          `numpy.fft.rfft`. If `False`, interpret the input as a a
          two-sided FFT. Defaults to `True`.
      boundary : bool, optional
          Specifies whether the input signal was extended at its
          boundaries by supplying a non-`None` ``boundary`` argument to
          `stft`. Defaults to `True`.
      time_axis : int, optional
          Where the time segments of the STFT is located; the default is
          the last axis (i.e. ``axis=-1``).
      freq_axis : int, optional
          Where the frequency axis of the STFT is located; the default is
          the penultimate axis (i.e. ``axis=-2``).
  
      Returns
      -------
      t : ndarray
          Array of output data times.
      x : ndarray
          iSTFT of `Zxx`.
  
      See Also
      --------
      stft: Short Time Fourier Transform
      check_COLA: Check whether the Constant OverLap Add (COLA) constraint
                  is met
  
      Notes
      -----
      In order to enable inversion of an STFT via the inverse STFT with
      `istft`, the signal windowing must obey 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. Whether a choice of `window`, `nperseg`, and
      `noverlap` satisfy this constraint can be tested with
      `check_COLA`, by using ``nperseg = Zxx.shape[freq_axis]``.
  
      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 esimation
      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. Limdt "Signal Estimation from
             Modified Short Fourier Transform", IEEE 1984,
             10.1109/TASSP.1984.1164317
  
      Examples
      --------
      >>> from scipy import signal
      >>> import matplotlib.pyplot as plt
  
      Generate a test signal, a 2 Vrms sine wave at 50Hz corrupted by
      0.001 V**2/Hz of white noise sampled at 1024 Hz.
  
      >>> fs = 1024
      >>> N = 10*fs
      >>> nperseg = 512
      >>> amp = 2 * np.sqrt(2)
      >>> noise_power = 0.001 * fs / 2
      >>> time = np.arange(N) / float(fs)
      >>> carrier = amp * np.sin(2*np.pi*50*time)
      >>> noise = np.random.normal(scale=np.sqrt(noise_power),
      ...                          size=time.shape)
      >>> x = carrier + noise
  
      Compute the STFT, and plot its magnitude
  
      >>> f, t, Zxx = signal.stft(x, fs=fs, nperseg=nperseg)
      >>> plt.figure()
      >>> plt.pcolormesh(t, f, np.abs(Zxx), vmin=0, vmax=amp)
      >>> plt.ylim([f[1], f[-1]])
      >>> plt.title('STFT Magnitude')
      >>> plt.ylabel('Frequency [Hz]')
      >>> plt.xlabel('Time [sec]')
      >>> plt.yscale('log')
      >>> plt.show()
  
      Zero the components that are 10% or less of the carrier magnitude,
      then convert back to a time series via inverse STFT
  
      >>> Zxx = np.where(np.abs(Zxx) >= amp/10, Zxx, 0)
      >>> _, xrec = signal.istft(Zxx, fs)
  
      Compare the cleaned signal with the original and true carrier signals.
  
      >>> plt.figure()
      >>> plt.plot(time, x, time, xrec, time, carrier)
      >>> plt.xlim([2, 2.1])
      >>> plt.xlabel('Time [sec]')
      >>> plt.ylabel('Signal')
      >>> plt.legend(['Carrier + Noise', 'Filtered via STFT', 'True Carrier'])
      >>> plt.show()
  
      Note that the cleaned signal does not start as abruptly as the original,
      since some of the coefficients of the transient were also removed:
  
      >>> plt.figure()
      >>> plt.plot(time, x, time, xrec, time, carrier)
      >>> plt.xlim([0, 0.1])
      >>> plt.xlabel('Time [sec]')
      >>> plt.ylabel('Signal')
      >>> plt.legend(['Carrier + Noise', 'Filtered via STFT', 'True Carrier'])
      >>> plt.show()
  
      """
  
      # 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
  
      if not check_COLA(window, nperseg, noverlap):
          raise ValueError('Window, STFT shape and noverlap do not satisfy the '
                           'COLA constraint.')
  
      # 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, string_types) 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 {0}'.format(nperseg))
  
      if input_onesided:
          ifunc = np.fft.irfft
      else:
          ifunc = fftpack.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)
  
      xsubs *= win.sum()  # This takes care of the 'spectrum' scaling
  
      # 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
  
      # Divide out normalization where non-tiny
      x /= np.where(norm > 1e-10, norm, 1.0)
  
      # Remove extension points
      if boundary:
          x = x[..., nperseg//2:-(nperseg//2)]
  
      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.rollaxis(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
      --------
      >>> from scipy import signal
      >>> import matplotlib.pyplot as plt
  
      Generate two test signals with some common features.
  
      >>> fs = 10e3
      >>> N = 1e5
      >>> amp = 20
      >>> freq = 1234.0
      >>> noise_power = 0.001 * fs / 2
      >>> time = np.arange(N) / fs
      >>> b, a = signal.butter(2, 0.25, 'low')
      >>> x = np.random.normal(scale=np.sqrt(noise_power), size=time.shape)
      >>> y = signal.lfilter(b, a, x)
      >>> x += amp*np.sin(2*np.pi*freq*time)
      >>> y += np.random.normal(scale=0.1*np.sqrt(noise_power), size=time.shape)
  
      Compute and plot the coherence.
  
      >>> f, Cxy = signal.coherence(x, y, fs, nperseg=1024)
      >>> plt.semilogy(f, Cxy)
      >>> plt.xlabel('frequency [Hz]')
      >>> plt.ylabel('Coherence')
      >>> plt.show()
      """
  
      freqs, Pxx = welch(x, fs, window, nperseg, noverlap, nfft, detrend,
                         axis=axis)
      _, Pyy = welch(y, fs, window, nperseg, noverlap, nfft, detrend, axis=axis)
      _, Pxy = csd(x, y, fs, window, nperseg, noverlap, nfft, 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='spectrum', 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. Note that 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("Unknown value for mode %s, must be one of: "
                           "{'psd', 'stft'}" % mode)
  
      boundary_funcs = {'even': even_ext,
                        'odd': odd_ext,
                        'constant': const_ext,
                        'zeros': zero_ext,
                        None: None}
  
      if boundary not in boundary_funcs:
          raise ValueError("Unknown boundary option '{0}', must be one of: {1}"
                            .format(boundary, 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:
              raise ValueError('x and y cannot be broadcast together.')
  
      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.rollaxis(np.empty(outshape), -1, axis)
              return emptyout, emptyout, emptyout
  
      if x.ndim > 1:
          if axis != -1:
              x = np.rollaxis(x, axis, len(x.shape))
              if not same_data and y.ndim > 1:
                  y = np.rollaxis(y, axis, len(y.shape))
  
      # 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.rollaxis(d, -1, axis)
              d = detrend(d)
              return np.rollaxis(d, axis, len(d.shape))
      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('Unknown scaling: %r' % scaling)
  
      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')
          else:
              sides = 'onesided'
              if not same_data:
                  if np.iscomplexobj(y):
                      sides = 'twosided'
                      warnings.warn('Input data is complex, switching to '
                                    'return_onesided=False')
      else:
          sides = 'twosided'
  
      if sides == 'twosided':
          freqs = fftpack.fftfreq(nfft, 1/fs)
      elif sides == 'onesided':
          freqs = np.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.rollaxis(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 strided array of data segments
      if nperseg == 1 and noverlap == 0:
          result = x[..., np.newaxis]
      else:
          # http://stackoverflow.com/a/5568169
          step = nperseg - noverlap
          shape = x.shape[:-1]+((x.shape[-1]-noverlap)//step, nperseg)
          strides = x.strides[:-1]+(step*x.strides[-1], x.strides[-1])
          result = np.lib.stride_tricks.as_strided(x, shape=shape,
                                                   strides=strides)
  
      # 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 = fftpack.fft
      else:
          result = result.real
          func = np.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
          6
          window.
      """
  
      #parse window; if array like, then set nperseg = win.shape
      if isinstance(window, string_types) or isinstance(window, tuple):
          # if nperseg not specified
          if nperseg is None:
              nperseg = 256  # then change to default
          if nperseg > input_length:
              warnings.warn('nperseg = {0:d} is greater than input length '
                                ' = {1:d}, using nperseg = {1:d}'
                                .format(nperseg, input_length))
              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
  
  