parameters: (M)
return_statement: return 0.5 - 0.5*cos(2.0*pi*n/(M-1))
docstring: Return the Hanning window.
  
      The Hanning window is a taper formed by using a weighted cosine.
  
      Parameters
      ----------
      M : int
          Number of points in the output window. If zero or less, an
          empty array is returned.
  
      Returns
      -------
      out : ndarray, shape(M,)
          The window, with the maximum value normalized to one (the value
          one appears only if `M` is odd).
  
      See Also
      --------
      bartlett, blackman, hamming, kaiser
  
      Notes
      -----
      The Hanning window is defined as
  
      .. math::  w(n) = 0.5 - 0.5cos\\left(\\frac{2\\pi{n}}{M-1}\\right)
                 \\qquad 0 \\leq n \\leq M-1
  
      The Hanning was named for Julius von Hann, an Austrian meteorologist.
      It is also known as the Cosine Bell. Some authors prefer that it be
      called a Hann window, to help avoid confusion with the very similar
      Hamming window.
  
      Most references to the Hanning window come from the signal processing
      literature, where it is used as one of many windowing functions for
      smoothing values.  It is also known as an apodization (which means
      "removing the foot", i.e. smoothing discontinuities at the beginning
      and end of the sampled signal) or tapering function.
  
      References
      ----------
      .. [1] Blackman, R.B. and Tukey, J.W., (1958) The measurement of power
             spectra, Dover Publications, New York.
      .. [2] E.R. Kanasewich, "Time Sequence Analysis in Geophysics",
             The University of Alberta Press, 1975, pp. 106-108.
      .. [3] Wikipedia, "Window function",
             https://en.wikipedia.org/wiki/Window_function
      .. [4] W.H. Press,  B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling,
             "Numerical Recipes", Cambridge University Press, 1986, page 425.
  
      Examples
      --------
      >>> np.hanning(12)
      array([0.        , 0.07937323, 0.29229249, 0.57115742, 0.82743037,
             0.97974649, 0.97974649, 0.82743037, 0.57115742, 0.29229249,
             0.07937323, 0.        ])
  
      Plot the window and its frequency response:
  
      >>> import matplotlib.pyplot as plt
      >>> from numpy.fft import fft, fftshift
      >>> window = np.hanning(51)
      >>> plt.plot(window)
      [<matplotlib.lines.Line2D object at 0x...>]
      >>> plt.title("Hann window")
      Text(0.5, 1.0, 'Hann window')
      >>> plt.ylabel("Amplitude")
      Text(0, 0.5, 'Amplitude')
      >>> plt.xlabel("Sample")
      Text(0.5, 0, 'Sample')
      >>> plt.show()
  
      >>> plt.figure()
      <Figure size 640x480 with 0 Axes>
      >>> A = fft(window, 2048) / 25.5
      >>> mag = np.abs(fftshift(A))
      >>> freq = np.linspace(-0.5, 0.5, len(A))
      >>> with np.errstate(divide='ignore', invalid='ignore'):
      ...     response = 20 * np.log10(mag)
      ...
      >>> response = np.clip(response, -100, 100)
      >>> plt.plot(freq, response)
      [<matplotlib.lines.Line2D object at 0x...>]
      >>> plt.title("Frequency response of the Hann window")
      Text(0.5, 1.0, 'Frequency response of the Hann window')
      >>> plt.ylabel("Magnitude [dB]")
      Text(0, 0.5, 'Magnitude [dB]')
      >>> plt.xlabel("Normalized frequency [cycles per sample]")
      Text(0.5, 0, 'Normalized frequency [cycles per sample]')
      >>> plt.axis('tight')
      ...
      >>> plt.show()
docstring_summary: Return the Hanning window.
docstring_tokens: Return the Hanning window .
function: def hanning(M):
      """
      Return the Hanning window.
  
      The Hanning window is a taper formed by using a weighted cosine.
  
      Parameters
      ----------
      M : int
          Number of points in the output window. If zero or less, an
          empty array is returned.
  
      Returns
      -------
      out : ndarray, shape(M,)
          The window, with the maximum value normalized to one (the value
          one appears only if `M` is odd).
  
      See Also
      --------
      bartlett, blackman, hamming, kaiser
  
      Notes
      -----
      The Hanning window is defined as
  
      .. math::  w(n) = 0.5 - 0.5cos\\left(\\frac{2\\pi{n}}{M-1}\\right)
                 \\qquad 0 \\leq n \\leq M-1
  
      The Hanning was named for Julius von Hann, an Austrian meteorologist.
      It is also known as the Cosine Bell. Some authors prefer that it be
      called a Hann window, to help avoid confusion with the very similar
      Hamming window.
  
      Most references to the Hanning window come from the signal processing
      literature, where it is used as one of many windowing functions for
      smoothing values.  It is also known as an apodization (which means
      "removing the foot", i.e. smoothing discontinuities at the beginning
      and end of the sampled signal) or tapering function.
  
      References
      ----------
      .. [1] Blackman, R.B. and Tukey, J.W., (1958) The measurement of power
             spectra, Dover Publications, New York.
      .. [2] E.R. Kanasewich, "Time Sequence Analysis in Geophysics",
             The University of Alberta Press, 1975, pp. 106-108.
      .. [3] Wikipedia, "Window function",
             https://en.wikipedia.org/wiki/Window_function
      .. [4] W.H. Press,  B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling,
             "Numerical Recipes", Cambridge University Press, 1986, page 425.
  
      Examples
      --------
      >>> np.hanning(12)
      array([0.        , 0.07937323, 0.29229249, 0.57115742, 0.82743037,
             0.97974649, 0.97974649, 0.82743037, 0.57115742, 0.29229249,
             0.07937323, 0.        ])
  
      Plot the window and its frequency response:
  
      >>> import matplotlib.pyplot as plt
      >>> from numpy.fft import fft, fftshift
      >>> window = np.hanning(51)
      >>> plt.plot(window)
      [<matplotlib.lines.Line2D object at 0x...>]
      >>> plt.title("Hann window")
      Text(0.5, 1.0, 'Hann window')
      >>> plt.ylabel("Amplitude")
      Text(0, 0.5, 'Amplitude')
      >>> plt.xlabel("Sample")
      Text(0.5, 0, 'Sample')
      >>> plt.show()
  
      >>> plt.figure()
      <Figure size 640x480 with 0 Axes>
      >>> A = fft(window, 2048) / 25.5
      >>> mag = np.abs(fftshift(A))
      >>> freq = np.linspace(-0.5, 0.5, len(A))
      >>> with np.errstate(divide='ignore', invalid='ignore'):
      ...     response = 20 * np.log10(mag)
      ...
      >>> response = np.clip(response, -100, 100)
      >>> plt.plot(freq, response)
      [<matplotlib.lines.Line2D object at 0x...>]
      >>> plt.title("Frequency response of the Hann window")
      Text(0.5, 1.0, 'Frequency response of the Hann window')
      >>> plt.ylabel("Magnitude [dB]")
      Text(0, 0.5, 'Magnitude [dB]')
      >>> plt.xlabel("Normalized frequency [cycles per sample]")
      Text(0.5, 0, 'Normalized frequency [cycles per sample]')
      >>> plt.axis('tight')
      ...
      >>> plt.show()
  
      """
      if M < 1:
          return array([])
      if M == 1:
          return ones(1, float)
      n = arange(0, M)
      return 0.5 - 0.5*cos(2.0*pi*n/(M-1))
function_tokens: def hanning ( M ) : if M < 1 : return array ( [ ] ) if M == 1 : return ones ( 1 , float ) n = arange ( 0 , M ) return 0.5 - 0.5 * cos ( 2.0 * pi * n / ( M - 1 ) )