parameters: (M)
return_statement: return 0.42 + 0.5*cos(pi*n/(M-1)) + 0.08*cos(2.0*pi*n/(M-1))
docstring: Return the Blackman window.
  
      The Blackman window is a taper formed by using the first three
      terms of a summation of cosines. It was designed to have close to the
      minimal leakage possible.  It is close to optimal, only slightly worse
      than a Kaiser window.
  
      Parameters
      ----------
      M : int
          Number of points in the output window. If zero or less, an empty
          array is returned.
  
      Returns
      -------
      out : ndarray
          The window, with the maximum value normalized to one (the value one
          appears only if the number of samples is odd).
  
      See Also
      --------
      bartlett, hamming, hanning, kaiser
  
      Notes
      -----
      The Blackman window is defined as
  
      .. math::  w(n) = 0.42 - 0.5 \\cos(2\\pi n/M) + 0.08 \\cos(4\\pi n/M)
  
      Most references to the Blackman 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. It is known as a
      "near optimal" tapering function, almost as good (by some measures)
      as the kaiser window.
  
      References
      ----------
      Blackman, R.B. and Tukey, J.W., (1958) The measurement of power spectra,
      Dover Publications, New York.
  
      Oppenheim, A.V., and R.W. Schafer. Discrete-Time Signal Processing.
      Upper Saddle River, NJ: Prentice-Hall, 1999, pp. 468-471.
  
      Examples
      --------
      >>> import matplotlib.pyplot as plt
      >>> np.blackman(12)
      array([-1.38777878e-17,   3.26064346e-02,   1.59903635e-01, # may vary
              4.14397981e-01,   7.36045180e-01,   9.67046769e-01,
              9.67046769e-01,   7.36045180e-01,   4.14397981e-01,
              1.59903635e-01,   3.26064346e-02,  -1.38777878e-17])
  
      Plot the window and the frequency response:
  
      >>> from numpy.fft import fft, fftshift
      >>> window = np.blackman(51)
      >>> plt.plot(window)
      [<matplotlib.lines.Line2D object at 0x...>]
      >>> plt.title("Blackman window")
      Text(0.5, 1.0, 'Blackman 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 Blackman window")
      Text(0.5, 1.0, 'Frequency response of Blackman 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 Blackman window.
docstring_tokens: Return the Blackman window .
function: def blackman(M):
      """
      Return the Blackman window.
  
      The Blackman window is a taper formed by using the first three
      terms of a summation of cosines. It was designed to have close to the
      minimal leakage possible.  It is close to optimal, only slightly worse
      than a Kaiser window.
  
      Parameters
      ----------
      M : int
          Number of points in the output window. If zero or less, an empty
          array is returned.
  
      Returns
      -------
      out : ndarray
          The window, with the maximum value normalized to one (the value one
          appears only if the number of samples is odd).
  
      See Also
      --------
      bartlett, hamming, hanning, kaiser
  
      Notes
      -----
      The Blackman window is defined as
  
      .. math::  w(n) = 0.42 - 0.5 \\cos(2\\pi n/M) + 0.08 \\cos(4\\pi n/M)
  
      Most references to the Blackman 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. It is known as a
      "near optimal" tapering function, almost as good (by some measures)
      as the kaiser window.
  
      References
      ----------
      Blackman, R.B. and Tukey, J.W., (1958) The measurement of power spectra,
      Dover Publications, New York.
  
      Oppenheim, A.V., and R.W. Schafer. Discrete-Time Signal Processing.
      Upper Saddle River, NJ: Prentice-Hall, 1999, pp. 468-471.
  
      Examples
      --------
      >>> import matplotlib.pyplot as plt
      >>> np.blackman(12)
      array([-1.38777878e-17,   3.26064346e-02,   1.59903635e-01, # may vary
              4.14397981e-01,   7.36045180e-01,   9.67046769e-01,
              9.67046769e-01,   7.36045180e-01,   4.14397981e-01,
              1.59903635e-01,   3.26064346e-02,  -1.38777878e-17])
  
      Plot the window and the frequency response:
  
      >>> from numpy.fft import fft, fftshift
      >>> window = np.blackman(51)
      >>> plt.plot(window)
      [<matplotlib.lines.Line2D object at 0x...>]
      >>> plt.title("Blackman window")
      Text(0.5, 1.0, 'Blackman 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 Blackman window")
      Text(0.5, 1.0, 'Frequency response of Blackman 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(1-M, M, 2)
      return 0.42 + 0.5*cos(pi*n/(M-1)) + 0.08*cos(2.0*pi*n/(M-1))
function_tokens: def blackman ( M ) : if M < 1 : return array ( [ ] ) if M == 1 : return ones ( 1 , float ) n = arange ( 1 - M , M , 2 ) return 0.42 + 0.5 * cos ( pi * n / ( M - 1 ) ) + 0.08 * cos ( 2.0 * pi * n / ( M - 1 ) )