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# scipy.signal.windows.lanczos

scipy.signal.windows.lanczos(M, *, sym=True)

  Return a Lanczos window also known as a sinc window.

  Parameters:
    M int

      Number of points in the output window. If zero, an empty array is returned. An exception is thrown when it is negative.

    sym bool, optional

      When True (default), generates a symmetric window, for use in filter design. When False, generates a periodic window, for use in spectral analysis.

  Returns:
    w ndarray

      The window, with the maximum value normalized to 1 (though the value 1 does not appear if M is even and sym is True).

  Notes

  The Lanczos window is defined as

  \[w(n) = sinc \left( \frac{2n}{M - 1} - 1 \right)\]

  where

  \[sinc(x) = \frac{\sin(\pi x)}{\pi x}\]

  The Lanczos window has reduced Gibbs oscillations and is widely used for filtering climate timeseries with good properties in the physical and spectral domains.

  New in version 1.10.

  References

  [1]

  Lanczos, C., and Teichmann, T. (1957). Applied analysis. Physics Today, 10, 44.

  [2]

  Duchon C. E. (1979) Lanczos Filtering in One and Two Dimensions. Journal of Applied Meteorology, Vol 18, pp 1016-1022.

  [3]

  Thomson, R. E. and Emery, W. J. (2014) Data Analysis Methods in Physical Oceanography (Third Edition), Elsevier, pp 593-637.

  [4]

  Wikipedia, “Window function”, http://en.wikipedia.org/wiki/Window_function

  Examples

  Plot the window

      >>> import numpy as np
      >>> from scipy.signal.windows import lanczos
      >>> from scipy.fft import fft, fftshift
      >>> import matplotlib.pyplot as plt
      >>> fig, ax = plt.subplots(1)
      >>> window = lanczos(51)
      >>> ax.plot(window)
      >>> ax.set_title("Lanczos window")
      >>> ax.set_ylabel("Amplitude")
      >>> ax.set_xlabel("Sample")
      >>> fig.tight_layout()
      >>> plt.show()

  and its frequency response:

      >>> fig, ax = plt.subplots(1)
      >>> A = fft(window, 2048) / (len(window)/2.0)
      >>> freq = np.linspace(-0.5, 0.5, len(A))
      >>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max())))
      >>> ax.plot(freq, response)
      >>> ax.set_xlim(-0.5, 0.5)
      >>> ax.set_ylim(-120, 0)
      >>> ax.set_title("Frequency response of the lanczos window")
      >>> ax.set_ylabel("Normalized magnitude [dB]")
      >>> ax.set_xlabel("Normalized frequency [cycles per sample]")
      >>> fig.tight_layout()
      >>> plt.show()