lanczos#, *, sym=True)[source]#

Return a Lanczos window also known as a sinc window.


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

symbool, optional

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


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


The Lanczos window is defined as

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


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

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

Added in version 1.10.



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


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


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


Wikipedia, “Window function”,


Plot the window

>>> import numpy as np
>>> from 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()

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()