, sym=True)[source]#

Return a minimum 4-term Blackman-Harris window according to Nuttall.

This variation is called “Nuttall4c” by Heinzel. 2

  • M (int) – Number of points in the output window. If zero or less, an empty array is returned.

  • 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.


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

Return type



For more information, see 1 and 2



A. Nuttall, “Some windows with very good sidelobe behavior,” IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 29, no. 1, pp. 84-91, Feb 1981. 10.1109/TASSP.1981.1163506


Heinzel G. et al., “Spectrum and spectral density estimation by the Discrete Fourier transform (DFT), including a comprehensive list of window functions and some new flat-top windows”, February 15, 2002


Plot the window and its frequency response:

>>> from import nuttall
>>> import cupy as cp
>>> from cupy.fft import fft, fftshift
>>> import matplotlib.pyplot as plt
>>> window = nuttall(51)
>>> plt.plot(cupy.asnumpy(window))
>>> plt.title("Nuttall window")
>>> plt.ylabel("Amplitude")
>>> plt.xlabel("Sample")
>>> plt.figure()
>>> A = fft(window, 2048) / (len(window)/2.0)
>>> freq = cupy.linspace(-0.5, 0.5, len(A))
>>> response = 20 * cupy.log10(cupy.abs(fftshift(A / cupy.abs(A).max())))
>>> plt.plot(cupy.asnumpy(freq), cupy.asnumpy(response))
>>> plt.axis([-0.5, 0.5, -120, 0])
>>> plt.title("Frequency response of the Nuttall window")
>>> plt.ylabel("Normalized magnitude [dB]")
>>> plt.xlabel("Normalized frequency [cycles per sample]")