Nuttall window

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The Nuttall window coefficients are given by the following formula

Nuttall window formula

where N is the length of the filter and k = 0, 1, …, N – 1.

Consider a finite impulse response (FIR) low pass filter of length N = 201. The following is the Nuttall window.

Nuttall window

Given a sampling frequency of 2000 Hz and a filter cutoff frequency of 40 Hz, the impulse response of the filter with a rectangular window (with no window) and with the Nuttall window is as follows.

Impulse response of a low pass filter with and without the Nuttall window

The magnitude response of the same filter is shown on the graph below.

Magnitude response of a low pass filter with and without the Nuttall window

Measures for the Nutall window

The following is a comparison of the discrete Fourier transform of the Nuttall window and the rectangular window.

Discrete Fourier transform of the Nuttall window

The Nuttall window measures are as follows.


Coherent gain0.36
Equivalent noise bandwidth2.03
Processing gain-3.06 dB
Scalloping loss-0.81 dB
Worst case processing loss-3.87 dB
Highest sidelobe level-93.3 dB
Sidelobe falloff-23.4 dB / octave, -77.6 dB / decade
Main lobe is -3 dB1.92 bins
Main lobe is -6 dB2.68 bins
Overlap correlation at 50% overlap0.035
Amplitude flatness at 50% overlap0.423
Overlap correlation at 75% overlap0.452
Amplitude flatness at 75% overlap1.000


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