Sine window

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The sine window coefficients are given by the following formula

Sine window formula

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

The sine window is a power of cosine window. The convolution of the sine window with itself produces the Bohman window.

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

Sine 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 sine window is as follows.

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

Measures for the sine window

The following graph compares the discrete Fourier transform of the sine window with that of the rectangular window.

Discrete Fourier transform of the sine window

The sine window measures are as follows.

Coherent gain0.64
Equivalent noise bandwidth1.24
Processing gain-0.92 dB
Scalloping loss-2.09 dB
Worst case processing loss-3.01 dB
Highest sidelobe level-23.0 dB
Sidelobe falloff-11.2 dB / octave, -37.1 dB / decade
Main lobe is -3 dB1.18 bins
Main lobe is -6 dB1.64 bins
Overlap correlation at 50% overlap0.317
Amplitude flatness at 50% overlap0.707
Overlap correlation at 75% overlap0.755
Amplitude flatness at 75% overlap0.924

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