Power of cosine window

Not Rated Yet

The power of cosine window coefficients are given by

Power of cosine window formula

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

At α = 0, the formula above produces the rectangular window. The sine window is the power of cosine window with α = 1. At α = 2, the power of cosine window becomes the Hann window by the standard double angle trigonometric identity.

Take a finite impulse response (FIR) low pass filter of length N = 201. The following is the power of cosine window for α = 3.

Power of cosine 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 power of cosine window is as follows.

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

With larger α, the window further depresses the side lobes of the impulse response of the window and the corresponding filter produces better stop band attenuation at the expense of a larger transition band. The following is the power of cosine window at three different values of α (1, 2, and 3).

Power of cosine window at three different values of alpha

The magnitude responses of the corresponding filters are as follows.

Magnitude response of a low pass filter with the power of cosine window at three different values of alpha

Measures for the power of cosine window

The following is a comparison of the discrete Fourier transform of the power of cosine window (α = 3) and the rectangular window.

Discrete Fourier transform of the power of cosine window

The power of cosine window measures are as follows.


α1.02.03.0
Coherent gain0.640.500.42
Equivalent noise bandwidth1.241.501.74
Processing gain-0.92 dB-1.77 dB-2.40 dB
Scalloping loss-2.09 dB-1.42 dB-1.07 dB
Worst case processing loss-3.01 dB-3.19 dB-3.47 dB
Highest sidelobe level-23.0 dB-31.5 dB-39.3 dB
Sidelobe falloff-11.2 dB / octave, -37.1 dB / decade-20.7 dB / octave, -68.9 dB / decade-21.3 dB / octave, -70.6 dB / decade
Main lobe is -3 dB1.18 bins1.44 bins1.66 bins
Main lobe is -6 dB1.64 bins2.00 bins2.32 bins
Overlap correlation at 50% overlap0.3170.1650.084
Amplitude flatness at 50% overlap0.7071.0000.707
Overlap correlation at 75% overlap0.7550.5680.565
Amplitude flatness at 75% overlap0.9241.0000.990


  Rating
Rate This Page: Poor Great   |  Rate Content |
Average rating:  No Ratings Yet   
Number of Ratings : 0
  Comments
Add Comment
No Comments Yet


Copyright 2006 by Kaliopa Publishing, LLC