Planck taper window

5

The Planck-taper window coefficients are given by the following formula

Planck-taper window formula

where N is the length of the filter and k = 0, 1, …, N – 1. ε controls the size of the top portion of the window (see below), where the window is equal to 1.

Consider a finite impulse response (FIR) low pass filter of length N = 201. The following is the Planck-taper window with ε = 0.4 .

Planck-taper 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 Planck-taper window is as follows.

Impulse response of a low pass filter with and without the Planck-taper 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 Planck-taper window

With larger ε, the top of the window becomes narrower and the transition band of the filter becomes wider, with better stop band attenuation. The following is the Planck-taper window with three different values of ε.

Planck-taper window at three different values of epsilon

The following are the corresponding magnitude responses for the same filter used above.

Magnitude responses of low pass filters with Planck-taper windows at three different values of epsilon

Measures for the Planck-taper window

The following is a comparison of the discrete Fourier transform of the Planck-taper window (ε = 0.4) and the rectangular window.

Discrete Fourier transform of the Planck-taper window

The Planck-taper window measures are as follows.


ε0.20.40.5
Coherent gain0.800.600.50
Equivalent noise bandwidth1.191.461.63
Processing gain-0.77 dB-1.64 dB-2.11 dB
Scalloping loss-2.46 dB-1.51 dB-1.20 dB
Worst case processing loss-3.22 dB-3.15 dB-3.31 dB
Highest sidelobe level-13.9 dB-17.5 dB-23.2 dB
Sidelobe falloff-27.4 dB / octave, -90.9 dB / decade-30.9 dB / octave, -102.6 dB / decade-30.8 dB / octave, -102.3 dB / decade
Main lobe is -3 dB1.10 bins1.40 bins1.56 bins
Main lobe is -6 dB1.50 bins1.92 bins2.16 bins
Overlap correlation at 50% overlap0.3920.2020.115
Amplitude flatness at 50% overlap0.5000.6721.000
Overlap correlation at 75% overlap0.7210.6660.614
Amplitude flatness at 75% overlap0.8600.9231.000


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