Parzen window

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The Parzen window coefficients are given by the following formula

Parzen window formula

where N is the length of the filter, M = (N – 1) / 2, and k = 0, 1, …, N – 1.

The Parzen window is a basic spline or B-spline window (see Triangular window). It is the fourth order convolution of the rectangular window with itself.

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

Parzen 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 Parzen window is as follows.

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

Measures for the Parzen window

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

Discrete Fourier transform of the Parzen window

The Parzen window measures are as follows.

Coherent gain0.69
Equivalent noise bandwidth1.12
Processing gain-0.50 dB
Scalloping loss-2.57 dB
Worst case processing loss-3.07 dB
Highest sidelobe level-24.0 dB
Sidelobe falloff-7.0 dB / octave, -23.3 dB / decade
Main lobe is -3 dB1.08 bins
Main lobe is -6 dB1.48 bins
Overlap correlation at 50% overlap0.393
Amplitude flatness at 50% overlap0.869
Overlap correlation at 75% overlap0.792
Amplitude flatness at 75% overlap0.966

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