The Parzen window coefficients are given by the following 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.
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.
The magnitude response of the same filter is shown on the graph below.
Measures for the Parzen window
The following graph compares the discrete Fourier transform of the Parzen window with that of the rectangular window.
The Parzen window measures are as follows.