Zeros of the Function of an FIR Filter from the Fourier Series plus Window.

Cuthbert Nyack
In the applet the parameter hm set by scrollbar 5 is used to show the zeros for different windows. The window changes continuously from one to the next as hm is changed from 0.42 to 1.5.
Fn = 0 show the zeros when the poles at the origin are included. Fn = 1 show the zeros without the poles at the origin.

To find numerical values of zeros, first locate the zero approximately with zr and zi, then zr2 znd zi2 are used to make finer adjustments. zg can be adjusted to increase visibility of zeros outside the unit circle or zeros close to the origin.

For parameters(7, 0.05, zr, zi, 0.54. 1.0, NA, 0, 0.0, 0.0), Hamming window.
Approximate location of zeros in the z plane are :-
-0.234449 ± j0.40247 at 0.334ws
-0.796379 ± j0.60479 at 0.397ws
-1.08068 ± j1.855152 at 0.334ws
All zeros are in the stop band. When a window is used, the zeros in the stop band may not be on the unit circle but the magnitude of the product of the 2 zeros at the same frequency = 1 ie 1/(-0.234449 + j0.40247) ~ -1.08068 - j1.855152.

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COPYRIGHT 2008 Cuthbert Nyack.