A study of pole-zero placement options for frequency sampling fir filter realizations

This thesis investigates finite impulse response (FIR) digital filters implemented using a frequency-sampling structure with a different radius for the poles than for the zeros. Even though it is well known that both the poles and zeros of the filter should ideally fall at the same locations on the unit circle, practical implementations move both the poles and zeros slightly inside to make the filter stable. Another benefit gained is that as the poles are moved farther inside the unit circle the error due to roundoff noise decreases. In a recent thesis both the poles and zeros were moved inside to the same radius. It was shown that the roundoff noise decreases as the poles are moved farther inside, but the frequency response in the passband also decreases in amplitude. It was also shown that the amplitude in the passband can be corrected by scaling, but it increases the amplitude in the stopband causing frequencies defined to have a zero gain to have a gain greater than zero.

Pole-zero separation distance and stopband zero compensation are investigated in this thesis for moving the poles inside the unit circle to a different radius than the zeros. These techniques were used in an attempt to retain the desirable effects of increased stability and reduced roundoff noise, while allowing the amplitude to be corrected in the passband by scaling without distorting the amplitude in the stopband. It is found that neither technique is useful for the design of low roundoff noise frequency sampling structure digital filters. Pole-zero separation cannot be used because both the magnitude and phase responses become too distorted for separation distances that are large enough to reduce roundoff noise. Stopband zero compensation is not practical because of the excessive computation involved in its implementation.

Since neither pole-zero separation nor stopband zero compensation proved to be useful it is concluded that the best technique for designing low roundoff noise frequency sampling filters is to move both the poles and zeros inside the unit circle to the same radius. The floating-point roundoff noise gain is analyzed for three different realizations of both lowpass and bandpass filters where this technique is used. This roundoff noise analysis takes the extended precision of floating-point digital signal processors into account.