An optimization approach to the design of frequency sampling filters implemented with finite word length architectures

Abstract

Many digital signal processing applications require linear phase filtering. Under certain conditions, a frequency sampling filter can implement a linear phase filter more efficiently than an equivalent filter implemented by a direct convolution structure. However, frequency sampling filter structures require pole-zero cancellations on the unit circle. For practical implementations of frequency sampling filters, finite word length effects usually prevent exact pole-zero cancellation. An uncancelled pole on the unit circle causes the filter to be unstable. Therefore, designers move the filter's poles and zeros off the unit circle and onto a circle of radius r where r < 1, by replacing z −1 with rz −1 in the filter's system function. As a result of this modification, the frequency response of the new filter using rz −1 where r < 1 is different from the frequency response of the original filterwhere r = 1. As r decreases, the frequency response of the modified frequency sampling filter increasingly differs from the frequency response of the original filter, and thus values of r close to 1 are usually chosen. However, it has been shown that the filter's output roundoff noise decreases as r decreases. Thus, there is a need for a design technique which chooses a value of r which renders an acceptable roundoff noise level while satisfying various frequency response design constraints. This paper develops an optimization method for designing a frequency sampling filter with r < 1 such that a linear combination of the mean square error between the desired and actual frequency responses in the passband and stopband and the sum of the square error of the impulse response symmetry is minimized while the stopband frequency samples are constrained to zero.