Fixed point roundoff noise in frequency sampling filters

A frequency sampling filter which is a finite impulse response digital filter uses a recursive structure that requires exact pole zero cancellation on the unit circle. When implemented with digital technology, errors due to finite length registers and finite precision arithmetic can prevent exact pole zero cancellation making the filter unstable. To guarantee stability, the poles and zeros on the unit circle are moved to a circle of radius r where 0 < r < 1. Most frequency sampling filter design techniques determine optimal frequency responses for r = 1 and then chose a value of r, 0 < 1 < 1, near to 1 so that the filter's frequency response does not differ significantly. Recently, other techniques have been presented that determine optimal frequency responses for 0 < r < 1. In this paper, the fixed point roundoff noise of Type 1 and Type 2 frequency sampling filters is determined as a function of register length, time and the value of r. An example demonstrates that for a fixed register length and output noise level, a frequency sampling filter designed with 0 < r < 1 can approximate a linear phase filter better than a frequency sampling filter designed with r = 1.