Roundoff noise minimization and pole-zero sensitivity in fixed-point digital filters using residue feedback

The problem of designing a finite wordlength fixed-point realization of an Nth-order digital filter, which uses residue feedback to minimize the output roundoff noise subject to l2-scaling, is considered. The new structures require N extra additions, but no more multiplications than the earlier low noise structures of Mullis and Roberts, and have lower roundoff noise for sufficiently narrow bandwidth filters. A new set of filter invariants, called the residue modes, are defined which characterize the new low noise structures and determine the output noise variance. If the sum of the residue modes is less than the sum of the second-order modes of Mullis and Roberts, then lower roundoff noise is achieved. Every filter structure is shown to define a unique (symmetrizing) matrix Q. Pole-zero sensitivities and a new noise measure are defined in terms of Q. Numerical results are included which compare the noise and pole-zero sensitivity characteristics of different filter structures.