Synthesis of minimum roundoff noise fixed point digital filters

Beginning with an external specification of a digital filter, structures which minimize roundoff noise are investigated. After fixing the probability of overflow through an
l_{2}
scaling procedure, roundoff noise is studied via the internal structure of the filter using a state variable formulation. An output noise variance formula in terms of the internal structure is derived. Conditions for minimizing this output noise are established and realizations which meet these conditions are constructed. A new set of filter invariants called second-order modes are defined and shown to play a definitive role in minimal noise realizations. From these invariants, for example, one can calculate the minimal output noise variance of a given external specification. Numerical results are given which compare these new filter structures with the usual parallel and cascade connections of second-order filters, both theoretically and through simulations. For narrow-band filters, these new structures can be orders of magnitude better (in terms of output noise variance). One drawback of these new structures is a large increase in the number of multipliers needed to realize them. However, by applying the theory to subfilters connected in parallel and cascade, a good compromise between output noise and number of multipliers is obtained.