Optimal Fixed-Point Implementation of Digital Filters

We consider the problem of finding optimal realizations of discrete-time digital filters w.r.t. implementation on a finite-memory machine with a fixed-point processor. A realization for which the performance of the fixed-point implementation under quantization effects is closest (in some sense) to the ideal performance (assuming precise arithmetic) is considered optimal. The transfer function corresponding to the optimal implementation is not necessarily the same as the transfer function of the original filter. Therefore, the optimal implementation cannot be obtained via a change of coordinates. This problem is inherently a nonlinear control problem due to the presence of the quantizer. We use a signal + noise model to linearize the system. After linearization, the problem of finding the optimal realization is still a nonconvex optimization problem. We show that under some mild technical assumptions the problem can be convexified losslessly, and transformed into an equivalent set of LMIs for which numerical solutions can be obtained efficiently.