Optimality and performance limitations of Analog to Digital Converters

The paper deals with the task of optimal design of Analog to Digital Converters (ADCs). A general ADC is modeled as a causal, discrete-time dynamical system with outputs taking values in a finite set. Its performance is defined as the worst-case average intensity of the filtered input matching error. The design task can be viewed as that of optimal quantized decision making with the objective of optimizing the performance measure. An algorithm based on principles of optimal control is presented for designing general m-dimensional ADCs. The design process involves numerical computation of the candidate value function of the underlying dynamic program, which is computed iteratively, in parallel with the quantization law. A procedure is presented for certifying the numerical solution and providing an upper bound for performance of the designed ADC. Furthermore, an exact analytical solution to the optimal one-dimensional ADC is presented. It is shown that the designed one-dimensional optimal ADC is identical to the classical Delta-Sigma Modulator (DSM) with uniform quantization spacing.