Frequency selective analog to digital converter design : optimality, fundamental limitations, and performance bounds

Abstract

In this thesis, the problem of analysis and design of Analog to Digital Converters (ADCs) is studied within an optimal feedback control framework. A general ADC is modeled as a causal, discrete-time dynamical system with outputs taking values in a finite set. The performance measure is defined as the worst-case average intensity of the filtered input-matching error, i.e., the frequency weighted difference between the input and output of the ADC. An exact analytic solution with conditions for optimality of a class of ADCs is presented in terms of the quantizer step size and range, resulting in a class of optimal ADCs that can be viewed as generalized Delta-Sigma Modulators (DSMs). An analytic expression for the performance of generalized DSMs is given. Furthermore, separation of quantization and control for this class of ADCs is proven under some technical conditions. When the technical conditions needed for establishing separation of quantization and control and subsequently optimality of the analytical solution to ADC design problem are not satisfied, suboptimal ADC designs are characterized in terms of solutions of a Bellman-type inequality. A computational framework is presented for designing suboptimal ADCs, providing certified upper and lower bounds on the performance.