Proceedings of the 25th Chinese Control Conference 7–10 August, 2006, Harbin, Heilongjiang Optimal AD-Conversion via Sampled-Data Receding Horizon Control Theory Milan S. Derpich, Daniel E. Quevedo and Graham C. Goodwin School of Electrical Engineering and Computer Science, The University of Newcastle, Australia E-mails: milan.derpich@studentmail.newcastle.edu.au; dquevedo@ieee.org; graham.goodwin@newcastle.edu.au Abstract: This paper presents novel results on the joint problem of sampling, reconstruction and quantization of analog signals. Existing literature on this topic deals exclusively with band-limited signals in sampled form. Our key departure from earlier results is that we deal with continuous time reconstruction of not necessarily band-limited signals. Our approach utilizes concepts and tools from optimal sampled-data and receding horizon control theory. The key conclusion from the work presented here is that, in the case under study, the optimal quantizer design problem can be partitioned into two sub-problems, namely (i) the design of an optimal analog pre-filter followed by sampling and (ii) an optimal quantizer, which works directly on the pre-sampled signals. Simulation results are presented which illustrate the performance of the optimal A-D converter designed via these principles. Key Words: Sampling, quantization, frames, signal processing, sampled-data control. 1 INTRODUCTION In many applications, one needs to convert analog, contin- uous time signals into quantized discrete time signals. This leads to an important set of questions regarding the best way to represent a signal by a sequence of sampled and quantized values, such that the information loss inherent in the sampling and quantization process is minimized in some sense. In the present work, we are interested in how to quantize a possible non band-limited signal to obtain the lowest possible reconstruction distortion. We will show that, for a given sampling rate and recon- struction filters, minimization of reconstruction error, in an L 2 sense, can be converted into a discrete time problem. It turns out that if an appropriate pre-filter is used, then all the information required to find the optimal quantized se- quence can always be extracted from discrete time samples of its output, even if the continuous time input signal is not band-limited. Solving the optimal quantization problem amounts to find- ing the solution of a combinatorial optimization pro- gramme, which is in general computationally intractable. Our proposal is to convert the optimal quantization problem into a sampled-data moving horizon optimization problem with quantized decision variables. The proposed method gives excellent results and incurs only limited computa- tional effort. It generalizes our previous work reported in [1][2][3][4] by concentrating on sampled-data signals rather than merely on discrete-time sequences. Background to the work described here arises from distinct streams. The first of these is associated with the problem of sampling in the absence of quantization [5][6]. The second related field of research is concerned with quantization of signals where the sampling strategy has been pre ordained [7][1][8][9]. The third stream of prior work arises in the area of sampled data control theory. Here, the emphasis has typically been on regulation (zero reference) problems with unconstrained decision variables [10] [11]. In the present work we extend these concepts to account for non zero reference signals and quantized deci- sion variables. Our approach differs from the work described above by virtue of the fact that we design the joint optimal sam- pler and quantizer using sampled data quantized moving horizon optimization. This leads to significant performance gains, compared with alternative approaches which do not take account of the interaction between sampling and quan- tization. The remainder of this work is organized as follows: In Sec- tion II we present the continuous time AD-conversion prob- lem and how it can be translated into discrete-time. Section III introduces the continuous time receding horizon quan- tizer. Simulation studies are included in Section IV. Section V draws conclusions. 2 PROBLEM FORMULATION The general form of the systems under study is illustrated in Fig. 1. Q U 〈·〉 H(s) Sampler-Quantizer Φ(s) - + Reconstruction Filter Error Weighting Filter a(t) u[k] ǫ(t) Figure 1: Block diagram of the general sampler-quantizer- reconstruction system.