Theory of multirate statistical signal processing

In many physical observations, data might be obtained using more than one measurement device. This thesis presents a statistical theory for merging information about an object contained in multiple discrete-time measurements. We allow the measurements to be acquired by different sampling devices which might have different resolutions. In mathematical terms, we consider the model where a continuous signal x(t ) is being measured (observed) indirectly using several linear discrete-time sensors. Through the development of a statistical theory, we address such questions as: How is it possible to combine the information provided by these multirate measurements and estimate samples of the original signal x(t) as could have been obtained by a single high-resolution sensor? Under what conditions can we emulate a high-precision (high-sampling-rate) measurement by several low-precision (low-sampling-rate) ones? How much information, in a multi-sensor setting, is gained by, say, sensor A and how much by sensors B and C? The main theory developed in this thesis consists of four sub-theories. The scope of each sub-theory is as follows: (1) Multirate Statistical Inference takes statistics of low-rate measured signals and estimates statistics of the original non-observable signal. (2) Information Theory of Multirate Systems provides a quantitative measure of the amount of information contained in the low-rate measurements about the original signal. This sub-theory enables one to compare different sensors in terms of the informativity or redundancy of their data. (3) Multirate Signal Estimation takes sample values of low-rate measured signals and estimates sample values of the original non-observable signal at a finer sampling rate. (4) Scalability of Multirate Systems provides a way of comparing different multi-rate measurement systems within a class of such systems. In addition, it defines what the optimal decomposition of a signal into low-rate componets means.