Local Average Of Random Signal Sampling

In this paper, we begin our research with discussing the upper error bound of de-terministic signals from local averages. Our main purpose is to study the upper errorbound of band-limited stationary stochastic processes and non band-limited stationarystochastic processes in wide sense from local averages, approximate of wide sense sta-tionary stochastic processes with generalized kernel functions, and use wavelet framesas a tool to investigate the local average sampling problems. We will also give someresults on the approximation of probabilistic operators and approximating complex sec-ond order moment processes by probabilistic operators.The sampling theorem introduced by Shannon C.E. in 1948 caused a revolutionin the field of engineering. Its basic idea is presenting a continuous bandlimited sig-nal by a sequence of discrete samples,which was developed by many mathematicianslater. The samples they used to reconstruct the signal are the exact values of the orig-inal signal at the sampling points. But due to physical reasons, e.g., the inertia of themeasurement apparatus, measured sampled values obtained in practice may not be val-ues of the original signal at the sampling points precisely, but only local averages ofthe signal near the sampling points. In 1992, Gro¨chenig K. used a sequence of weightfunctions to average the original signal locally near the sampling points, and then usedthe obtained local averages to reconstruct the original signal. experiments show thatthe local average sampling method can suppress the high frequency noise effectively.Later, Aldroubi A., Butzer P. L. and Lei J., Sun W. and Zhou X. gave a lot of interest-ing results on the local average sampling and reconstruction of deterministic signals inabout 2002.On the other hand, there is a so called white noise phenomenon in the fields ofengineering and physical examination, which in?uences the accuracy of measurement.White noise is one kind of wide sense stationary stochastic processes. Guided by theoutstanding probability expert Kolmogorov A. N. and as a Ph.D. student, Belyaev Y. K.was engaged in the study Shannon sampling theorem for stationary stochastic processes in 1956. Balakrishnan A.V., Butzer P.L., Splettsto¨sser W. et. gave a lots of results onthe sampling theorem of wide sense stationary stochastic processes. Their research wasmore completely and systematically. The first result on the stability of wavelet framescomplex wide sense stationary stochastic processes was gave by Seip I. in 1990. SeipI. is president of the Norwegian Mathematical Society now. The result was publishedon《IEEE Transactions on Information Theory》which is a authority journal of signalprocessing.Based on the results forementioned, we give a new local average sampling method,which is more close to true applications, and can be used to sampling both deterministicsignals and stochastic processes in mean square sense precisely. The innovation of thispaper is summed as four points.1. The explicit upper error bounds of deterministic signals for our new localaverage sampling method with modulus of continuity is given(Chapter 2 in thisPh.D. thesis).We also give concrete examples to illustrate that our new estimate in special casesis only 1/6 times of the estimate by Butzer and Lei in 2000 in the same special cases.This results is published by《Appleid Mathematics Letters》.2. The explicit upper error bounds of band-limited and non band-limitedstationary stochastic processes in wide sense for our new local average samplingmethod with new modulus of continuity in mean square sense is presented (Chap-ter 3 in this Ph.D. thesis).The tow results are submitted to《Science in China Ser. A Mathematics》and《Lecture Notes in Computer Science》, respectively . The late is accepted for publish-ing and will be published in May 2006.3. The generalized Shannon wavelet series approximation of real station-ary stochastic processes in wide sense and reconstruction of complex band-limitedstationary stochastic processes in wide sense from local average samples and thestability of wavelet frames are given(Chapter 4 in this Ph.D. thesis).The tow results above are submitted to《Transactions of Tianjin University》and《IEEE Transactions on Information Theory》, respectively. Using the second results,we can improve the results of Seip I., Gro¨chenig, K., Feicgtinger H., Sun W. and Zhou X., which is published in《Constructive Approximation》and《IEEE Transactions onInformation Theory》etc.4. The approximation of continuous signals by linear combination of proba-bilistic operators and the approximation of stochastic signals by classical proba-bilistic operators using local average sampling data (Chapter 5 in this Ph.D. the-sis).The first result above was published in《Journal of Tianjin University》(2005.11).The second result above make a dent in the research of second order moment processesby probabilistic operators, and give a application of probabilistic operator approxima-tion, which have been searched by mathematicians for many years. The result will besubmitted to《Journal of Approximation Theory》.