摘要
本文主要从研究确定性信号的局部平均采样产生的误差上界入手,旨在系统研究Lp (1≤p≤∞)空间上实频谱有限和非频谱有限宽平稳随机过程局部平均采样产生的误差上界的明确估计,用广义核函数来逼近L2上复值宽平稳随机过程,以及利用小波框架作为工具来研究局部平均采样问题.同时对概率型算子逼近和利用概率型算子逼近复值二阶矩过程也给出了部分结果.
Shannon C.E.于1948年提出的采样定理引起电子工程和通信领域的巨大变革.其基本思想是利用一组离散的采样值来表示一个连续的频谱有限信号.这一思想后来被许多数学家发展.他们重构信号所利用的采样值是信号在采样点处的准确取值.但是在实际处理中,比如由于测量仪器的属性,实际中得到的采样值并不是信号在该点的精确值,而是在该点附近的局部平均值. 1992年, Gro¨chenig K.首先利用一列权函数对信号取局部积分平均,然后用所得到的平均采样来重构原信号.试验表明这种局部平均的采样方法可以有效抑制噪声.随后, Aldroubi A.,Butzer P. L.和Lei J.以及孙文昌教授和周性伟教授分别利用局部平均在2002年前后连续发表了一系列文章对确定性信号的局部平均采样与重构问题进行了深入的研究.
另一方面,在工程和物理实验等领域时常出现一种称为“白噪声”的现象影响着测量数据的准确性,白噪声实际上是一种宽平稳过程.受此影响,著名概率专家Kolmogorov A. N. 1956年指导博士生Belyaev Y. K.从事随机过程的Shannon采样定理的研究工作. Balakrishnan A.V., Butzer P.L., Splettsto¨sser W.等人先后发表了一系列文章比较完整的解决了宽平稳过程的采样问题.宽平稳过程的框架稳定性的第一个结果是由挪威国家数学会主席Seip I.在1990年首次给出的.结果发表在信号处理的权威杂志《IEEE Transactions on Information Theory》上.
本文在总结前人工作的基础上,给出了更符合实际情况的新的局部平均,以便更准确地刻划函数局部平均以及随机过程的局部均方积分平均.所得主要结果可以概括为四个创新点.
1.以新构造的的局部平均为工具,用新的光滑模处理技巧给出了确定性信号局部平均采样代入Shannon采样公式后产生的明确的误差上界估计(本文第二章).
文中通过计算实例验证新的估计的特殊情况上界估计值只有Butzer和Lei2000年所得结果的1/6左右.这一结果已经发表在《Appleid Mathematics Letters》.
2.以新构造的局部平均为工具,用新的随机过程均方光滑模处理技巧给出了频谱有限实宽平稳过程和非频谱有限实宽平稳过程局部平均采样代入Shannon采样公式后产生的均方意义下的明确的误差上界估计(本文第三章).两项结果已经分别投往《中国科学》(A辑,英文版)和《Lecture Notes in Com-puter Science》.第二项结果已经被接受并将于2006年5月发表.
3.以新构造的局部平均采样为工具,给出了实宽平稳过程的广义Shannon小波级数逼近和频谱有限复值宽平稳过程的重构及小波框架的稳定性(本文第四章).
上述两项结果已经分别投往《TransactionsofTianjinUniversity》和《IEEETrans-actions on Information Theory》.以第二项结果为工具可以推广Seip I., Gro¨chenig,K., Feicgtinger H.以及孙文昌教授和周性伟教授发表在《Constructive Approxima-tion》和《IEEE Transactions on Information Theory》等杂志刊出的多项结果.4.以局部平均为工具,给出了连续信号概率型算子线性组合的点态逼近和随机信号的经典概率型算子逼近(本文第五章).
上述第一项结果已经发表在《天津大学学报》(2005年11期).第二项结果打开了用局部平均概率型算子研究二阶矩过程的一个缺口,也为几十年来许多数学家考虑算子逼近应用问题找到了一个落脚点.结果整理后投《Journal of Approxi-mation Theory》.
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》.
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