Research On The Error Of Sampling Non-band-limited Stochastic Signals

The signal, a tool of communication, is an indispensable part in our life andwork. Reducing the error of a signal’s reconstruction from its samples makesthe signaling more accurate and efcient. After Shannon’s publication of sam-pling theorem in1949, the theories of sampling developed at a fairly high speed,which contribute to the fields of information communication, medical engineer-ing, geological exploration and etc on a large scale. While the limitation ofpractical condition results in big errors when putting Shannon’s sampling theoryinto practice inevitably. Specifically, the facts that realistic signals are usuallynon-band-limited and that stochastic signals are much more common in signalprocessing, set a higher requirement to the sampling theory.In order to sampling non-band-limited stochastic signals more accurately,two sampling series are given, as well as their property of convergence and trun-cation errors are discussed in this thesis. Firstly, based on available result ofsampling theories at home and abroad, the research goal of reducing truncationerror is determined, and proper sampling series are selected. Secondly, the spacesof research objects and their norms are defined. Thirdly, the definition of a sig-nal’s decaying exponentially in frequency domain is introduced for the purposeof building a specific mathematical model of non-band-limited signals. Finally,we transform the stochastic item of error into a definitive function by spectraldecomposition, and give respective conditions of convergence in square mean andalmost everywhere convergence of sampling series of unidimensional and multi-dimensional situations with the skills of approximation in mathematical analysisand residue theorem. At the same time, their truncation error in square mean isgiven.