Sampling Approximation Of Stationary Stochastic Processes And Its Estimate Of Truncation Error

It is well known that, in order to obtain useful information, we need to carryon e?ective processing to plenty of data and signals. For a long time, both inthe signal processing and the mathematics domains, people all seek methods toexpress any signal. In the nearly 30 years, as a result of the computer technologyprogress, the signal processing technology already obtained a fast development. Inorder to use the computer to carry on the rapid calculation, the signals must besampled. It is a very important step in the signal processing. The rule which thesampling process follows, is named sampling theorem which was first presentedby an American telecommunication engineer H.Nyquistin in the year 1928[1]. Itis also called Shannon Sampling Theorem because it was first showed and quotedformally by C.E.Shannon in the year 1949[2]. It tells the relation between thesampling frequency and the signal frequency spectrum. After Shannon's work, itwas used broadly. Actually, because of the limits of the condition, precise samplingvalues cannot be obtained, so it is needed to carry on processing the observationvalues which we have. Usually, the integral to make the local average method isused. Recent years, the local average theory has a quick development. The signalscan be reconstructed based on the local average sampling theory to the signal.But, the use of the sampling theory needs the sampling value in the infinite timewhich cannot be obtained, only the limited observation value can be used to carryon to the signal approaches. So it is necessary to discuss the truncation error.This paper is mainly to approach the stochastic signals using the local averagesampling theory, and to obtain its estimate of the truncation error.