Research On Analog Signal Sampling And Reconstruction Based On A Maximal Pointwise Reconstruction Tolerance Of Time Domain

In this digital era,it has almost been a standard practice to convert any measurable analog signal to its digital form for later tasks such as processing and storage.Sampling and reconstruction are the most fundamental steps in the process.So far the cornerstone for sampling rate determination and reconstruction is the classical Shannon sampling theory for band-limited signals,which is also known as Nyquist sampling theorem.The band-limited signals can be reconstructed perfectly by the Shannon theory.With the high speed DSP applications becoming ever more widespread,it is often desirable to sample with as few samples as possible while maintaining the desired accuracy.Thus the pointwise stability can be obtained,and it is convenient for studying transient signal behavior.But there are few simple and efficient methods suitable for engineering practice.So we propose a group of theorems based on a maximal pointwise reconstruction tolerance through some classical interpolations.In some cases,we cannot obtain the spectral information about the analog signal to be sampled beforehand,so we can not use Shannon directly.Thus in this paper,we make the research in the time domain.Firstly,we introduce several classical interpolations and the corresponding interpolation errors.Then for the constant and linear interpolation which are often used in reconstructing analog signals,we propose the sampling and reconstruction algorithm AT0 and AT1.Both the theorems can strictly approximate the pointwise reconstruction error tolerance.Secondly,based on the improvement of the above two low-order interpolation algorithms,we also present a group of high-order interpolation algorithms for Taylor interpolation,Lagrange interpolation and Newton interpolation.We get the sampling and reconstruction theorem ATPA through Taylor interpolation.With the Taylor interpolation polynomial order increasing,the actual reconstruction error shows a decreasing trend.Theorem ATPA is only suitable for differential continuous signals,and there is no continuity existing in the signals reconstructed by ATPA.Therefore,we get the theorem ATPB and ATPC by improving this two disadvantages of ATPA.Wealso propose the sampling and reconstruction algorithm ATPD through Lagrange and Newton interpolation.Compared with the algorithms based on Taylor interpolation,ATPD need not sample the derivatives of original signals.What's more,when reconstructing the signal of each interpolation period [(7)1,(10)(8)TnnT],we can use any m continuous samples including x(7)nT(8)and[(7)(10)1(8)Tnx] to finish the m-1 order polynomial.Compared with the Shannon sampling theorem,the algorithms we proposed based on the maximal pointwise reconstruction tolerance of time domain have three main advantages: the first is the application range.It is suitable for differential and piecewise differential continuous signals,no matter non-band-limited signals or signals which are difficult to obtain the spectrum information beforehand.Second,it can guarantee the reconstruction accuracy of each point in the time domain,and it also allows local adaptive sampling.The third advantage is that the samplers are easy to design,because all the related calculations can be completed by simple circuits.