tailieunhanh - Phân tích tín hiệu P5

Transforms and Filters for Stochastic Processes In this chapter, we consider the optimal processing of random signals. We start with transforms that have optimal approximation properties, in the least-squares sense, for continuous and discrete-time signals, respectively. Then we discuss the relationships between discrete transforms, optimal linear estimators, and optimal linear filters. | Signal Analysis Wavelets Filter Banks Time-Frequency Transforms and Applications. Alfred Mertins Copyright 1999 John Wiley Sons Ltd Print ISBN 0-471-98626-7 Electronic ISBN 0-470-84183-4 Chapter 5 Transforms and Filters for Stochastic Processes In this chapter we consider the optimal processing of random signals. We start with transforms that have optimal approximation properties in the least-squares sense for continuous and discrete-time signals respectively. Then we discuss the relationships between discrete transforms optimal linear estimators and optimal linear filters. The Continuous-Time Karhunen-Lo eve Transform Among all linear transforms the Karhunen-Lo eve transform KLT is the one which best approximates a stochastic process in the least squares sense. Furthermore the KLT is a signal expansion with uncorrelated coefficients. These properties make it interesting for many signal processing applications such as coding and pattern recognition. The transform can be formulated for continuous-time and discrete-time processes. In this section we sketch the continuous-time case 81 149 .The discrete-time case will be discussed in the next section in greater detail. Consider a real-valued continuous-time random process x t a t b. 101 102 Chapter 5. Transforms and Filters for Stochastic Processes We may not assume that every sample function of the random process lies in L-iÇa b and can be represented exactly via a series expansion. Therefore a weaker condition is formulated which states that we are looking for a series expansion that represents the stochastic process in the mean 1 N x t yj t 5-1 The unknown orthonormal basis t i 1 2 . has to be derived from the properties of the stochastic process. For this we require that the coefficients Xi i x t ipi f dt J a of the series expansion are uncorrelated. This can be expressed as E xiXj E x j 5-3 The kernel of the integral representation in is the autocorrelation function rxx t u E x t x u . We see