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