Tham khảo tài liệu 'stochastic control part 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 272 Stochastic Control 2. Random signal expansion 1-D discrete-time signals Let S be a zero mean stationary discrete-time random signal made of M successive samples and let si S2 . SM be a zero mean uncorrelated random variable sequence . E snsm s j 3n m 1 where ỗn m denotes the Kronecker symbol. It is possible to expand signal S into series of the form M S smYm 2 m 1 where Ym m i M corresponds to a M-dimensional deterministic basis. Vectors Ym are linked to the choice of random variables sequence sm so there are many decompositions 2 . These vectors are determined by considering the mathematical expectation of the product of sm with the random signal S. It comes 1 Ym E S2J E Sm S . 3 Classically and using a M-dimensional deterministic basis m m i M the random variables Sm can be expressed by the following relation sm S m. 4 The determination of these random variables depends on the choice of the basis m m i M We will use a basis which provides the uncorrelation of the random variables. Using relations 1 and 4 we can show that the uncorrelation is ensured when vectors m are solution of the following quadratic form mTTssu E S2m 5n m 5 where Tss represents the signal covariance. There is an infinity of sets of vectors obtained by solving the previous equation. Assuming that a basis m m i M is chosen we can find random variables using relation 4 . Taking into account relations 3 and 4 we obtain as new expression for Ym 1 Ym Ef b rss . 6 Furthermore using relations 5 and 6 we can show that vectors Ym and m are linked by the following bi-orthogonality relation m T Yn ỗn m. 7 The stochastic matched filter and its applications to detection and de-noising 273 Approximation error When the discrete sum describing the signal expansion relation 2 is reduced to Q random variables sm only an approximation Sq of the signal is obtained - Q Sq SmTm. 8 m 1 To evaluate the error induced by the restitution let us consider the mean square error e between signal S and its .
