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1、Statistical and Adaptive Signal Processing Chapter 7,Ju Liu ,Chapter 7 Fundaments of Order-recursive Algorithms,In this chapter, we present different algorithms for the solution of the normal equations, the computation of the minimum mean square error (MMSE), and the implementation of the optimum fi
2、lter. We start in Section 7.1 with a discussion of some results from matrix algebra that are useful for the development of order-recursive algorithms and introduce an algorithm for the order-recursive computation of the LDLH decomposition, the MMSE, and the optimum estimate in the general case.In Se
3、ction 7.2, we present some interesting interpretations for the various introduced algorithmic quantities and procedures that provide additional insight into the optimum filtering problem.,In Section 7.3, we explore the shift invariance of the input data vector to develop a time-varying lattice-ladde
4、r structure for the optimum filter. However, to derive an order-recursive algorithm for the computation of either the direct or lattice-ladder structure parameters of the optimum time-varying filter.we need an analytical description of the changing second-order statistics of the nonstationary input
5、process. Recall that in the simplest case of stationary processes, the correlation matrix is constant and Toeplitz. As a result, the optimum FIR filters and predictors are time-invariant, and their direct or lattice-ladder structure parameters can be computed (only once) using efficient, order-recur
6、sive algorithms due to Levinson and Durbin (Section 7.4) or Schr (Section 7.6).,Chapter 7 Fundaments of Order-recursive Algorithms,Section 7.5 provides a derivation of the lattice-ladder structures foroptimum filtering and prediction, their structural and statistical properties, and algorithms for t
7、ransformations between the various sets of parameters. Section 7.7 deals with efficient,order-recursive algorithms for the triangularization and inversion of Toeplitz matrices.The chapter concludes with Section 7.8 which provides a concise introduction to the Kalman filtering algorithm. The Kalman f
8、ilter provides a recursive solution to the minimum MSE filtering problem when the input stochastic process is described by a known state spacemodel. This is possible because the state space model leads to a recursive formula for the updating of the required second-order moments.,Chapter 7 Fundaments
9、 of Order-recursive Algorithms,7.1 Fundamentals of order-recursive algorithm,When the order of the estimator becomes a design variable, we need to modify our notation to take this into account. For example, the mth-order estimator cm(n) is obtained by minimizing E|em(n)|2, whereIf the mth-order esti
10、mator cm(n) has been computed by solving the normal equations, it seems to be a waste of computational power to start from scratch to compute the (m+1)st-order estimator cm+1(n) Thus, we would like to arrange the computations so that the results for order m, that is, cm(n) orcan be used to compute t
11、he estimates for order m + 1, that is, cm+1(n) or. The resulting procedures are called order-recursive algorithms or order-updating relations. Similarly, procedures that compute cm(n+1) from cm(n) or from are called time-recursive algorithms or time-updating relations. Combined order and time update
12、s are also possible. All these updates play a central role in the design and implementation of many optimum and adaptive filters.,7.1 Fundamentals of order-recursive algorithm,7.1.1 Matrix Partitioning and Optimum Nesting Matrix PartitioningNotice that if the order of the estimator increases from m
13、to m + 1, then the input data vector is augmented with one additional observation .The matrix , obtained by the intersection of the first m rows and columns of , is known as the mth-order leading principal submatrix ofIn other words, if are the elements of , then the elements of are ，1 i, j m.Since
14、we can easily see that the correlation matrix can be partitioned as(7.1.6)where (7.1.7)and (7.1.8),7.1.1 Matrix Partitioning and Optimum Nesting Optimum Nesting The result (7.1.9)is known as the optimum nesting property and is instrumental in the development of order recursive algorithms. Similarly,
15、 we can show thatimplies(7.1.10) or (7.1.11)that is, the right-hand side of the normal equations also has the optimum nesting property.the correlation matrix and the cross-correlation vector contain the information for the computation of all the optimum estimators for 1 m M.,7.1 Fundamentals of orde
16、r-recursive algorithm,7.1.2 Inversion of Partitioned Hermitian Matrices Backward Linear Predictor(BLP)We note that (7.1.19)through (7.1.21)express the parts of the inverse matrix in terms of known quantities. For our purposes, we express the above equations in a more convenient form, using the quant
17、ities (7.1.22) (7.1.23)where (7.1.19) (7.1.21)Thus, if matrix is invertible and ,we obtain(7.1.24)which determines from by using a simple rank-one modification known as the matrix inversion by partitioning lemma .Another useful expression for is(7.1.25)which reinforces the importance of the quantity for the invertibility of matrix .,
