By Sophocles J Orfanidis
This revised variation is an unabridged and corrected republication of the second one variation of this booklet released by way of McGraw-Hill Publishing Company,New York, big apple, in 1988 (ISBN 0-07-047794-9), and in addition released earlierby Macmillan, Inc., manhattan, big apple, 1988 (ISBN 0-02-389380-X). Allcopyrights to this paintings reverted to Sophocles J. Orfanidis in 1996.The content material of the 2007 republication continues to be almost like that of the1988 variation, apart from a few corrections, the deletion from theAppendix of the Fortran and C functionality listings, that are now availableonline, and the addition of MATLAB types of the entire services. A pdfversion of the booklet, in addition to the entire desktop courses, can bedownloaded freely from the net page:http://www.ece.rutgers.edu/~orfanidi/osp2e
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Extra info for Optimum Signal Processing
5), it is easily shown that a necessary and sufﬁcient condition for R ¯ −1 rb > 0; alternatively, ¯ be positive deﬁnite and ρb − rT R to be positive deﬁnite is that R b −1 ˜ ˜ be positive deﬁnite and ρa − rT that R r > 0. 12) 26 1. Random Signals Backward Prediction and LU Factorization Next, we discuss the connection of the forward and backward predictors to the GramSchmidt procedure and to the Cholesky factorizations of the covariance matrix R. 13) Then, it follows from Eq. 15) ¯y ¯. It follows that LRLT is the covariance matrix of the transformed vector where ¯ eb = L eb .
38) are already part of the right-hand sides of Eq. 12), and therefore, the solutions of Eq. 12). 42) The γs are known as the reflection or PARCOR coefﬁcients. 7. Forward/Backward Prediction and LU/UL Factorization 33 The two Δs are equal, Δa = Δb , as seen from the following considerations. 44b) Noting that dT u and dT v are equal to the ﬁrst and last components of a vector d, we T T T ˜ ]u = 0 and [0, b ˜ ]v = 1 because the ﬁrst and last components of [0, b ˜ ] are have [0, b zero and one, respectively.
16). 7 Forward/Backward Prediction and LU/UL Factorization The Gram-Schmidt orthogonalization procedure discussed in the previous sections was a forward procedure in the sense that the successive orthogonalization of the components of a random vector y proceeded forward from the ﬁrst component to the last. It was given a linear prediction interpretation, that is, at each orthogonalization step, a prediction of the present component of y is made in terms of all the past ones. The procedure was seen to be mathematically equivalent to the LU Cholesky factorization of the covariance matrix R = E[yyT ] (or, the UL factorization with respect to the reversed basis).
Optimum Signal Processing by Sophocles J Orfanidis