KLtransform - Karhunen-Loeve expansion Let X be a zero-mean...

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Karhunen-Loeve expansion Let X be a zero-mean m x 1 random vector, and let { } H R E = XX . Let R have the factorization H R U U = Λ , where the columns of U are the normalized eigenvectors of R. Let H U = Y X . Then Y is a zero-mean random vector with uncorrelated components: { } { } { } H H H H H H H E E U U U E U U U U U = = = Λ = Λ YY XX XX (1) We can write 1 m i i i U y = = = X Y u (2) where i u is the i th column vector of Y . This says that we can construct X as a linear combination of orthogonal vectors i u , where the coefficients i y are uncorrelated random variables. This representation is called Karhunen-Loeve (K-L) expansion of X . In this case, the eigenvectors of R are used as the basis vectors. K-L expansion can be used in signal coding (data compression). In the encoder, the encoding signal X is K-L transformed by calculating H U = Y X . (3) Then, i y are encoded and transmitted. In the decoder, the reconstructed X is obtained by 1 ˆ ˆ m i i i y = = X u . (4).
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KLtransform - Karhunen-Loeve expansion Let X be a zero-mean...

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