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

This preview shows pages 1–2. Sign up to view the full content.

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).

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online