SC_I_04_KLT_4 - Signal Compression 114 Signal Compression...

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Signal Compression 114 Copyright © 2005 – 2008 Hayder Radha The Karhunen-Loeve Transform The KLT is the leading transform in terms of theoretical importance and in being optimal under certain criteria. In particular, the KLT is an orthogonal, normal, linear transform that generates an uncorrelated random vector. Signal Compression 115 Copyright © 2005 – 2008 Hayder Radha As before, we have the linear system: [ ] [ ] [ ] 11 nn n n YT X ×× × = , where [ ] 12 t n X XX X = " is a random source vector of jointly random variables (in general, dependent and correlated), and which is being transformed into another random vector [ ] t n YY Y Y = " . For a particular instance X x = Signal Compression 116 Copyright © 2005 – 2008 Hayder Radha of the source, we have a particular instance of the output Yy = . Hence, we also have: [ ] [ ] [ ] n n yT x × = . In short, we will use both expressions X = and x = , interchangeably. The KLT only assumes knowledge of the covariance matrix X K of the source. More importantly, the Signal Compression 117 Copyright © 2005 – 2008 Hayder Radha development of the KLT is tightly coupled with the basic linear algebraic tools for the decomposition of symmetric matrices. In other words, the KLT exploits the symmetric nature of the covariance matrix X K to generate the desired diagonal matrix Y K .
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Signal Compression 118 Copyright © 2005 – 2008 Hayder Radha Independent of our objective to have a diagonal covariance matrix Y K , we have already seen that we can express Y K using the basic definition of the covariance: () t Yy y KE Y m Y m ⎡⎤ =− ⎢⎥ ⎣⎦ t Yx x K E TX Tm TX Tm Signal Compression 119 Copyright © 2005 – 2008 Hayder Radha t tt t x T X T m X T m T t t x KT E X mX mT t YX K T = The symmetric covariance matrix X K can be expressed using Singular Value Decomposition (SVD): t X KQ Q , Signal Compression 120 Copyright © 2005 – 2008 Hayder Radha where Λ is a diagonal matrix consisting of the eigenvalues of the matrix X K : 1 2 00 n λ Λ= " ## % # " Q is a matrix of eigenvectors that form the columns of Q : Signal Compression 121 Copyright © 2005 – 2008 Hayder Radha [ ] 12 n Qe e e = " . where t ii i n i ee e e = " is the i th eigenvector. Therefore, we can express the matrix Q more explicitly:
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Signal Compression 122 Copyright © 2005 – 2008 Hayder Radha 11 12 1 21 22 2 12 n n nn n n ee e e Q e ⎡⎤ ⎢⎥ = ⎣⎦ " ## % # " . Signal Compression 123 Copyright © 2005 – 2008 Hayder Radha It is important to recall the relationship between the eigenvalues and their corresponding eigenvectors of a given matrix (in this case X K ); X ii i Ke e λ = [ ] 0 Xi i KI e −= Signal Compression 124 Copyright © 2005 – 2008 Hayder Radha It is also important to recall that Q is an orthonormal matrix since eigenvectors are orthogonal and have a unit norm: 22 2 1, 1, 2, , i n i e e i n =+ + = = "" 0, ij i j =∀ i Signal Compression 125 Copyright © 2005 – 2008 Hayder Radha Therefore, the transpose of Q have orthonormal (eigenvectors) rows: 1 2 t t t t n e e Q e = # ,
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Signal Compression 126 Copyright © 2005 – 2008 Hayder Radha where t i e is a row vector representing the transpose of the i th eigenvector.
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SC_I_04_KLT_4 - Signal Compression 114 Signal Compression...

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