KLT - E [ X t X s ] = E [ X t X s ] = R X ( s,t ) , and so...

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ECE514 Random Processes Fall 2010Karhunen-Loeve Transform November 30 Here is the analysis that seemed confusing in class. Recall that φ n ( t ) is a basis function (not a basis vector), it is an eigen-function from the interval [ - T/ 2 , + T/ 2] to R . Similarly, X ( t ) is the original random process, it too is a function that maps the interval [ - T/ 2 , + T/ 2] to R . The coefficients X n are an infinite sequence in R . Finally, the autocorrelation R X ( · , · ) is a two-dimensional function, it maps [ - T/ 2 , + T/ 2] × [ - T/ 2 , + T/ 2] to R . We begin by understanding the concept of an eigen-function. To see this, let us take the inner product between φ m ( · ) and the correlation function R X ( · ,s ) , h φ m ,R X ( · , · ) i = Z - T 2 - T 2 φ m ( s ) R X ( t,s ) ds = λ m φ m ( t ) . Note that R X ( t,s ) =
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Unformatted text preview: E [ X t X s ] = E [ X t X s ] = R X ( s,t ) , and so we can write m m ( t ) = Z-T 2-T 2 m ( s ) R X ( t,s ) ds. Let us now resume our study of the correlation between X n and X m , which are coefcients of X ( ) according to the eigen-function , E [ X m X n ] = E h h x, m i h x, n i i = E " Z-T 2-T 2 x ( s ) m ( s ) ds Z-T 2-T 2 x ( t ) n ( t ) dt # = Z-T 2-T 2 Z-T 2-T 2 m ( s ) E [ X s X t ] n ( t ) ds dt = Z-T 2-T 2 Z-T 2-T 2 m ( s ) R X ( s,t ) n ( t ) ds dt = Z-T 2-T 2 m m ( t ) n ( t ) dt = m ( m-n ) . 1...
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