SC_I_03_Linear-Transform_4

SC_I_03_Linear-Transform_4 - Signal Compression 68 Signal...

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Signal Compression 68 Copyright © 2005 – 2008 Hayder Radha Linear Transformation of Random Sources Here, we briefly outline the basic properties of linear transformations of jointly random variables. To start with the most simple linear operation, if we assume that we have a single random variable X that relates to an output random variable Y through a simple real scalar a : Ya X = , where 0 a , then we have the following: Signal Compression 69 Copyright © 2005 – 2008 Hayder Radha 1. The probability distribution () Y F y of Y is related to the distribution X F x of X as follows: 0 1 0 X Y X y a F a Fy y F a a ⎛⎞ > ⎜⎟ ⎪⎝ = < ⎝⎠ Signal Compression 70 Copyright © 2005 – 2008 Hayder Radha 2. For continuous random variables, the probability density Y f y can be expressed as follows: 1 || X Y f ay fy a = . 3. The mean and variance of Y can be expressed in terms of the mean and variance of X : Signal Compression 71 Copyright © 2005 – 2008 Hayder Radha [ ] [ ] EY aEX = [ ] 22 2 var YX σ = ± The above results can be generalized for the case where the random source X is a vector of n jointly random variables: [ ] [ ] [ ] 11 nn n n YT X ×× × = . In this case, we have the following key properties:
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Signal Compression 72 Copyright © 2005 – 2008 Hayder Radha 4. For continuous joint random variables, the joint probability density () 12 ,, n YY Y Y n f yf y y y = " " can be expressed as follows in terms of the joint density n X XX X n f xf x x x = " " : 1 || X Y f Ty fy T = . Signal Compression 73 Copyright © 2005 – 2008 Hayder Radha where 1 T and T are the inverse and determinant of the transform matrix T , respectively. Note that we assume here that the transform T is invertible, and hence, its determinant has a non-zero value: ||0 T . Signal Compression 74 Copyright © 2005 – 2008 Hayder Radha 5. The mean vector, n t Y Y mm m m ⎡⎤ = ⎣⎦ " , where [ ] i Yi mE Y = , can be expressed in terms of the mean vector n t X X m m = " of the input source: [ ] [ ] [ ] 11 YX nn n n mT m ×× × = . Or, m = . Signal Compression 75 Copyright © 2005 – 2008 Hayder Radha 6. The covariance matrix Y K relates to the covariance matrix X K as follows: t KT K T =
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Signal Compression 76 Copyright © 2005 – 2008 Hayder Radha Now let assume that the input random source has uncorrelated random variables with unit variances. Then, the covariance matrix is the identity matrix X KI = . Therefore, in this case, we have: tt YX KT K TT I T == . Thus: t Y T = Signal Compression 77 Copyright © 2005 – 2008 Hayder Radha This is an important result that leads to key conclusions about the transformation of random sources: ± We can generate a random vector Y with a “desired” covariance matrix t Y T = by transforming an uncorrelated unit-variance source using a transform matrix T : YT X = . Signal Compression 78 Copyright © 2005 – 2008 Hayder Radha ± For a given desired Y K , we can find the required transform matrix T by using basic methods from linear algebra. In particular, since the covariance matrix Y K is a symmetric matrix, then we can express using a diagonal matrix Λ : t Y KQ Q Signal Compression 79 Copyright © 2005 – 2008 Hayder Radha Where Q
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This note was uploaded on 10/27/2009 for the course ECE 802 taught by Professor Staff during the Fall '08 term at Michigan State University.

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SC_I_03_Linear-Transform_4 - Signal Compression 68 Signal...

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