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Unformatted text preview: ECE 534: Elements of Information Theory, Fall 2010 Homework: 8 Solutions Exercise 8.9 (Johnson Jonaris GadElkarim) Gaussian mutual information . Suppose that ( X,Y,Z ) are jointly Gaussian and that X Y Z forms a Markov chain. Let X and Y have correlation coefficient 1 and le t Y and Z have correla- tion coefficient 2 . Find I ( X ; Z ). Solution I ( X ; Z ) = h ( X ) + h ( Z ) h ( X,Z ) Since X, Y, Z are jointly Gaussian, hence X and Z are jointly Gaussian, their covariance matrix will be: K = bracketleftBigg 2 x x z xz x z xz 2 z bracketrightBigg Hence I ( X ; Z ) = 0 . 5log(2 e 2 x ) + 0 . 5log(2 e 2 z ) . 5log(2 e | K | ) | K | = 2 x 2 z (1 2 xz ) I ( X ; Y ) = . 5log(1 2 xz ) Now we need to compute xz , using markotivity p(x,zy) = p(xy)p(zy) we can get xz = E ( xz ) x z = E ( ( xz | y ) ) x z = E ( E ( x | y ) E ( z | y ) ) x z Since X, Y and Z are jointly Gaussian: E ( x | y ) = x xy y Y , we can do the same for E ( z | y ) xz = xy zy I ( X ; Z ) = . 5log(1 ( xy zy ) 2 ) Exercise 9.2 (Johnson Jonaris GadElkarim) Two-look Gaussian channel . Consider the ordinary Gaussian channel with two correlated looks at X , that is, Y = ( Y 1 ,Y 2 ), where Y 1 = X + Z 1 Y 2 = X + Z 2 1 with a power constraint P on X , and ( Z 1 ,Z 2 ) N 2 (0 ,K ), where K = bracketleftbigg N N N N bracketrightbigg . Find the capacity C for (a) = 1 (b) = 0 (c) = 1 Solution The capacity will be C = max I ( X ; Y 1 ,Y 2 ) I ( X ; Y 1 ,Y 2 ) = h ( Y 1 ,Y 2 ) h ( Y 1 ,Y 2 | X ) = h ( Y 1 ,Y 2 ) h ( Z 1 ,Z 2 ) h ( Z 1 ,Z 2 ) = 0 . 5log(2 e ) 2 | k | = 0 . 5log(2 e ) 2 N 2 (1 2 ) The mutual information will be maximized when Y 1 ,Y 2 are jointly Gaussian with covariance matrix K y = P.I 2 X 2 + K z where I 2 X...
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This note was uploaded on 01/19/2012 for the course ECE 534 taught by Professor Natashadevroye during the Fall '10 term at Ill. Chicago.
- Fall '10