HW8s[1] - ECE 534: Elements of Information Theory, Fall...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

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.

Page1 / 6

HW8s[1] - ECE 534: Elements of Information Theory, Fall...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online