This is just a mathematical way of saying EE 562a Homework Solutions 3 February

This is just a mathematical way of saying ee 562a

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This is just a mathematical way of saying:
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EE 562a Homework Solutions 3 February 16, 2011 5 The first component of Re { x ( u, t ) } is a Gaussian random variable with mean m and variance σ 2 . The second component of Re { x ( u, t ) } is zero with probability one. The first component of Im { x ( u, t ) } is zero with probability one. The second component of Im { x ( u, t ) } is equal to 2 m - (the first component of Re { x ( u, t ) } ) with probability one. (c) Our solution for part (a) was x o ( u ) = σa ( u, 1) 1 - i , so the pseudo-covariance (unconjugated covariance) is e K x = σ 2 1 - i - i - 1 , which is not the all-zeros matrix O . This suggests that we need more than one real random variable to generate x ( u ) with e K x = O . Necessary and sufficient conditions for e K x = O are K r = K y = 1 2 Re { K x } = σ 2 2 1 0 0 1 K yr = 1 2 Im { K x } = σ 2 2 0 1 - 1 0 . The requirement on the correlation matrices means that the components of both r o ( u ) and y o ( u ) are uncorrelated, while the cross correlation matrix implies E { y 0 ( u, 1) r 0 ( u, 1) } = E { y 0 ( u, 2) r 0 ( u, 2) } = 0 E { y 0 ( u, 1) r 0 ( u, 2) } = σ 2 2 = y 0 ( u, 1) = r 0 ( u, 2) with probability 1 E { y 0 ( u, 2) r 0 ( u, 1) } = - σ 2 2 = y 0 ( u, 2) = - r 0 ( u, 1) with probability 1 . It is clear that two real random variables are required. r o ( u ) = σ 2 a ( u, 1) a ( u, 2) y o ( u ) = 0 1 - 1 0 r o ( u ) = σ 2 a ( u, 2) - a ( u, 1) . Adding in the mean value gives r ( u ) = r o ( u ) + m 0 y ( u ) = y o ( u ) + 0 m . It is now clear that x ( u ) = r ( u ) + i y ( u ) has the desired second order description. The number of real random variables required to simulate the vector x ( u ) in this case is two .
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6 EE 562a Homework Solutions 3 February 16, 2011 (d) Let me call the complex random vector x ( u ) for this part of the problem as well, and again x ( u ) = r ( u ) + i y ( u ) . For this part of the problem K x is nonsingular and the pseudo-covariance is O , so it will require 4 real random variables to generate x ( u ). The design follows the standard method. Let x o ( u ) = Hw ( u ), where w ( u ) is a 2-dimensional complex random vector with m w = 0 , K w = I and e K w = O . Such a w ( u ) can be constructed as follows: w ( u ) = w ( u, 1) w ( u, 2) = 1 2 a ( u, 1) + i a ( u, 3) a ( u, 2) + i a ( u, 4) . We solve for H from the relation K x = HH . You can perform this using either the direct, eigen or LDL methods. Here is the sequence of eliminations for the LDL method. σ 2 3 i - i 3 σ 2 3 i 0 8 / 3 = DL . We take H = LD - 1 / 2 to get H = σ 3 0 - i / 3 p 8 / 3 . In order to find the probability density function, you need to have read the notes. In particular equation (4.2-54) on page 88 of the notes yields p x ( u ) ( z ) = 1 π 2 | K x | exp[( z - m x ) K - 1 x ( z - m x )] , where z C 2 . This holds only when e K x = O . The inverse is simple. K - 1 x = 1 8 σ 2 3 - i i 3 . 4. (a) This question requires you to interpret random variables as elements in an inner product space. Using the given inner product definition and the metric that it induces, it follows that i. || x (1) || = ( x (1) , x (1)) 1 / 2 = [ E { x 2 ( u, 1) } ] 1 2 = ( 1 2 ) 1 / 2 = 1 2 ii. mean-squared length of x ( u ) = E { x 2 ( u ) } = Tr ( K x ) = 2 1 8 (b) o 1 = x 1 w 1 = 1 ( o 1 , o 1 ) 1 2 · o 1 o 2 = x 2 - ( x 2 , w 1 ) w 1 = x 2 - ( x 2 , o 1 ) ( o 1 , o 1 ) · o 1 w 2 = 1 ( o 2 , o 2 ) 1 2 · o 2 o 3 = x 3 - ( x 3 , w 1 ) w 1 - ( x 3 , w 2 ) w 2 = x 3 - ( x 3 , o 1 ) ( o 1 , o 1 ) · o 1 - ( x 3 , o 2 ) ( o 2 , o 2 ) · o 2 w 3 = 1 ( o 3 , o 3 ) 1 2 · o 3
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EE 562a Homework Solutions 3 February 16, 2011 7
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