mid2(07)sol - nctuee07f Stochastic Processes Midterm 2...

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nctuee07f Stochastic Processes Midterm 2 Solutions 1. Circularly Symmetric Complex Gaussian (5+10+10=25 points) (a) X is circularly symmetric if e X has the same distribution as X for any θ . (b) ( ) We know X is complex Gaussian, its real part and imaginary part are jointly Gaussian. To show they both are identically distributed, we need to show their means and variances are the same. X is circularly symmetric, then E [ X ] = 0 and E [ XX T ] = 0 . Since E [ X ] = E [ X r ] + jE [ X i ] , E [ X ] = 0 implies the real and imaginary part have the same mean; i.e. E [ X r ] = E [ X i ] = 0 . Next, carrying out the pseudo-covariance yields E [ XX T ] = E [( X r + jX i ) 2 ] = E [ X 2 r - X 2 i ] + 2 jE [ X r X i ] , and E [ XX T ] = 0 suggests that the real part and imaginary part of E [ XX T ] are both zero. That is E [ X 2 r - X 2 i ] = 0 and E [ X r X i ] = 0 . From the above, we have reached the conclusion that X r and X i have the same variance E [ X 2 r ] = E [ X 2 i ]. And, E [ X r X i ] = 0 indicates that X r and X i are statistically independent. Therefore, X r and X i are zero mean i.i.d. Gaussian random variables. ( ) For this part, we need to prove e X follows complex Gaussian density with the same mean, covariance, and pseudo-covariance as X . — Justify e X is complex Gaussian Let Y , e X . Carrying out Y gives Y = X r cos θ - X i sin θ | {z } , Y r + j ( X r sin θ + X i cos θ | {z } , Y i ) , for which the real part Y r and the imaginary part Y i are linear combination of X r and X i . Thus, Y is indeed complex Gaussian ( i.e. Y r and Y i are jointly Gaussian). — Justify e X has the same mean, covariance, and pseudo-covariance as X i. Since X r and X i both have zero mean. It is clear that Y r and Y i also both have zero mean. 1
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ii. E [ Y Y H ] = E [ e X ( e X ) H ] = E [ XX H ]. iii. The pseudo-covariance of
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mid2(07)sol - nctuee07f Stochastic Processes Midterm 2...

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