mid06v2-sol - 921 U1050, Stochastic Signals and Systems,...

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921 U1050, Stochastic Signals and Systems, Midterm Solution, Spring 2006 (1) (3%; 1% each) Determine if each of the following function can be the autocorrelation function of a real-valued wide-sense stationary random process? Explain your answer. (Any correct answer without explanation will result in zero credit.) (a) R 1 ( τ )= τ 4 τ 3 (b) R 2 ( τ )= | τ | 1+ | τ | (c) R 3 ( τ )= sin( τ ) τ 2 Sol: R X ( τ ) of a real-valued WSS X ( μ ,t ) has to satisfy three conditions: (i) R X (0) | R X ( τ ) | 0, for all τ , (ii) R X ( τ )= R X ( τ ), for all τ , and (iii) R X ( τ ) is nonnegative de f nite. All three functions can not be the autocorrelation function of a real-valued wide-sense stationary random process because (a) R 1 (0) = 0 < | R 1 ( τ ) | for τ 9 =0 ,1 , (b) R 2 (0) = 0 <R 2 ( τ )for τ 9 =0, (c) R 3 (0) < 0. (2) (3%; 1% each) Let X ( μ )and Y ( μ ) be two real-valued random variables (rv’s) with probability density functions f X ( x )= 1 2 π exp { 1 2 ( x m X ) 2 } f Y ( y )= 1 2 π exp { 1 2 ( y m Y ) 2 } . Determine if each of the following statements is true or not. Explain your answer. (Any correct answer without explanation will result in zero credit.) (a) If E { X ( μ ) Y ( μ ) } = E { X ( μ ) } E { Y ( μ ) } ,then X ( μ )and Y ( μ ) are jointly Gaussian. (b) X ( μ )+ Y ( μ ) is a Gaussian rv. (c) If conditioned on Y ( μ )th erv X ( μ ) is Gaussian distributed, then X ( μ )and Y ( μ ) are jointly Gaussian. Sol: All are false. (a) False. This is because ”uncorrelated marginally Gaussian” does NOT imply ”jointly Gaussian.” (b) Fa lse .Th isisbecause X ( μ
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(c) Fa lse .Th isisbecause f X,Y ( x,y )= f X | Y ( x | y ) f Y ( y ) = 1 t 2 πσ 2 X | Y exp { 1 2 σ 2 X | Y ( x m X | Y ( y )) 2 } · 1 2 π exp { 1 2 ( y m Y ) 2 } may not have a quadratic exponent in x and y . (3) (6%, 3% each) Let X 1 ( μ ), X 2 ( μ ) ,...,X N ( μ ) be independent and identically distrib- uted (i.i.d.) continuous rv’s with a common continuous probability density function (p.d.f.) f X ( x ) and cumulative probability distribution function (c.d.f.) F X ( x ). Now, form Y 1 ( μ ), Y 2 ( μ ) ,..., Y N ( μ ) in a way that for a given outcome μ Y 1 ( μ )= X k 1 ( μ ) Y 2 ( μ )= X k 2 ( μ ) ... Y N ( μ )= X k N ( μ ) where X k 1 ( μ ) ,...,X k N ( μ ) are the N numbers X 1 ( μ ) ,X 2 ( μ ) ,...,X N ( μ
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mid06v2-sol - 921 U1050, Stochastic Signals and Systems,...

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