e3q - v t is a given real signal such that V f = 1 for | f...

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ECE 514 , Fall 2007 Exam 3: Due ECE front desk, 3:00pm, December 7, 2007 Name: 75 mins.; Total 50 pts. 1. (15 pts.) Suppose we wish to estimate a quantity represented by a real-valued random variable X with mean m and variance σ 2 X . For this estimation, we take n measurements of X represented by Y i = X + W i , i = 1 , . . . , n , where W 1 , . . . , W n represent i.i.d. noise, independent of X , with mean 0 and variance σ 2 W . a. Let e = [1 , . . . , 1] . Show that e ( σ 2 X ee + σ 2 W I ) - 1 = 1 2 X + σ 2 W e . b. Find the Wiener filter for estimating X based on Y 1 , . . . , Y n . (Part a helps.) c. What exactly is the difference between the filter in part b and simply averaging the measure- ments? What is the effect of very large n , or very large σ 2 X , or very large σ 2 W ? 1
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2. (10 pts.) Suppose we pass an n th-order strictly stationary random process through a linear time invariant system. Is the output necessarily n th-order stationary? Explain fully. (For your analysis you may assume either discrete or continuous time.) 3
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3. (10 pts.) Consider the setting of the matched-filter problem (Section 10.7), where
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Unformatted text preview: v ( t ) is a given real signal such that V ( f ) = 1 for | f | ∈ [1 , 2] and V ( f ) = 0 elsewhere. Suppose you treat the random process X t as a jamming signal that you are free to design (as an adversary). The design objective is to minimize the best-case SNR at the output of the receiver ( best-case here means maximum over all possible receiver designs). The design constraint is that you have only a fixed amount of power P to allocate to your jammer. Describe your design of the jamming signal. (It suffices to design the power spectral density S X .) 5 6 4. (15 pts.) Let B be a Bernoulli random variable taking values on { , 1 } . Define the discrete-time process X 1 , X 2 , . . . by X t = (-1) B + t , t = 1 , 2 , . . . . Let p = P { B = 0 } . a. For what values of p is the process X t strictly stationary? b. For what values of p in part a is the process also ergodic? Explain fully. 7 8...
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