Chapter 15 Problems

Chapter 15 Problems - Exercises m 433 B—measuiable and...

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Unformatted text preview: - Exercises m. 433 B—measuiable and that agrees almost surely with Y5 (P (Z = Y3) = 1) will also satisfy the defini- tion of being a conditional expectation Hence, theie are-many versions of conditional expectation, but each pair agrees almost surely. Obseive how the awkward case 'of' conditioning on an event, say, Be 6 B, that is null (Pale) = 0) is handled. with apparent ease in this definition of conditional expectationw-there is no division by zero” It is clear from the definition that any choice of Y3(co) for a) 6 Bo is acceptable. Conditional expectation E (Y 13) can be defined arbitrarily on a null set in 8“ However, we still need constructive means to identify Y3? EXERCISES _ E15..1 at If fig y (x. y) = e“""” U (x)U (y), then evaluate E (Y |X)‘, . b.. If many) = %1e‘x"U(x)U(y), then evaluate E(Y1_X).. E152 If (Vi)EX,-2 < co and (Vi aé j)E‘(X,- 1X3) = 0, then evaluate E311 and E(X1X2)t E153 We observe voltages V1 and Y2 having a joint density that is uniform over the unit disk; ' Evaluate E (V2] V1)“ _ _- I - . 1315.4 a. If we know that E(V2|V1) = V13 and that V1 '9 L“ _—--_1; 1), then evaluate the cone. lation between V1 and V2. by. Are V1 and V2 independent? E155 a. IfX - m0, 1) and E(Z IX) = X2, then evaluate EZ.‘ b.. For the specification of part (a), evaluate the 'couelation E‘(XZ).. E156 a“ If, in a system with output Y and input)! ~ 8(a), we know that E(Y IX) 2: X2, evaluate EY t. - b.‘ Evaluate the input—output correlation E(YX)U . E153 It is known that X ~ ‘N'(O, 1) and 'E(Y_lX) = X4” Giving reasons for your answers, are X and Y independent or uncoirelated? - E153 It is knOWn that; for all x, E_(l’ {X _=x) > x‘. . a. If' EX is finite, what can you conclude about E_Y and EX? b.. If EX is infinite, what can you conclude about EY‘?‘ E153 A receiver whose antenna is pointed at the ‘sky yields output voltage measurements V1 ' and V2, made at different times, that are jointly normally distn'buted- with mean zero, equal valiances of 1/2, and a covariance of 1/4. :1. Evaluate the conditional density ‘ fvzlvl y. b.. Evaluate the minimum mean square estimator V2(V1) of V2” 1215.10 If _ _ _ _ _ __ me,» = (x +y)U(x)U(y)Ut1--x)U(1-y). . then evaluate the minimum mean square estimator f7 (X )- of Y .. . - 1315.11 We obsetve the product)! = SN , where the signal S ~uc1, 2) and the independent noise N has the Pareto model ' - 7 he) = fizzle — n, - 434 Chapter 15 Conditional Expectation a: Determine ‘ fxls (xii). (Hint: Consider the cdeF'xls- ‘) . b.‘ Determinefstflslx)‘. Be explicit about the range of' s and the cases J: < 2 and on. What is the minimum mean square estimator .§' (X) of the signal given the obser- vation; for observations x Z 2? - r d.‘ What is the performance of this estimator? ‘- e.‘ Determine the asymptotic behavior (large sample size n) of E (S; -— S)? E1512 at. If Y = ex +-N and XJLN, with X «Mon, a2), N ~ Mo, 1), provide an ex.- pression for the MMSE estimator )2 (Y) of X given Y M . b.‘ What is E}? (Y )? (This can be done with very little calculation.) E1513 A thermal noise voltage V appears across an independent random resistor R N JV(R0, '1), R0 >> 1, and it induces a noise current 1.. - ' at. If EV?- = 2, what is fy? b.‘ Evaluate the joint pdffygfiv, r)‘. . . Evaluate the average power E(V2/R) dissipated in R4. Evaluatefm (v, i).for the current I through Re. Are V and I independent? ' ' ' What is the MMSE I‘m!) that estimates R from V? E1514 If ’9’ 2-bit?!“ ' fscx) = 2sU(s)U(1-— s),N.._LLS,N ~ £_(1),X 4—. s +N, Y = 52, then find the minimum mean square estimator 17' (X) of Y .i b. Evaluate the performance of this estimator” .' . " £315.15 A system transforms its input X ~ 8(a) into an output Y through ' . ' 2 - _ We wish to br'edict theoutput Y from the input X i _ an Evaluate the minimum meansquar'e error predictor 1711’)“ .b.‘ .. Evaluate the mean square error performance of this estimator. 1315.16 at. If' 1 -(y-~xzxz)2, - fissurin Ix: . x2) =. 7-52 with {X.-} it'd N‘(0; a2), thendeterrnine the least mean square estimator )9 (X1, X2) of Y ., . - ‘- . : - bi, Evaluate the performance of 1‘". (Hint: Consider subconditioning.) E1517 a. A noisy measurement Y =X +N of X ~ 190.1) arises from additive indepen-- - » dent noiseN' ~ Poo); Evaluate PU = 35).: ' ’ ' ' [1. Describe what we know about X after observing Y i. c.‘ Evaluate the least meansquare estimator J? (Y) of X given Y i. d.‘ Provide an expression for the performance of'1i(Y).. /\ Exercises 1315.18 13115.19 1315.20 E1521 E1522 435 . at It XJLY, X ~N(1,2.), r ~.N‘(—1, 1) and z =XY, then VAR(Z)H - bi. Evaluate COV(X,Z), - ‘ c1. Evaluate the linear least mean square estimator XL(Z) of X H d.‘ Evaluate the least mean square estimator 20:) 012 given X. (Caution: Note the changes from part (c)..) ‘ - ‘. _ '- _ - .- rr Y = mile, where N.lL{X,—},' {X.-} are stag; ~ mo, 1), and N ~ 190.), then evaluate E (YIN) and HY” ' a. Using the matrix operation of trace, express the error criterion E(X —— iFM (X -- ii) in terms of the positive definite weighting matrix M and the unweighted error covariance matrix ]P = E (X - It)? (X -—_ 12)., bh Simplify the expression when M is a diagonal matrix with diagonal elements {di}« - . ‘_ . Consider the emitter location problem of Exarhple 155, only now we have just a single sensor X1 at (0, 0)‘. Provide an expression for-the MMSE'estimator if of the emitter ' location Y. A dynamical system with state vector Xk at time 1;; is specified by ¢=randn{3.3), B=randn(3,3), evaluate EZ ' and with initial condition Xo satisfying ' ' nxo=o, C0V(X0,Xo)=iPo=]I and input process {Uk} satisfying _ ' o flj¢k EU}: : 0. COVtUj.Ur) =.= {E if]. =k I The observation process 215k _>__1, is specified by- 0211 if“ j = k . _ 0 ' if j :1; Once ‘seiect'ed; <1), B, IHI should-be treated as nonrandom, The initial condition X0, mea- 1111': ranan(2,3), EVk =0. COV(Vr.Vri) ={ ' surernent'noises,'_and input noises are also uncorrelated with each other: ' a What is the leastmean square estimator of _ bi. Using.__Matlab or any other computing environment, for k = 1 : 100, iteratively evaluate _ .._ Pr = coves “taxi -— it)“ c.,~ Using your computing environment, fer k = I : 100; iteratively evaluate 12,; (Zr, ---~- ’11-)" - , . '_ , . d. For each of the three compenents of the state vector, plot the true state and its ' Kalman estimate, bath as a function of the time index k” ' ...
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