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### HW9_Johnson_Jonaris

Course: ECE 534, Fall 2010
School: Ill. Chicago
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534: ECE Elements of Information Theory, Fall 2010 Homework 9 Name: Johnson Jonaris GadElkarim UIN 656 312 844 Problem 9.14 a) C = h(Y ) h(Y |X ) = h(Y ) h(Z ) But Z has a discrete component, hence h(Z) = -; Since h(Y) is , hence C = . b) We can transmit zero followed by the message: 0 m1 0 m2 ..., hence we will be able to detect the noise presence, however we have to follow the power constraint. P(noise-free)...

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534: ECE Elements of Information Theory, Fall 2010 Homework 9 Name: Johnson Jonaris GadElkarim UIN 656 312 844 Problem 9.14 a) C = h(Y ) h(Y |X ) = h(Y ) h(Z ) But Z has a discrete component, hence h(Z) = -; Since h(Y) is , hence C = . b) We can transmit zero followed by the message: 0 m1 0 m2 ..., hence we will be able to detect the noise presence, however we have to follow the power constraint. P(noise-free) = 1 - (0.9)n Problem 9.16 I (X ; Z ) = h(X ) + h(Z ) h(X, Z ) Since X, Y, Z are jointly Gaussian, hence X and Z are jointly Gaussian, their covariance matrix will be: K= Hence 2 x x z xz x z xz 2 z 2 2 I (X ; Z ) = 0.5 log(2ex ) + 0.5 log(2ez ) 0.5 log(2e|K |) 22 |K | = x z (1 2 ) xz I (X ; Y ) = 0.5 log(1 2 ) xz Now we need to compute xz , using markotivity p(x,zy) = p(xy)p(zy) we can get xz = E (xz |y ) E E (x|y )E (z |y ) E (xz ) = = x z x z x z Since X, Y and Z are jointly Gaussian: E (x|y ) = x Y , xy y we can do the same for E (z |y ) xz = xy zy I (X ; Z ) = 0.5 log(1 (xy zy )2 ) Problem 9.18 a) When i = 0 capacity will be C = 0.5 log(1 + P/N ) b) When i = ei , since its know at both the transmitter and the receiver, the receiver will be able to subtract it while decoding and then we will go back the the zero mean case like in (a), hence C = 0.5 log(1 + P/N ) c) When i N (0, N 1), the density distribution of Z: fZ (z ) = N (0, N ) N (0, N 1) = N (0, N + N 1); where * denotes P convolution, hence C = 0.5 log(1 + N +N 1 ) Problem 9.22 In the achievability proof, we pointed that the capacity is the max achievable rate, we dened an achievable rate to (n) be able to recover X n under the power constraint with a small Pe that tend to zero as n go to innity. If we know X n and since Z n = Y n X n , hence recovering Z n is equivalent to recover X n , hence the maximum rate of X n is 0.5 log(1 + P/N ). 1
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Ill. Chicago - ECE - 534
ECE 534: Elements of Information Theory, Fall 2010Homework 10Name: Johnson Jonaris GadElkarimUIN 656 312 844Problem 10.1From the lecture we reached that x( 1 ) = E [X |X &gt; 0]x( 1 ) =xf (x)dx =0The above by settingx22202x2 2x2e 2 2 dx =0
Ill. Chicago - ECE - 534
ECE 534: Elements of Information Theory, Fall 2010Homework 11Name: Johnson Jonaris GadElkarimUIN 656 312 844Problem 10.14a) Since X and Y are independent we have:p(x, y, x, y ) = p(x)p(y )p(, y |x, y )xHenceI (X, Y ; X, Y ) = H (X, Y ) H (X, Y, X
Ill. Chicago - ECE - 534
ECE 534: Elements of Information Theory, Fall 2010Homework 12Name: Johnson Jonaris GadElkarimUIN 656 312 844Problem 15.7(1)(1)nLet R1 and R2 be achievable rate pairs, hence we are sure of the existence of a (2nR1 , 2nR2 ), n) codes with Pe1 0(2)
Ill. Chicago - ECE - 534
ECE 534: Elements of Information Theory, Fall 2010Homework 13Name: Johnson Jonaris GadElkarimUIN 656 312 844Problem 15.20a) Exploring the possibilities for the output we can construct the following table:X12424X21122Y24416It is clear
Ill. Chicago - ECE - 534
ECE 534: Elements of Information Theory, Fall 2009Homework 2Johnson Jonaris GadElkarimUIN 656 312 844Problem 2.28p +pp +pLet pmf P1 (p1 , . . . , pi , . . . , pj , . . . , pm ) and pmf P2 (p1 , . . . , i 2 j , . . . , i 2 j , . . . , pm )H (P2 ) H
Ill. Chicago - ECE - 534
ECE 534: Elements of Information Theory, Fall 2010Name: Johnson Jonaris GadElkarimUIN 656 312 844Information theory and Neuroscience:Topic Covered: Neural coding and neural models Tractography fMRI Brain Network Analysis1. C. E. Shannon. A mathem
Ill. Chicago - ECE - 534
12Entropy, Relative Entropy and Mutual Informationsince t log t 0 for 0 t 1 , and is strictly p ositive for t not equal to 0 or 1.Therefore the conditional entropy H (Y |X ) is 0 if and only if Y is a function of X .6. Conditional mutual information v
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Cha pte r10Technology, Production,and CostsP re pa re d by:Fe rna ndo &amp; Yvonn Quija no 2008 Prentice Hall Business Publishing Economics R. Glenn Hubbard, Anthony Patrick OBrien, 2e.Sony Uses a CostCurve to Determine thePrice of RadiosLearning Ob
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Cha pte r6Elasticity: TheResponsiveness ofDemand and SupplyP re pa re d by:Fe rna ndo &amp; Yvonn Quija no 2008 Prentice Hall Business Publishing Economics R. Glenn Hubbard, Anthony Patrick OBrien, 2e.Do People Care about thePrices of Books?Learning
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W. Connecticut - ECON - 100-206
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W. Connecticut - ECON - 100-206
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W. Connecticut - ECON - 100-206
CHAPTER2Trade-offs,ComparativeAdvantage, andthe Market SystemTo compete in theautomobile market,the managers ofBMW must makemany strategicdecisions, such aswhether to introducenew car models.Prepared by:e pah CtFernando QuijanoCopyright
W. Connecticut - ECON - 100-206
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CHAPTER15Fiscal PolicyThe tax laws arecomplicatedbecause thegovernmentchanges themrepeatedly,trying to achievesometimesconflictingeconomic,social, andpolitical goals.Prepared by:e pah CtFernando QuijanoCopyright 2010 Pearson Education,
W. Connecticut - ECON - 100-206
CHAPTER10Long-Run EconomicGrowth: Sources andPoliciesGoogles problemshighlight one of theparadoxes of Chinain recent years:very rapid economicgrowth occurring inthe context ofgovernmentregulations that mayultimately stifle thatgrowth.Prepa
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W. Connecticut - ECON - 100-206
W. Connecticut - ECON - 100-206
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W. Connecticut - ECON - 100-206
W. Connecticut - ECON - 100-206
W. Connecticut - ECON - 100-206
CHAPTER1A Tour of the WorldPrepared by:Fernando Quijano and Yvonn Quijano 2003 Prentice Hall Business Publishing2003Macroeconomics, 3/eOlivier Blanchard1-1The United States, 2000The 2003 Prentice Hall Business Publishing2003Macroeconomics, 3
W. Connecticut - ECON - 100-206
W. Connecticut - ECON - 100-206
W. Connecticut - ECON - 100-206
W. Connecticut - ECON - 100-206
W. Connecticut - ECON - 100-206
W. Connecticut - ECON - 100-206
W. Connecticut - ECON - 100-206
W. Connecticut - ECON - 100-206
W. Connecticut - ECON - 100-206
Exam IV (Final): ECO 101 - Principles of MicroeconomicsStudent: _1. The labor demand curve of a firm:A. will shift to the left if the price of the product the labor is producing falls.B. is perfectly elastic if the firm is selling its product in a pur
W. Connecticut - ECON - 100-206
W. Connecticut - ECON - 100-206
Chapter 8Inflation: ItsCauses andCuresCopyright 2009 Pearson Addison-Wesley. All rights reserved.Definition: Inflation Inflation is a sustained upward movement in theaggregate price level that is shared by mostproducts. The price level is notated
W. Connecticut - ECON - 100-206
W. Connecticut - ECON - 100-206
W. Connecticut - ECON - 100-206
Chapter 4Monetary andFiscal Policy inthe IS-LM ModelCopyright 2009 Pearson Addison-Wesley. All rights reserved.The Definition of Money Money is defined as any good or asset that serves thefollowing three functions: Medium of Exchange Store of Val
W. Connecticut - ECON - 100-206
Chapter 5The GovernmentBudget, ForeignBorrowing, andthe Twin DeficitsCopyright 2009 Pearson Addison-Wesley. All rights reserved.Key Questions How does fiscal policy affect the budget surplus ordeficit? How is the government budget interact with n
W. Connecticut - ECON - 100-206