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The Hong Kong University of Science and Technology**We aren't endorsed by this school

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ELEC 5360

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Electrical Engineering

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Dec 18, 2024

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Uploaded by DrWhaleMaster1226

ELEC5360 2024 Fall Assignment 1Prof. Hengtao HeDept. of Electronic and Computer EngineeringThe Hong Kong University of Science and TechnologySeptember 29, 2024Problem 11. Use the Chernoff bound to prove thatQ(x)≤e−x2/2, x >02. Apply integration by parts toR∞xey2/2y2dyand show thatx(1 +x2)1√2πe−x2/2< Q(x)<1x√2πe−x2/23. Based on the results of 2. show that, for largex,Q(x)≈1x√2πe−x2/2Problem 2LetX1, X2, . . . , Xn, . . .be a sequence of independent identically distributed (i.i.d.) contin-uous random variales (rvs) with the common probability density functionfX(x). Note thatPr{Xn=α}= 0 for allαand that Pr{Xn=Xm}= 0 form̸=n.1. Find Pr{X1≤X2}.2. Find Pr{X1≤X2X1≤X3}.(In other words, find the probability thatX1is thesmallest of{X1, X2, X3}.)3. Let the rvNbe the index of the first rv in the sequence to be less thanX1; that is,Pr{N=n}= Pr{X1≤X2X1≤X3;. . .;X1≤Xn−1;X1> Xn}. Find Pr{N≥n}asa function ofn.4. For a non-negative integer-valued rvY, show thatE[Y] =∑y>0Pr(Y≥y).5. Show thatE[N] =∞.1

Problem 3A random process is defined asX(t) =Xcos(2πf0t) +Ysin(2πf0t),(1)whereXandYare independent Gaussian random variables with zero mean and varianceσ2.1. Determine the mean function of the random processX(t).2. Determine the autocorrelation function ofX(t) and show whetherX(t) is a wide-sensestationary (WSS) stochastic process.3. Determine the power spectral density (PSD) ofX(t).Problem 4Consider the following four signals:Problem 4 A random process is defined as , where is a positive constant. Random variables obeys a Rayleigh distribution: where, and . and are independent with each other.a)Find the mean and variance of this random process. b)Find its autocorrelation function. c)Find the variance of this random process. d)Is the process wide-sense stationary? Problem 5 Consider the following four signals: ( )()cos,xbj=+- ¥<<¥Xtttbx( )2222,00, otherwisexss->=ìïíïîxxfxxe0s>(0,2 )jpUxjtS1(t)3tS4(t)13121tS3(t)13tS2(t)21-111. What is the dimension of the signal space?2. Find an orthonormal basis for representing the four signals.3. Use the orthonormal basis to represent the four waveforms by vectorsS1,S2,S3, andS4.4. Plot the constellation of the four signals. Using the constellation, find the minimumdistances between any pair of vectors.5. Determine the energy in each signal and compute the average signal energy.6. Compute the inner products between any pair of signals.2

Problem 5Consider a baseband digital communication system, which employs the following two wave-forms for transmission. The channel has no attenuation and the noise is AWGN with double-sided power spectral densityN0/2.d)Plot the constellation of the four signals. Using the constellation, find the minimum distances between any pair of vectors. e)Determine the energy in each signal and compute the average signal energy. f)Compute the inner products between any pair of signals. Problem 6 Consider a baseband digital communication system, which employs the following two waveforms for transmission. The channel has no attenuation and the noise is AWGN with double-sided power spectral density N0/2. (a)Find an appropriate orthonormal basis for the representation of the signals. (b)Design a matched filter for the two signals and give the general block diagram for the matched filter implementation of the optimum receiver. (c)Sketch the waveform at the output of the matched filter. (d)Now assume that a correlation detector is used instead, give the general block diagram for the correlator receiver implementation of the optimum receiver and sketch the waveform at the output of the correlation detector. (e)Compute the Probability of error for this signaling scheme assuming the optimum receiver is employed. (f)Assume that s1(t) and s2(t) are equiprobable, draw the decision regions. 19 February 2022K. B. LETAIEF 0AT/2tS1(t)TAT/2tS2(t)01. Find an appropriate orthonormal basis for the representation of the signals.2. Design a matched filter for the two signals and give the general block diagram for thematched filter implementation of the optimum receiver.3. Sketch the waveform at the output of the matched filter.4. Now assume that a correlation detector is used instead, give the general block diagramfor the correlator receiver implementation of the optimum receiver and sketch thewaveform at the output of the correlation detector.5. Compute the Probability of error for this signaling scheme assuming the optimumreceiver is employed.6. Assume thatS1(t) andS2(t) are equiprobable, draw the decision regions.3