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# Statistical Communicat Theory ECE 630

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This 6 page Class Notes was uploaded by Antonina Wuckert on Monday September 28, 2015. The Class Notes belongs to ECE 630 at George Mason University taught by Bernd-Peter Paris in Fall. Since its upload, it has received 20 views. For similar materials see /class/215016/ece-630-george-mason-university in ELECTRICAL AND COMPUTER ENGINEERING at George Mason University.

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Date Created: 09/28/15

ECE 630 Statistical Communication Theory Prof BP Paris Practice Problems 1 Amplitude Variations ln realistic communication systems the amplitude of signals in a signal set used for digital communication may be known by the transmitter but are usually scaled by an unknown amount prior to reception Per formance can become poor when the received signals are small and are contaminated by additive white Gaussian noise Assume that a binary orthogonal signal set having equally likely equal energy components is chosen for this channel The received signal is of the form RAsitNt 091 212 where E a Find the minimum probability of error receiver for this signal set b A clever engineer wants to estimate the unknown amplitude A by evaluating A T A Ttgt dt 0 when the signal 51t is present What single choice for gt both 0 minimizes the percentage error of the estimate as expressed by the coefficient of variation de ned as the ratio of the standard deviation of the estimate to its mean and 0 results in an unbiased estimate ie A A O V The engineer decides that the receiver should shut off77 when the estimate of A suggests that errors are occurring too frequently What threshold value for A should be used in an attempt to guarantee that the error rate is less than 10 Assume ENO equals unity in this part to Jamming and Football For those who remember the days of the bandwagon the secret to the Washington Redskins success in the 199192 season was the installa tion of a new digital communication system for relaying messages from the press box to the eld A former ECE 630 student designed the following binary signal set 50t Mcos wfot 51t 7Mcos27rfot The duration of each transmission interval is 1f0 and the frequency is 1 0 is 1 MHZ The communication channel is modeled as an additive white Gaussian noise channel with spectral amplitude 1 Assume the signals are equally likely a What is the minimum probability of error that any receiver can achieve when this presumably well designed communication sys tem is used b When visiting RFK stadium the Houston Oilers decided to jam this system and nearly managed to beat the Redskins I know this was a long time ago but you may recall that they missed the game winning eld goal with time running out What they did is to transmit a constant amplitude cosine wave of frequency f0 In the presence of this jammer the received signal can be modeled as Rt 525 mcosmfot Nt The noise has the same characteristics as described above What is the probability of error when the receiver from part a is used c In this game the famous Redskins7 half time adjustments in cluded a redesign of the receiver for the communication system Find the optimum receiver for communication in the presence of the jamming signal and the corresponding probability of error d The then lowly Dallas Cowboys on their visit to RFK stadium used a smarter jammer They transmitted a cosine wave whose amplitude alternates between km and 7m This jamming signal has a xed amplitude over each signalling interval and the amplitude is equally likely to be positive or negative Assuming the sign of the jamming signal during a particular transmission interval is statistically independent of the sign in any other in terval and the noise characteristic is as before explain why the Cowboys brought the Redskins winning streak to an end despite all efforts to adjust at halftime 3 Diversity Channels Diversity signalling is the transmission of the same message over N distinct channels simultaneously to a receiver If the statistical char acteristics of each channel are independent of each other potentially an improvement in performance can be obtained F Assume that an on off signalling scheme is used over each of N Rayleigh channels The received signals are of the form H0 332N517 ogth 2 1N H1 RA mtcos27rft0Ntlt gt ogth i1N The amplitude A are statistically independent Rayleigh random vari ables having variance 02 and the phases 0 are statistically independent uniform random variables distributed over 7713 7139 The additive noise N51 on channel i is white Gaussian noise of spectral height N02 the noise in one channel is independent of the noise in other channels The energy of each of the signals milt is EN a Find the optimum receiver which uses the output of only one channel b Find the optimum receiver which uses the output of all channels c Find an expression for the probability of error when the hypothe ses are equally likely for each receiver Warning This problem is very hard d Compare the results for the rst receiver N 1 and the second receiver N gt 1 Does diversity signalling result in improved performance Neural Networks as Receivers Arti cial Neural Networks are being explored in a variety of problem In this problem we will demonstrate that a neural network can be used as a vector receiver in communication applications An important feature of neural networks is their learning ability which we do not consider here areas In a vector communication problem the vector receiver observes a vector X of N random variables and tries to determine which of several hypotheses H X is most likely to have produced this observation The fundamental element of arti cial neural networks is the neuron typically depicted as U A neuron receives N inputs X17 XN and yields the single output Y by computing N Y sign 2 uan 7 o n1 where wn is a set of weights and 9 is a threshold The output of this neuron is bi valued 1 or 1 and can thus be used to indicate which of two signal vectors7 so or l is present a Find the optimum set of weights and threshold used by a neuron to distinguish between equally likely signals7 0 and 1 when they are presented to the neuron in additive Gaussian noise with uncorrelated components One problem with neural networks is unknown signal amplitudes they can be sensitive to scaling Under what conditions will the single neuron be insensitive to the size of the signal How can an optimum receiver for ternary signal sets be con structed from neurons Hint You will have to interconnect neu rons into layers The rst layer is responsible for distinguishing between each possible pair of hypotheses and the second layer combines the results from the rst layer The following signal set is used to transmit three equally likely sym bols 8005 8105 5205 2 2 2 1 1 1 1 2 t 1 2 t 1 2 t a 7 The channel adds white Gaussian noise of spectral height to the transmitted signal a Draw and accurately label a block diagram of the optimum re ceiver7 ie7 the receiver that minimizes the probability of a wrong decision b Find the appropriate signal space and indicate the decision re gions of the optimum receiver c Compute the minimum probability of error attainable with this signal set d Repeat part b under the assumption that the amplitude of sig nal 51t is increased to e Find another signal set with 50t Acos27rft that achieves the same performance as the signal set sketched above Mary Signal Sets The following signal set is used to transmit equally likely messages over an additive white Gaussian noise channel with spectral height 2E 5M 1Ticos27rfctjsin27rfct for 0 g t T7 Lj 71071 Thus7 this signal set consists of M 9 signals a Draw and accurately label the signal constellation in an appro priately chosen signal space and indicate the decision boundaries formed by the optimum receiver b Compute the probability of error achieved by the optimum re ceiver c Assume that the energy of the transmitted signal can never ex ceed 2E ls it possible to modify the above signal set in such a way that the probability of error is reduced without exceeding the limit on the signal energy Explain why or why not Mary Signal Sets The following signal set is used to transmit equally likely messages over an additive white Gaussian noise channel with spectral height QE 4 20711 sil T21cos T Thus7 this signal set consists of M 3 signals forOgthil7101 a Draw and accurately label a block diagram for the optimum re ceiver for this signal set b Draw and accurately label the signal constellation in an appro priately chosen signal space and indicate the decision boundaries formed by the optimum receiver Then compute the probability of error achieved by the optimum receiver c Repeat part b for the following signal set 8h i1COSlt gt 2 2 cos for 0 g t g T 2M2 7101 d Repeat part b for the following signal set 5mm V i1 008 i2cos i3 cos for 0 g t g T i1i2i3 710 e Derive a general expression for the probability of error of the N dimensional signal set N siIWJN Z EincosW forOgth in7101 71 T T 8 Mary Signal Sets The following signal set is used to transmit equally likely messages over an additive white Gaussian noise channel with spectral height SHOE coswt j sinwt for 0 g t g T i 7101andj 7271012 Thus this signal set consists of M 15 signals a Draw and accurately label the signal constellation in an appro priately chosen signal space and indicate the decision boundaries formed by the optimum receiver b Compute the probability of error achieved by the optimum re ceiver

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