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Lecture+11+Capture - 1/26 EEL 5544 Lectures 11 EXAMPLES...

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Unformatted text preview: 1/26 EEL 5544 Lectures 11 EXAMPLES USING CONDITIONAL DENSITIES fl Ex Binary Communication System 1 A binary signal (0 or 1) is sent through a channel with additive white Gaussian noise. 2/26 EEL 5544 Lectures 11 EXAMPLES USING CONDITIONAL DENSITIES xi? Ex Binary Communication System 1 A binary signal (0 or 1) is sent through a channel with additive white Gaussian noise. The received signal is processed to form a decision statistic X, which can be used to estimate the transmitted signal. EEL 5544 Lectures 11 EXAMPLES USING CONDITIONAL DENSITIES xi? Ex Binary Communication System 1 A binary signal (0 or 1) is sent through a channel with additive white Gaussian noise. The received signal is processed to form a decision statistic X, which can be used to estimate the transmitted signal. X is a random variable, which can be specified as follows: EEL 5544 L11-1 3/26 4/26 {X N Gaussian(+1,02), 0 transmitted 5 [26 ()‘gkcmox 6w {X X N G N a G us a S. U la 33' n( 13 + n 1 (—1102) ’02), 0t 9 r 1 an tra 8m n it S t m. ed Itte d 6/26 X N Gaussian(+1,02), 0 transmitted X N Gaussian(—1,02), 1 transmitted The demodulation system generates decisions according to the following figure: generate decide 1 erasure decide () ‘l—l—l—l—l— l 0.25 0 0.25 H Thus, the demodulator provides one of three decisions at its output: {erasure, 0, 1}. 7/26 X m Gaussian(+1,02), 0 transmitted X N Gaussian(—1,02), 1 transmitted The demodulation system generates decisions according to the following figure: generate decide 1 erasure decide () ‘i—l—l—l—l— l 0.25 0 0.25 H Thus, the demodulator provides one of three decisions at its output: {erasure, 0, 1}. Note that in what follows, an erasure is not an error. EEL 5544 L11-2 8/26 a) Determine the probability that a symbol is erased if EEL 5544 L11-3 9/26 L11-4 COVSQL ‘ _. z .5— Q bah): 9(X7/21U‘0)- @(J‘Trffi == w—g?) :- Wei) NH P(Er¢;J€lZ3TX): 395— Q L_____,._\ whoch / T3; 61*6‘1g/ kaleQ) 1 Q0 €\ #Q (2 5—) EEL5544 L11-5 11/26 EEL 5544 L11—6 12/26 b) Determine the probability that a symbol is in error if 0220.25. [re-kg E '7 CVMJC Qua cccofiS an? 01'? _H M Chile“ (fig wen-=— Q L25)- (we): $2“ 5' EEL 5544 L1 1‘7 13/26 0) Determine the value 0@ required to achieve a probability of error of 10—”. Me? @032 i =* i6" kzs— :_ @400“ch EEL 5544 L11-8 14/26 «‘9 Ex Binary Communication System 2 Consider the following ogtimal detection problem: 15/26 Hie? Ex Binary Communication System 2 Consider the following optimal detection problem: A binary signal (0 or 1) is sent through a channel with additive white Gaussian noise. 16/26 Hie? Ex Binary Communication System 2 Consider the following optimal detection problem: A binary signal (0 or 1) is sent through a channel with additive white Gaussian noise. The received signal is processed to form a decision statistic X, which can be used to estimate the transmitted signal. Hie? Ex Binary Communication System 2 Consider the following optimal detection problem: A binary signal (0 or 1) is sent through a channel with additive white Gaussian noise. The received signal is processed to form a decision statistic X, which can be used to estimate the transmitted signal. X is a random variable, which can be specified as follows: EEL 5544 L11-9 17/26 18/26 {X m GaUSSian(p0,02), 0 transmitted 19/26 X N GaussiaanpQ), 0 transmitted X m Gaussian(p,1,02), 1transmitted 20/26 .w X m Gaussianmomg), 0 transmitted 'WBJ . 9 . \‘ I X N Gaussuan(p,1,a~), 1transmltted QM / '7 Find the MAP and ML decision 4' . _ : ‘34 Q (’3, Tx 1%) EEL 5544 L11-10 21/26 2:1?5' O .. L 74 J“ Qx€(_(qfl' P1 1 267* M N? .‘ <:s:———‘—~—+ 7 VDIAE Curve ‘Tflw 06¢ 3,6 ’3, “:7 gee 36C) EEL 5544 L11-11 22/26 MM} MIA/V" “Lle— 1 chunk no\n<c)\evéx NM {"7 olbfiesgr EEL 5544 L11-12 23/26 m $86 ’3. 5 (96639 0 /MW anflw My! Mt; EEL 5544 L11-13 ...
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Lecture+11+Capture - 1/26 EEL 5544 Lectures 11 EXAMPLES...

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