EE4512 Analog and Digital Communications Chapter 4Chapter 4Chapter 4Receiver DesignReceiver DesignEE4512 Analog and Digital Communications Chapter 4Chapter 4Chapter 4Receiver DesignReceiver Design••Probability of Bit ErrorProbability of Bit Error••Pages 124Pages 124--149149EE4512 Analog and Digital Communications Chapter 4••Probability of Bit ErrorProbability of Bit ErrorThe low pass filtered and sampled PAM signal results in The low pass filtered and sampled PAM signal results in an expression for the an expression for the probability of bit errorprobability of bit errorPPb b (S&M p. (S&M p. 124124--127). 127). AAis the amplitude at the sampling point and is the amplitude at the sampling point and γγis the attenuation of the channel (0 is the attenuation of the channel (0 ≤≤γγ≤≤11))P{ P{ ithithbit in error } = P(bbit in error } = P(bii= 0) P{ n= 0) P{ noo[ (i[ (i--1)1)TTbb+ + TTbb/2 ] /2 ] < < ––γγA }A }+ + P(bP(bi i = 1) P{ n= 1) P{ noo[ (i[ (i--1)1)TTbb+ + TTbb/2 ] /2 ] ≥≥γγA }A }EE4512 Analog and Digital Communications Chapter 4••Review of Probability and Stochastic ProcessesReview of Probability and Stochastic Processes(S&M p. 127(S&M p. 127--132)132)Probability distribution functionProbability distribution functionFFXX(a) = P{ (a) = P{ XX= a }= a }Probability density functionProbability density functionffXX(x) = d F(x) = d FXX(x) / (x) / dxdxMean (or expected value)Mean (or expected value)µµX X = = ∫∫x x ffXX(x)(x)dxdx∞∞VarianceVarianceσσXX22==∫∫(x (x ––µµXX)ffXX(x)(x)dxdx––∞∞EE{ ({ (X X ––µµXX)2 2 }}EE4512 Analog and Digital Communications Chapter 4••Review of Probability and Stochastic ProcessesReview of Probability and Stochastic Processes(S&M p. 127(S&M p. 127--132)132)Joint probabilityJoint probabilitydistribution functiondistribution functionFFX,YX,Y(a(a, b) = P{ , b) = P{ XX= a and = a and YY= b }= b }Joint probabilityJoint probabilitydensity functiondensity functionffX,YX,Y(x, y)) = (x, y)) = ∂∂22FFX,YX,Y(x, y) / (x, y) / ∂∂xx∂∂yyConditional probabilitiesConditional probabilitiesP { P { XX> a and event Z } => a and event Z } =P { event Z } P{ X > a | event Z }P { event Z } P{ X > a | event Z }EE4512 Analog and Digital Communications Chapter 4Chapter 4Chapter 4Receiver DesignReceiver Design••Examining Thermal NoiseExamining Thermal Noise••Pages 132Pages 132--136136EE4512 Analog and Digital Communications Chapter 4••JohnsonJohnson––Nyquist noiseNyquist noiseor or thermal noisethermal noiseis the is the electronicelectronicnoisenoisegenerated by the thermal agitation of the charge generated by the thermal agitation of the charge carriers (usually the carriers (usually the electronselectrons) inside an ) inside an electrical electrical conductorconductorat equilibriumat equilibrium.This thermal noise was first measured by This thermal noise was first measured by John B. JohnsonJohn B. Johnsonat at Bell LabsBell Labsin in 19281928. He described his findings to . He described his findings to Harry Harry NyquistNyquist, also at Bell Labs, who was able to explain the , also at Bell Labs, who was able to explain the results. results. 1984-1995Harry Nyquist Harry Nyquist 18891889--19761976EE4512 Analog and Digital Communications Chapter 4••Thermal (or Gaussian) noise is approximately Thermal (or Gaussian) noise is approximately whitewhite,,meaning that the meaning that the power spectral densitypower spectral densityis equal is equal throughout the throughout the frequency spectrumfrequency spectrum. Additionally, the . Additionally, the amplitude of the signal has very nearly a amplitude of the signal has very nearly a GaussianGaussianprobability density functionprobability density functionwith mean with mean µµnn= 0.= 0.S&M Figure 4S&M Figure 4--33µµnn= 0 = 0 σσnn= 1= 1EE4512 Analog and Digital Communications Chapter 4••Since thermal noise has a Since thermal noise has a GaussianGaussianprobability density probability density functionfunctionthe probability that a noise voltage the probability that a noise voltage n(tn(t) at time t) at time toowill be will be less than or equal to a thresholdless than or equal to a threshold––γγA is (S&M Eq. A is (S&M Eq. 4.27):4.27):and the probability that and the probability that aanoise voltage noise voltage n(tn(t) at time t) at time to o will be will be greater than a threshold greater than a threshold γγAAis (S&M Eq. 4.28):is (S&M Eq. 4.28):)(−−∞≤− −∫γA2noX2nn(x -µ )1P{ n(t γA } = F γA) = exp dx2σ2π σ)(∞>−∫2noX2nγAn(x -µ )1P{ n(t γA } = 1 F γA) = exp dx2σ2π σEE4512 Analog and Digital Communications Chapter 4••The probabilistic properties ofThe probabilistic properties ofthermal noise do not change withthermal noise do not change withtime (time (stationaritystationarity). Thermal noise). Thermal noiseis an is an insidious propertyinsidious propertyofofcommunication systems that limitscommunication systems that limitsthe speed of reliable data transmissionthe speed of reliable data transmissionand the detection of weak signals. and the detection of weak signals.EE4512 Analog and Digital Communications Chapter 4••A A MMATLABATLABand and Simulink Simulink simulation verifies the spectral simulation verifies the spectral characteristics of thermal noise and the performance of characteristics of thermal noise and the performance of lowlow--pass filtered pass filtered ffcutoff cutoff = 11.25 kHz thermal noise. = 11.25 kHz thermal noise. MS Figure 1.11MS Figure 1.11EE4512 Analog and Digital Communications Chapter 4••Thermal noise PSD = Thermal noise PSD = NNoo, | f | , | f | →→∞∞MS Figure 1.12MS Figure 1.12•Thermal noise LPF PSD = Thermal noise LPF PSD = NNoo, | f | < 11.25 kHz, | f | < 11.25 kHzNNooNoNNoo(single(single--sided spectrum)sided spectrum)11.25 kHz11.25 kHzNoEE4512 Analog and Digital Communications Chapter 4••Thermal noise PSD = Thermal noise PSD = NNoo, | f | , | f | →→∞∞MS Figure 1.12 MS Figure 1.12 •Thermal noise autocorrelation MS Figure 1.14Thermal noise autocorrelation MS Figure 1.14uncorrelateduncorrelatedEE4512 Analog and Digital Communications Chapter 4••Thermal noise LPF PSD = Thermal noise LPF PSD = NNoo, | f | < 11.25