ECE 4331, Fall, 2009Receiver StructureMatched FilterMatched filter exampleSlide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Timing ExtractionExampleTiming/Synchronization Block DiagramTiming JitterSlide 16Slide 17Slide 19Slide 20Error function and its complementSlide 22Slide 23Slide 24Slide 25Slide 26Bit error rate with error function complementBit error rate for unipolar and antipodal transmissionSlide 29Slide 30Slide 31ECE 4331, Fall, 2009Zhu HanDepartment of Electrical and Computer EngineeringClass 13Oct. 6th, 2009Receiver StructureReceiver StructureMatched filter: match source impulse and maximize SNR–gRx to maximize the SNR at the sampling time/outputEqualizer: remove ISITiming –When to sample. Eye diagramDecision –d(i) is 0 or 1d(i)gTx(t)Noise na(t)?)()()(0iTniTriTr maxNSTi gRx(t)Matched FilterMatched FilterInput signal s(t)+n(t)Maximize the sampled SNR=s(T0)/n(T0) at time T0Matched filter exampleMatched filter exampleReceived SNR is maximized at time T00Texample:transmit filterReceive filter (mathed filter)t)()(0tgtTgRxTx0Tt)( tgTx0Tt)(tgTxMatched Filter: optimal receive filter for maximizedNSMatched Filter  (4.7) )(2 )()( (4.6) )(2)( is n(t) of )( PSD the, whiteis w(t)Since(4.5) )2exp()()()((4.4) )2exp()()()()( and )( of ransforms Fourier t thedenote )( and )(Let eperformanc optimal obtain to maximaze tohave Wepower signaloutput ousinstantane theis )( where (4.3) )()( as ratio noise-to-signal pulsepeak theDefine20220220222dffHNdffStnEfHNfSfSdf fTjfGfHTgdf ftjfGfHtgthtgfHfGTgtnETgNNNoooMatched Filter(4.10) )()( iff holds (4.9) inequality The )( )( . variablereal in functionscomplex are )( and )( where(4.9) )()()()(inequality sSchwarz' theRecallmaximum.a makes that )( find ),( Given(4.8) )(2)2exp()()(*21-22-212122212-21202xkxdxxdxxxxxdxxdxxdxxxfHfGdffHNdf fTjfGfH Matched FilterSince g(t)Matched Filter  ratio) PSDnoise energy to signal( waveformoft independen is which(4.20) 2)2()( (4.19) 2 )(2 )()( ispower noiseoutput average the(4.14) and (4.7) From(4.18) )( )( )2exp()()( (4.17) )2exp()( )2exp()()( )()()( , )( signal knowna Consider 0002max02202-20 orem)energy the sh'by Rayleig ( energy siganl2002*opt0NENEEkNkEENkdffGNk dffStnEkETgdffGkdf fTjfGTg fTjfGk fTjfGfkGfGfHfGtgNEProperties of Matched FiltersTiming ExtractionTiming ExtractionReceived digital signal needs to be sampled at precise instants. Otherwise, the SNR reduced. The reason, eye diagramThree general methods–Derivation from a primary or a secondary standard. GPS, atomic clockTower of base stationBackbone of Internet–Transmitting a separate synchronizing signal, (pilot clock, beacon)Satellite –Self-synchronization, where the timing information is extracted from the received signal itselfWirelessCable, FiberExampleExampleSelf Clocking, RZContain some clocking information. PLLTiming/Synchronization Block DiagramTiming/Synchronization Block DiagramAfter equalizer, rectifier and clipperTiming extractor to get the edge and then amplifierTrain the phase shifter which is usually PLLLimiter gets the square wave of the signalPulse generator gets the impulse responsesTiming JitterTiming JitterRandom forms of jitter: noise, interferences, and mistuning of the clock circuits.Pattern-dependent jitter results from clock mistuning and, amplitude-to-phase conversion in the clock circuit, and ISI, which alters the position of the peaks of the input signal according to the pattern.Pattern-dependent jitter propagates Jitter reduction–Anti-jitter circuits –Jitter buffers –DejitterizerError Rate Due to the Noise2YError Rate Due to the Noise(4.27) )(exp1)0|(is of pdf lconditiona The(4.26) 2 )(21(4.25) )(2),( 2 PSD withnoise whiteis )( Since02000 002200 bbYbT TbYW/TNAy/TNyfYTN dt duutNTσutNutR/Ntwb bError Rate Due to the NoiseFigure 4.5 Figure 4.5 Noise analysis of PCM system. (Noise analysis of PCM system. (aa) Probability density function of random variable ) Probability density function of random variable YY at matched filter output when 0 is transmitted. (at matched filter output when 0 is transmitted. (bb) Probability density function of ) Probability density function of YY when 1 is when 1 is transmitted.transmitted.Error Rate Due to the NoiseEE 541/451 Fall 2007Error Rate Due to the NoiseEE 541/451 Fall 2007Error function and its complementError function and its complementfunction y = Q(x); y = 0.5*erfc(x/sqrt(2)); Only scale change in x and y-3 -2 -1 0 1 2 3-1.5-1-0.500.511.522.5xerf(x), erfc(x)erf(x) erfc(x)EE 541/451 Fall 200700)(210)exp(1/TN/λAdzzPEE 541/451 Fall 2007EE 541/451 Fall 2007Figure 4.6 Figure 4.6 Probability of error in a PCM receiver.Probability of error in a PCM receiver.PCM receiver exhibits an exponential improvement in with increase in eP0/NEbBit error rate with error function complementBit error rate with error function complementExpressions with andSE0Nantipodal:unipolar1d0d;ddNbP22erfc210d1d1d0d;d00erfc21NEPSb01erfc2 2SbEPN� �� �=� �� �1 0221 1erfc erfc2 22 2 21 1 SNRerfc erfc2 2 2 2bN NNd d dPds ss� � � �-= =� � � �� � � �� � � �� �� �= =� �� �� �� �� �� �22matched0SNR/ 2SNEdNs= =22matched0/ 2SNR/ 2SNEdNs= =22221 1erfc erfc2 2 82 21 / 2 1 SNRerfc erfc2 4 2 4bNNNd dPdsss� �� �= =� �� �� �� �� �� �� �� �= =� �� �� �� �� �� �12202201 2 11 e d22NbNdxdxPsp s--� �� �= -� �� �� ��Q functionBit error rate for unipolar and antipodal transmissionBit error rate