ECE 4331, Fall, 2009MidtermReceiver StructureMatched filter exampleSlide 5Slide 6Bit error rate with error function complementBit error rate for unipolar and antipodal transmissionBaseband binary data transmission system.Slide 10Slide 11ISIISI ExampleSlide 141st Nyquist Criterion: Time domainSlide 161st Nyquist Criterion: Frequency domainProofSlide 19Sample rate vs. bandwidthSlide 21Slide 22Cosine rolloff/Raised cosine filterRaised cosine shapingSlide 25Cosine rolloff filter: Bandwidth efficiency2nd Nyquist CriterionExample3rd Nyquist CriterionCosine rolloff filter: Eye patternECE 4331, Fall, 2009Zhu HanDepartment of Electrical and Computer EngineeringClass 14Oct. 13th, 2009MidtermMidtermDistribution–Mean 79–Variance 8.950 60 70 80 90 100 11000.511.522.533.54Receiver 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 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 maximizedNSError 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 NoiseBit 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 for unipolar and antipodal transmissionBER vs. SNRCoherent (antipodal) and noncoherent (unipolar) detection -2 0 2 4 6 8 1010-410-310-210-1dBinNE0SBERtheoreticalsimulationunipolarantipodalBaseband binary data transmission system.Baseband binary data transmission system.ISI arises when the channel is dispersive Frequency limited -> time unlimited -> ISITime limited -> bandwidth unlimited -> bandpass channel -> time unlimited -> ISIp(t)(Polar form)TX Filter Channel RX FilterISIISIFirst term : contribution of the i-th transmitted bit.Second term : ISI – residual effect of all other transmitted bits.We wish to design transmit and receiver filters to minimize the ISI.When the signal-to-noise ratio is high, as is the case in a telephone system, the operation of the system is largely limited by ISI rather than noise.ISI ExampleISI Example5T0tSequence of three pulses (1, 0, 1)sent at a rate 1/T sequence sent 1 0 1 sequence received 1 1(!) 1Signal receivedThreshold 4T3T2T T 0-T-2T-3TISIISINyquist three criteria–Pulse amplitudes can be detected correctly despite pulse spreading or overlapping, if there is no ISI at the decision-making instants 1: At sampling points, no ISI2: At threshold, no ISI3: Areas within symbol period is zero, then no ISI–At least 14 points in the finals4 point for questions10 point like the homework1st Nyquist Criterion: Time domain1st Nyquist Criterion: Time domainp(t): impulse response of a transmission system (infinite length)Equally spaced zeros, intervalTfn21TfN2102t0tt01p(t)-1 shaping functionno ISI !1st Nyquist Criterion: Time domain1st Nyquist Criterion: Time domainSuppose 1/T is the sample rateThe necessary and sufficient condition for p(t) to satisfy   0,00,1nnnTpIs that its Fourier transform P(f) satisfy TTmfPm1st Nyquist Criterion: Frequency domain1st Nyquist Criterion: Frequency domain2a Nf f= 4Nff0(limited bandwidth) TTmfPmProofProof                TTTTmmTTmTmTmdffnTjfBdffnTjTmfPdffnTjTmfPdffnTjfPnTp2121212121212122122exp2exp2exp2exp      dfftjfPtp2exp      dffnTjfPnTp2expAt t=T   mTmfPfBFourier TransformProofProof   nnnfTjbfB2exp   TTnnfTjfBTb21212exp   mTmfPfB nTTpbn  000nnTbn TfB  TTmfPmSample rate vs. bandwidthSample rate vs. bandwidthW is the bandwidth of P(f)When 1/T > 2W, no function to satisfy Nyquist condition.P(f)Sample rate vs. bandwidthSample rate vs. bandwidthWhen 1/T = 2W, rectangular function satisfy Nyquist condition      ,,0,sincsinotherwiseWfTfPTttTttp0 1 2 3 4 5 6-0.4-0.200.20.40.60.81Subcarrier Number kSpectraSample rate vs. bandwidthSample rate vs. bandwidthWhen 1/T < 2W, numbers of choices to satisfy Nyquist condition A typical one is the raised cosine functionCosine rolloff/Cosine rolloff/Raised cosineRaised cosine filter filterSlightly notation different from the book. But it is the same20)2(1)cos()sin()(TtTtTtTtrctprrr: rolloff factor 10 r)1()1(2121rfrTTTrf21)1( )1(21rfT)2(0fjPrc10ifrr ))1(cos(1221TfRaised cosine shapingRaised cosine shapingWW ωP(ω)r=0r = 0.25r = 0.50r = 0.75r = 1.00 Wπ0t0p(t)Wπ2wTradeoff: higher r, higher bandwidth, but