9Determine the correlator outputs at the sampling instants. Example 5.1: suppose the signal waveforms s0(t) and s1(t) are the ones shown in figure 5.2, and let s0(t) be the transmitted signal. Then, the received signal isAtA-Figure 5.2: Signal waveforms s0(t) and s1(t) for a binary communication systemDetermine the correlator outputs at the sampling instants.Answerip_05_01
10Figure 5.3 illustrates the two noise-free correlator outputs in the interval for each of the two cases-I.e., when s0(t) is transmitted and when s1(t) is transmitted.tOutput of correlator 0tOutput of correlator 1tOutput of correlator 1tOutput of correlator 0(b)(a)Figure 5.3:Noise-free correlator outputs.(a) s0(t) was transmitted.(b) s1(t) was transmitted.
11r Probability density function p(r0|0) and p(r1|0) rProbability density function p(r0|0) and p(r1|0)when s0(t) is transmitted
12Matched Filter Provides an alternative to the signal correlator for demodulating the received signal r(t).A filter that is matched to the signal waveform s(t)has an impulse response;
13The matched filter output at the sampling instant t=Tb is identical to the output of the signal correlator.
14Example 5.2:Consider the use of matched filters for the demodulation of the two signal waveforms shown in the figure and determine the outputsAtA-Answerip_05_02
15tAtAA-Figure 5.5:Impulse responses of matched filters for signals s0(t) and s1(t).(a)(b)Figure 5.6:Signal outputs of matched filters when s0(t) is transmitted
16The DetectorThe detector observes the correlator or the matched filter outputr0 and r1 and decided on whether the transmitted signal waveformis s1(t) or s0(t), which corresponding to “1” or “0”, respectively.The optimum detector is defined the detector that minimizesthe probability of error.
17Example 5.3:Let us consider the detector for the signals shown in Figure 5.2 which are equally probable and have equal energies. The optimum detector for these signals compares r0 and r1 and decides that a 0 was transmitted when r0>r1 and that a 1 was transmitted when r0>r1 . Determine the probability of error.AAttA-Answerip_05_03
18Monte Carlo Simulation Communication System Monte Carlo computer simulations are usually performed inpractice to estimate the probability of error of a digitalcommunication system, especially in cases where the analysisof the detector performance is difficult to perform.
19Example 5.4: use Monte Carlo simulation to estimate an plot Pe versus SNR for a binary communication system that employs correlators or matched filters. The model of the system is illustrated in figure 5.8.Uniform random number generatorBinary data sourcedetectorOutput dataCompareError counterGaussian random number generatorFigure 5.8: Simulation model for IllustrativeAnswerip_05_04
22Antipodal signal If one signal waveform is negative of the other. Antipodal Signal for Binary Signal TransmissionAntipodal signal If one signal waveformis negative of the other.
23AtA-(a) A pair of antipodal signalAtA-(b) Another pair of antipodal signal
24The received signal isMatched filter demodulatorCorrelator demodulator
25probability density function for the input to the detector rprobability density function for the input to the detector
26For antipodal signal we have : The DetectorThe detector observes the correlator or the matched filter outputr0 and r1 and decided on whether the transmitted signal waveformis s1(t) or s0(t), which corresponding to “1” or “0”, respectively.The optimum detector is defined the detector that minimizesthe probability of error.For antipodal signal we have :
27Example 5.5: use Monte Carlo simulation to estime and plot the error probability performance of binary communication system. The model of the system is illustrated in Figure 5.13.Uniform randomnumber generatorBinary datasourceCompareError counterdetectorrnGaussian randomOutput dataFigure 5.13: Model of binary communication system employing antipodal signalAnswerip_05_05
28On-Off Signal for Binary Signal Transmission Binary information sequence may also betransmitted by use of ON-OFF signalsThe received signal is:
29rFigure 5.15: The probability density function for the received signal at the output of te correlator for on-off signal.
30Probability density function for ON-OFF signals rProbability density function for ON-OFF signals
31For antipodal signal we have : For On-OFF signal we have : The DetectorFor antipodal signal we have :For On-OFF signal we have :
32Example 5.6:use Monte Carlo simulation to estimate and plot the performance of a binary communication system employing on-off signalingUniform randomnumber generatorBinary datasourceCompareError counterdetectorrnGaussian randomOutput dataAnswerip_05_06
33Signal Constellation diagrams for Binary Signals(a)(b)(b)Figure 5.17: signal point constellation for binary signal.(a) Antipodal signal.(b) On-off signals.(c) Orthogonal signals.
34Example 5.7:The effect of noise on the performance of a binary communication system can be observed from the received signal plus noise at the input to the detector. For example, let us consider binary orthogonal signals, for which the input to the detector consists of the pair of random variables (r0,r1), where either.The noise random variables n0 and n1 re zero-mean, independent Gaussian random variables with variance .as in Illustrative Problam 5.4 use Monte Carlo simulation to generate 100 samples of (r0,r1) for each value of =0.1, =0.3, and =0.5, and plot these 100 samples for each on different two-dimensional plots. The energy E of the signal may by normalized to unity.Answerip_05_07
35Receiver signal points at input to the selector for orthogonal signals
36Multiamplitude Signal transmission Transmitting multiple bits per signal waveformSymbol = several bits in a single waveform
37Signal Waveforms with Four Amplitude Levels TtTTTttFigure 5.19: Multi amplitude signal waveforms.-3d d d d
38Optimum receiver for AWGN Channel Signal correlator
39The detector Observes the correlator output r and decides which of the four PAM signals was transmitted in the signalinterval.The optimum amplitude detector computes the distancesThe detector selects the amplitude correspondingto the smallest distance.
40Example 5.8:Perform a Monte Carlo simulation of four - level PAM communicationsystem that employs a signal correlator, followed by an amplitudedetector. The model for the system to be simulated is shown in Fig 5.2.UniformRGGaussian randomNumber GeneratorcompareError counterdetectorMapping toAmplitude levels+XAmrFigure 5.22: Block diagram of four level PAM for Monte Carlo SimulationAnswerip_05_08Example 5.8:
42Example 5.9: perform a Monte Carlo simulation of a 16-level PAM digital communication system and measure its error-rate performance.Answerip_05_09
43We able to transmit k=log2(M) bits of information Multidimensional signalsSignal waveform having M=2k amplitude levelsWe able to transmit k=log2(M) bits of informationper signal waveform.Multidimensional Orthogonal signals
48Mapping to signal points Example 5.10: perform a Monte Carlo simulation of a digital communication system that employs M=4 orthogonal signals. The model of the system to be simulated is illustrated in Figure 5.30.Gaussian RNGCompare si with ^siError counterMapping to signal pointsUniform RNGdetectorOutput decisionAnswerip_05_10Figure 5.30: Block diagram of system with m=4 orthogonal signals for Monte Carlo simulation
50Mapping to signal points Example 5.11: perform a Monte Carlo simulation of a digital communication system that employs M=4 orthogonal signals. The model of the system to be simulated is illustrated in Figure 5.30.Gaussian RNGOutput decisiondetectorUniform RNGMapping to signal pointsGaussian RNGCompare si with ^siError conterFigure 5.30: Block diagram of system with m=4 orthogonal signals for Monte Carlo simulationAnswerip_05_11