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Dr. Uri Mahlabn

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Binary Signal Transmission Binary data consisting of a sequence of 1 ’ s and 0 ’ s. T b - Bit time interval

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Dr. Uri Mahlabn + Noise PSD AWGN AWGN - Channel

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Dr. Uri Mahlabn + Receiver The receiver task is to decide whether a O or 1 was transmitter The receiver is designed to minimize the error probability. Such receiver is called the Optimum receiver.

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Dr. Uri Mahlabn Optimum Receiver for the AWGN Channel

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Dr. Uri Mahlabn Signal Correlator Output data t=T b

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Dr. Uri Mahlabn detecto r Output data

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Dr. Uri Mahlabn Example 5.1: suppose the signal waveforms s 0 (t) and s 1 (t) are the ones shown in figure 5.2, and let s 0 (t) be the transmitted signal. Then, the received signal is Answer ip_05_01 0 A t 0 A t A- Figure 5.2: Signal waveforms s 0 (t) and s 1 (t) for a binary communication system Determine the correlator outputs at the sampling instants.

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Dr. Uri Mahlabn Figure 5.3 illustrates the two noise-free correlator outputs in the interval for each of the two cases-I.e., when s 0 (t) is transmitted and when s 1 (t) is transmitted. 0 t Output of correlator 0 0 t Output of correlator 1 0 t 0 t Output of correlator 0 (a)(a) (b) Figure 5.3:Noise-free correlator outputs.(a) s 0 (t) was transmitted.(b) s 1 (t) was transmitted.

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Dr. Uri Mahlabn 0 r Probability density function p(r0|0) and p(r1|0) when s0(t) is transmitted

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Dr. Uri Mahlabn Matched 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;

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Dr. Uri Mahlabn The matched filter output at the sampling instant t=Tb is identical to the output of the signal correlator.

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Dr. Uri Mahlabn Answer ip_05_02 Example 5.2: Consider the use of matched filters for the demodulation of the two signal waveforms shown in the figure and determine the outputs 0 A t 0 A t A-

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Dr. Uri Mahlabn 0 A t 0 A A- t Figure 5.5:Impulse responses of matched filters for signals s 0 (t) and s 1 (t). 00 (a)(a) (b) Figure 5.6:Signal outputs of matched filters when s 0 (t) is transmitted

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Dr. Uri Mahlabn The Detector The detector observes the correlator or the matched filter output r0 and r1 and decided on whether the transmitted signal waveform is s1(t) or s0(t), which corresponding to “ 1 ” or “ 0 ”, respectively. The optimum detector is defined the detector that minimizes the probability of error.

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Dr. Uri Mahlabn Example 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 r 0 and r 1 and decides that a 0 was transmitted when r 0 >r 1 and that a 1 was transmitted when r 0 >r 1. Determine the probability of error. Answer ip_05_03 0 A t 0 A t A-

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Dr. Uri Mahlabn Monte Carlo Simulation Communication System Monte Carlo computer simulations are usually performed in practice to estimate the probability of error of a digital communication system, especially in cases where the analysis of the detector performance is difficult to perform.

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Dr. Uri Mahlabn Example 5.4: use Monte Carlo simulation to estimate an plot P e versus SNR for a binary communication system that employs correlators or matched filters. The model of the system is illustrated in figure 5.8. Answer ip_05_04 Uniform random number generator Binary data source detector Output data Compare Error counter Gaussian random number generator Figure 5.8: Simulation model for Illustrative

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Dr. Uri Mahlabn

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Other Binary Signal Transmission Methods

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Dr. Uri Mahlabn Antipodal Signal for Binary Signal Transmission Antipodal signal If one signal waveform is negative of the other.

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Dr. Uri Mahlabn 0 A t 0 A- t (a) A pair of antipodal signal 0 A t A- 0 A t (b) Another pair of antipodal signal

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Dr. Uri Mahlabn Matched filter demodulator Correlator demodulator The received signal is

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Dr. Uri Mahlabn 0 r probability density function for the input to the detector

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Dr. Uri Mahlabn The Detector The detector observes the correlator or the matched filter output r0 and r1 and decided on whether the transmitted signal waveform is s1(t) or s0(t), which corresponding to “ 1 ” or “ 0 ”, respectively. The optimum detector is defined the detector that minimizes the probability of error. For antipodal signal we have :

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Dr. Uri Mahlabn Answer ip_05_05 Example 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 Uniform random number generator Binary data source Compare Error counter detector r n Gaussian random number generator Output data Figure 5.13: Model of binary communication system employing antipodal signal

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Dr. Uri Mahlabn On-Off Signal for Binary Signal Transmission The received signal is: Binary information sequence may also be transmitted by use of ON-OFF signals

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Dr. Uri Mahlabn 0 r Figure 5.15: The probability density function for the received signal at the output of te correlator for on-off signal.

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Dr. Uri Mahlabn 0 r Probability density function for ON-OFF signals

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Dr. Uri Mahlabn The Detector For antipodal signal we have : For On-OFF signal we have :

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Dr. Uri Mahlabn Answer ip_05_06 Example 5.6:use Monte Carlo simulation to estimate and plot the performance of a binary communication system employing on-off signaling Uniform random number generator Binary data source Compare Error counter detector r n Gaussian random number generator Output data

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Dr. Uri Mahlabn 0 (a)(a) 0 (b) 0 Figure 5.17: signal point constellation for binary signal.(a) Antipodal signal.(b) On-off signals.(c) Orthogonal signals. Signal Constellation diagrams for Binary Signals

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Dr. Uri Mahlabn Answer ip_05_07 Example 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 (r 0, r 1 ), where either. The noise random variables n 0 and n 1 re zero-mean, independent Gaussian random variables with variance.as in Illustrative Problam 5.4 use Monte Carlo simulation to generate 100 samples of (r 0, r 1 ) 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.

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Dr. Uri Mahlabn Receiver signal points at input to the selector for orthogonal signals

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Dr. Uri Mahlabn Multiamplitude Signal transmission Transmitting multiple bits per signal waveform Symbol = several bits in a single waveform

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Dr. Uri Mahlabn t 0 T 0 t T 0 t T t 0 T Figure 5.19: Multi amplitude signal waveforms. -3d -d 0 d 3d Signal Waveforms with Four Amplitude Levels

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Dr. Uri Mahlabn Optimum receiver for AWGN Channel Signal correlator

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Dr. Uri Mahlabn The detector Observes the correlator output r and decides which of the four PAM signals was transmitted in the signal interval. The optimum amplitude detector computes the distances The detector selects the amplitude corresponding to the smallest distance.

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Dr. Uri Mahlabn Example 5.8: Perform a Monte Carlo simulation of four - level PAM communication system that employs a signal correlator, followed by an amplitude detector. The model for the system to be simulated is shown in Fig 5.2. Answer ip_05_08 Uniform RG Gaussian random Number Generator compare Error counter detector Mapping to Amplitude levels + XAmAm r Figure 5.22: Block diagram of four level PAM for Monte Carlo Simulation Example 5.8:

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Dr. Uri Mahlabn Signal Waveforms with Multiple Amplitude Levels

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Dr. Uri Mahlabn Answer ip_05_09 Example 5.9: perform a Monte Carlo simulation of a 16-level PAM digital communication system and measure its error-rate performance.

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Dr. Uri Mahlabn Multidimensional signals Signal waveform having M=2 k amplitude levels We able to transmit k=log2(M) bits of information per signal waveform. Multidimensional Orthogonal signals

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Dr. Uri Mahlabn A T t A T t T A t A T t

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M=2 M=3 Figure 5.27: Signal constellation for M=2 and M=3 orthogonal signals.

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Dr. Uri Mahlabn detecto r Optimum receiver for multidimensional orthogonal signals.

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Dr. Uri Mahlabn Detector algorithm:

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Dr. Uri Mahlabn Answer ip_05_10 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 Gaussian RNG Compar e s i with ^ s i Error counter Mapping to signal points Uniform RNG detecto r Output decision Figure 5.30: Block diagram of system with m=4 orthogonal signals for Monte Carlo simulation

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Dr. Uri Mahlabn A t A- t T A t T A t

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Dr. Uri Mahlabn Answer ip_05_11 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 Gaussian RNG 0 Compar e s i with ^ s i Error conter Mapping to signal points Uniform RNG detecto r Output decision Figure 5.30: Block diagram of system with m=4 orthogonal signals for Monte Carlo simulation

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