Digital Communication Simulation in Digital Communication By: Dr. Uri Mahlab

Baseband Digital Transmission chapter # 5 Baseband Digital Transmission By: Dr. Uri Mahlab

Binary Signal Transmission Binary data consisting of a sequence of 1’s and 0’s. Tb - Bit time interval

AWGN - Channel + AWGN Noise PSD

+ 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.

Optimum Receiver for the AWGN Channel

Signal Correlator Output data Sampling @ t=Tb

detector Output data

Determine 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 is A t A- Figure 5.2: Signal waveforms s0(t) and s1(t) for a binary communication system Determine the correlator outputs at the sampling instants. Answer ip_05_01

Figure 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. t Output of correlator 0 t Output of correlator 1 t Output of correlator 1 t Output of correlator 0 (b) (a) Figure 5.3:Noise-free correlator outputs.(a) s0(t) was transmitted.(b) s1(t) was transmitted.

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

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;

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

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 A t A- Answer ip_05_02

t A t A A- 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

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.

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 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. A A t t A- Answer ip_05_03

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.

Example 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 generator Binary data source detector Output data Compare Error counter Gaussian random number generator Figure 5.8: Simulation model for Illustrative Answer ip_05_04

Other Binary Signal Transmission Methods

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

A t A- (a) A pair of antipodal signal A t A- (b) Another pair of antipodal signal

The received signal is Matched filter demodulator Correlator demodulator

probability density function for the input to the detector r probability density function for the input to the detector

For antipodal signal we have : 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 :

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

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

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

Probability density function for ON-OFF signals r Probability density function for ON-OFF signals

For antipodal signal we have : For On-OFF signal we have : The Detector For antipodal signal we have : For On-OFF signal we have :

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 Output data Answer ip_05_06

Signal 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.

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 (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. Answer ip_05_07

Receiver signal points at input to the selector for orthogonal signals

Multiamplitude Signal transmission Transmitting multiple bits per signal waveform Symbol = several bits in a single waveform

Signal Waveforms with Four Amplitude Levels T t T T T t t Figure 5.19: Multi amplitude signal waveforms. 00 01 11 10 -3d -d 0 d 3d

Optimum receiver for AWGN Channel Signal correlator

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.

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. Uniform RG Gaussian random Number Generator compare Error counter detector Mapping to Amplitude levels + X Am r Figure 5.22: Block diagram of four level PAM for Monte Carlo Simulation Answer ip_05_08 Example 5.8:

Signal Waveforms with Multiple Amplitude Levels

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

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

A T t T A t A T t A T t

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

Optimum receiver for multidimensional orthogonal signals. detector

Detector algorithm:

Mapping 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 RNG Compare si with ^si Error counter Mapping to signal points Uniform RNG detector Output decision Answer ip_05_10 Figure 5.30: Block diagram of system with m=4 orthogonal signals for Monte Carlo simulation

A t T A t T A t A- t

Mapping 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 RNG Output decision detector Uniform RNG Mapping to signal points Gaussian RNG Compare si with ^si Error conter Figure 5.30: Block diagram of system with m=4 orthogonal signals for Monte Carlo simulation Answer ip_05_11