# Performance Evaluation Of A Binary Signaling

## Question

Performance Evaluation of a binary signaling in a linear distortive channel
In this part, we are interested in evaluating the performance of a BPSK-modulated communication system over a multi-path channel. For the purpose of simulation, we consider a discrete-time baseband system. It means that we simulate the system assuming that each bit is represented by a single sample (+1 for bit “1” and -1 for bit “0”). It is a valid assumption when the sampling is perfectly done at the receiver (without timing error). The underlying channel impulse response is h = [-0.09 0.64 0.95 -0.25 0.14] (assumed to be fixed during the transmission) and the AWGN is also added to the received signal. Therefore, the discrete channel model can be conveniently written as r[k] = h ∗ x[k] + n[k]where x[k] is the transmit sample vector of size 5×1, and r[k] and n[k] are the received signal and noise samples at time instant k, respectively.
1) Write a MATLAB code to simulate the probability of error versus different values of Eb/N0 (Pe-vs-Eb/N0) using Monte Carlo method (explained in the previous computer assignment) assuming the above channel. Plot the BER versus Eb/N0 (or SNR) curve for Eb/N0 from 0 to 10dB.
2) Design a 3-tap zero-forcing equalizer (ZFE) for the above channel and then repeat part one but this time, assume that the zero forcing equalizing is done at the receiver. Calculate the resulting MSE of this equalizer. Plot the BER versus Eb/N0 curve for Eb/N0 from 0 to 10dB.
3) Design a 3-tap MMSE equalizer for the above channel and then repeat part one but this time, assume that the zero forcing equalizing is done at the receiver. Calculate the resulting MSE of this equalizer and compare it with that of ZFE in (2). Plot the BER versus Eb/N0 curve for Eb/N0 from 0 to 10dB.
4) Plot the above three curves as well as the theoretical BER curve of BPSK signaling in AWGN (ideal case) in one plot. Discuss the result. How much gain is obtained in each case?

Performance Evaluation of a coded binary signaling
In this part, we are interested in the performance evaluation of the above system when a coding scheme is added. Consider a simple (3,1) repetition code. In this code, each bit is repeated for transmission, i.e. for a 0 we transmit 000 and for a 1 we transmit 111. At the receiver, after detecting each received bit, a simple majority vote is done. The following table summarized the decoding rule:
000                                        0
001                                        0
010                                        0
011                                          1
100                                        0
101                                          1
110                                          1
111                                          1

1) Write a MATLAB code to plot the BER versus Eb/N0 curve for Eb/N0 from 0 to 10dB for this coded system. For this case assume that the channel is AWGN.
2) Plot the above curve as well as the theoretical BER curve of (uncoded) BPSK signaling in AWGN (ideal case) in one plot. Compare the results. How much gain is obtained using the code? Discuss your observations.

## Solution Preview

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Matlab code for problem 1.
Main code
clc;
clear all;
close all;

EbNodb=0:10;%Eb/No from 0 to 10db
z=10.^(EbNodb/10);% convertion to linear scale
delay=0;
FilterSwitch=1;
BER=zeros(1,length(z));% initialisation of BER
Errors=zeros(1,length(z));%intialisation of errors
BER_T=qfunc(sqrt(2*z));% calucation of BER in analytic way
N=round(20./BER_T);
for k=1:length(z)
N(k)=max(1000,N(k));
[BER(k),Errors(k)]=MCBPSKrun(N(k),z(k),delay,FilterSwitch);
end...
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