CCS

Optimum Receiver for the AWGN Channel

The optimum receiver for the AWGN channel consists of two building blocks.

One is either a signal correlator or a matched filter . The other is a detector.

Signal Correlator

The signal correlator cross-correlates the received signal r ( t) with the two possible
transmitted signals s0 ( t) and s1( t), as illustrated in Figure 5 .1. That is, the signal
correlator computes the two outputs,

 in the interval 0 <= t <= Tb, samples the two outputs at t = Tb, and feeds the sampled

outputs to the detector.

Suppose that 

Matlab coding

% MATLAB script

% Initialization:
K=20;       % Number of samples
A=1;        % Signal amplitude
l=0:K;

% Defining signal waveforms:
s_0=A*ones(1,K);
s_1=[A*ones(1,K/2) -A*ones(1,K/2)];

% Initializing output signals:
r_0=zeros(1,K);
r_1=zeros(1,K);

% Case 1: noise~N(0,0)     sigma^2= 0
noise=random('Normal',0,0,1,K);
    % Sub-case s = s_0:
    s=s_0;
    r=s+noise;  % received signal
    for n=1:K
        r_0(n)=sum(r(1:n).*s_0(1:n));    % Signal Correlator 
        r_1(n)=sum(r(1:n).*s_1(1:n));
    end
    % Plotting the results:
    subplot(3,2,1)
    plot(l,[0 r_0],'-',l,[0 r_1],'--')
    set(gca,'XTickLabel',{'0','5Tb','10Tb','15Tb','20Tb'})
    axis([0 20 -5 30])
    xlabel('(a) sigma^2= 0 & S_{0} is transmitted','fontsize',10)
    % Sub-case s = s_1:
    s=s_1;
    r=s+noise;  % received signal
    for n=1:K
        r_0(n)=sum(r(1:n).*s_0(1:n));
        r_1(n)=sum(r(1:n).*s_1(1:n));
    end
    % Plotting the results:
    subplot(3,2,2)
    plot(l,[0 r_0],'-',l,[0 r_1],'--')
    set(gca,'XTickLabel',{'0','5Tb','10Tb','15Tb','20Tb'})
    axis([0 20 -5 30])
    xlabel('(b) sigma^2= 0 & S_{1} is transmitted','fontsize',10)

% Case 2: noise~N(0,0.1)      sigma^2= 0.1
    noise=random('Normal',0,0.1,1,K);
    % Sub-case s = s_0:
    s=s_0;
    r=s+noise;  % received signal
    for n=1:K
        r_0(n)=sum(r(1:n).*s_0(1:n));
        r_1(n)=sum(r(1:n).*s_1(1:n));
    end
    % Plotting the results:
    subplot(3,2,3)
    plot(l,[0 r_0],'-',l,[0 r_1],'--')
    set(gca,'XTickLabel',{'0','5Tb','10Tb','15Tb','20Tb'})
    axis([0 20 -5 30])
    xlabel('(c) sigma^2= 0.1 & S_{0} is transmitted','fontsize',10)
    % Sub-case s = s_1:
    s=s_1;
    r=s+noise;  % received signal
    for n=1:K
        r_0(n)=sum(r(1:n).*s_0(1:n));
        r_1(n)=sum(r(1:n).*s_1(1:n));
    end
    % Plotting the results:
    subplot(3,2,4)
    plot(l,[0 r_0],'-',l,[0 r_1],'--')
    set(gca,'XTickLabel',{'0','5Tb','10Tb','15Tb','20Tb'})
    axis([0 20 -5 30])
    xlabel('(d) sigma^2= 0.1 & S_{1} is transmitted','fontsize',10)

% Case 3: noise~N(0,1)    sigma^2= 1
    noise=random('Normal',0,1,1,K);
    % Sub-case s = s_0:
    s=s_0;
    r=s+noise;  % received signal
    for n=1:K
        r_0(n)=sum(r(1:n).*s_0(1:n));          
        r_1(n)=sum(r(1:n).*s_1(1:n));
    end
    % Plotting the results:
    subplot(3,2,5)
    plot(l,[0 r_0],'-',l,[0 r_1],'--')
    set(gca,'XTickLabel',{'0','5Tb','10Tb','15Tb','20Tb'})
    axis([0 20 -5 30])
    xlabel('(e) sigma^2= 1 & S_{0} is transmitted','fontsize',10)
    % Sub-case s = s_1:
    s=s_1;
    r=s+noise;  % received signal
    for n=1:K
        r_0(n)=sum(r(1:n).*s_0(1:n));
        r_1(n)=sum(r(1:n).*s_1(1:n));
    end
    % Plotting the results:
    subplot(3,2,6)
    plot(l,[0 r_0],'-',l,[0 r_1],'--')
    set(gca,'XTickLabel',{'0','5Tb','10Tb','15Tb','20Tb'})
    axis([0 20 -5 30])
    xlabel('(f) sigma^2= 1 & S_{1} is transmitted','fontsize',10)


Simulation Results

Reference,

　　1. <<Contemporary Communication System using MATLAB>> - John G. Proakis