The Optimum Detector and Probability of Error of Binary PSK

Matlab Coding,

Note,The detector maks a decision which is based onselecting the signal point corresponding in the largest projection.

  1 % MATLAB script
  2 echo on
  3 SNRindB1=0:2:10;
  4 SNRindB2=0:0.1:10;
  5 for i=1:length(SNRindB1),
 
  6     [pb,ps]=cm_sm32(SNRindB1(i));           % simulated bit and symbol error rates
  7     smld_bit_err_prb(i)=pb;
  8     smld_symbol_err_prb(i)=ps;
  9     echo off ;
 10 end;
 11 echo on;
 12 for i=1:length(SNRindB2),
 13     SNR=exp(SNRindB2(i)*log(10)/10);         % signal-to-noise ratio
 14     theo_err_prb(i)=Qfunct(sqrt(2
 
*SNR));     % theoretical bit-error rate
 15     echo off ;
 16 end;
 17 echo on ;
 18 % Plotting commands follow
 19 semilogy(SNRindB1,smld_bit_err_prb,'*');
 20 xlabel('dB');
 21 hold
 22 semilogy(SNRindB1,smld_symbol_err_prb,'o');
 23 semilogy(SNRindB2,theo_err_prb);
 24 
 25 
 26 function [pb,ps]=cm_sm32(snr_in_dB)
 
 27 % [pb,ps]=cm_sm32(snr_in_dB)
 28 %        CM_SM32  finds the probability of bit error and symbol error for the 
 29 %           given value of snr_in_dB, signal-to-noise ratio in dB.
 30 N=10000;
 31 E=1;                      % energy per symbol
 32 snr=10^(snr_in_dB/10);               % signal-to-noise ratio
 33 sgma=sqrt(E/snr)/2;                  % noise variance

 34 % the signal mapping
 35 s00=[1 0];
 36 s01=[0 1];
 37 s11=[-1 0];
 38 s10=[0 -1];

 39 % generation of the data source
 40 for i=1:N,    
 41   temp=rand;                   % a uniform random variable between 0 and 1
 42   if (temp<0.25),              % With probability 1/4, source output is "00."
 43     dsource1(i)=0;
 44     dsource2(i)=0;           
 45   elseif (temp<0.5),              % With probability 1/4, source output is "01."
 46     dsource1(i)=0;
 47     dsource2(i)=1;
 48   elseif (temp<0.75),               % With probability 1/4, source output is "10."
 49     dsource1(i)=1;    
 50     dsource2(i)=0;
 51   else                          % With probability 1/4, source output is "11."
 52     dsource1(i)=1;
 53     dsource2(i)=1;
 54   end;
 55 end;

 56 % detection and the probability of error calculation
 57 numofsymbolerror=0;
 58 numofbiterror=0;
 59 for i=1:N,
 60   % The received signal at the detector, for the ith symbol, is:
 61   n(1)=gngauss(sgma);            
 62   n(2)=gngauss(sgma);
 63   if ((dsource1(i)==0) & (dsource2(i)==0)),
 64     r=s00+n;
 65   elseif ((dsource1(i)==0) & (dsource2(i)==1)),
 66     r=s01+n;
 67   elseif ((dsource1(i)==1) & (dsource2(i)==0)),
 68     r=s10+n;
 69   else
 70     r=s11+n;
 71   end;

 72   % The correlation metrics are computed below.
 73   c00=dot(r,s00);
 74   c01=dot(r,s01);
 75   c10=dot(r,s10);
 76   c11=dot(r,s11);

 77   % The decision on the ith symbol is made next.
 78   c_max=max([c00 c01 c10 c11]);
 79   if (c00==c_max),
 80     decis1=0; decis2=0;
 81   elseif (c01==c_max),
 82     decis1=0; decis2=1;
 83   elseif (c10==c_max),
 84     decis1=1; decis2=0;
 85   else
 86     decis1=1; decis2=1;
 87   end;

 88   % Increment the error counter, if the decision is not correct.
 89   symbolerror=0;
 90   if (decis1~=dsource1(i)),
 91     numofbiterror=numofbiterror+1;
 92     symbolerror=1;
 93   end;
 94   if (decis2~=dsource2(i)),
 95     numofbiterror=numofbiterror+1;
 96     symbolerror=1;
 97   end;
 98   if (symbolerror==1),
 99     numofsymbolerror = numofsymbolerror+1;
100   end;
101 end;

102 ps=numofsymbolerror/N;                  % since there are totally N symbols
103 pb=numofbiterror/(2*N);                 % since 2N bits are transmitted


Simulation result

Reference,

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