CCS - Baseband Digital Transmission - Optimum Receiver for the AWGN Channel - The Detector

 

The Detector

The detector observes the correlator or matched filter outputs ro and r1 and decides
on whether the transmitted signal waveform is s0 ( t) or s1 ( t) , which correspond to the
transmission of either a 0 or a 1, respectively. The optimum detector is defined as the
detector that minimizes the probability of error.

 

 

 

 

 

 

 

  Note that,  https://en.wikipedia.org/wiki/Q-function

　　In statistics, the Q-function is the tail distribution function of the standard normal distribution.^[1][2] In other words, {\displaystyle Q(x)} is the probability that a normal (Gaussian) random variable will obtain a value larger than {\displaystyle x} standard deviations. Equivalently, {\displaystyle Q(x)} is the probability that a standard normal random variable takes a value larger than {\displaystyle x}.

 

The derivation of the detector performance given in the example was based on the

transmission of the signal waveform s0 ( t). The reader may verify that the probability
of error that is obtained when s1 ( t) is transmitted is identical to that obtained when
so (t) is transmitted. Because the 0's and 1 's in the data sequence are equally probable,
the average probability of error is that given by (5.2.23). The probability of error decreases exponentially as the SNR increases.

 

Matlab Coding

% The Matlab script that generates the probability of error versus the
% signal-to-noise ratio.

initial_snr = 0;
final_snr = 15;
snr_step = 0.25;
snr_in_dB = initial_snr:snr_step:final_snr;
for i = 1:length(snr_in_dB),
    snr = 10^(snr_in_dB(i)/10);
    Pe(i) = Qfunct(sqrt(snr));
    echo off;
end
echo on;
semilogy(snr_in_dB,Pe)
xlabel('SNR')
ylabel('Probability of error')

Simulation Result

 

 

 

Reference,

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