Why does the theoretical BER not coincide with the simulation one?

clc;
clear all;
close all;

M = 4;                 % Modulation order
K = log2(M);            % Bits per symbol
EbNoVec = (-10:1:10);      % Eb/No values (dB)
numSymPerFrame = 100;   % Number of PSK symbols per frame
N=10^6;
berEst = zeros(size(EbNoVec));

for i = 1:length(EbNoVec)
    snrdB = EbNoVec(i) + 10*log10(K);
    % Reset the error and bit counters
    numErrs = 0;
    numBits = 0;
    while numErrs < 200 && numBits < 1e6
        % Generate binary data and convert to symbols
        dataIn = randi([0 1],N,1);
        n = 7; % Kolichestvo vsekh bitov na simvol.
k = 4;
txEncoded = encode(dataIn,n,k, 'hamming');%%%%%%%%%%%%%%
        dataSym = bit2int(txEncoded,K);
        % QAM modulate using 'Gray' symbol mapping
        txSig = pskmod(dataSym,M);
        % Pass through AWGN channel
        rxSig = awgn(txSig,snrdB,'measured');
        % Demodulate the noisy signal
        rxSym = pskdemod(rxSig,M);
        % Convert received symbols to bits
        dataOut = int2bit(rxSym,K);
        rxDecoded = decode(dataOut,n,k,'hamming');
        % Calculate the number of bit errors
        nErrors = biterr(dataIn,rxDecoded);
        % Increment the error and bit counters
        numErrs = numErrs + nErrors;
        numBits = numBits + N;
    end
    % Estimate the BER
    berEst(i) = numErrs/numBits;
end

berTheory = berawgn(EbNoVec,'psk',M, 'nondiff');

semilogy(EbNoVec,berEst,'*')
hold on
semilogy(EbNoVec,berTheory)
grid
legend('Estimated BER','Theoretical BER')
xlabel('Eb/No (dB)')
ylabel('Bit Error Rate')

Answer 1

[Rangesh]

Hi Vtoroy,

I understand that you are seeking to identify why the theoretical value of BER does not align with the simulation results.

It's important to note that the use of a hamming code aims to reduce the BER by correcting the error. However, the function "berawgn" used for the theoretical value calculation does not account for the application of the hamming code. Consequently, there is a decrease in the BER rate compared to the theoretical values. That’s why there is a discrepancy in the simulation and the theoretical results.

To observe how the results compare, you can eliminate the encoding performed through the hamming code. Here is the modified code without the hamming code encoding:

M = 4;                 % Modulation order
K = log2(M);            % Bits per symbol
EbNoVec = (-10:1:10);      % Eb/No values (dB)
numSymPerFrame = 100;   % Number of PSK symbols per frame
N=10^6;
berEst = zeros(size(EbNoVec));

for i = 1:length(EbNoVec)
    snrdB = EbNoVec(i) + 10*log10(K);
    % Reset the error and bit counter 
    numErrs = 0;
    numBits = 0;
    
    while numErrs < 200 && numBits < 1e6
        % Generate binary data and convert to symbols
        dataIn = randi([0 1],N,1);
        n = 7; % Kolichestvo vsekh bitov na simvol.
        k = 4;
        % txEncoded = encode(dataIn,n,k, 'hamming');%%%%%%%%%%%%%%
        dataSym = bit2int(dataIn,K);
        
        % QAM modulate using 'Gray' symbol mapping
        txSig = pskmod(dataSym,M);
        % Pass through AWGN channel
        
        rxSig = awgn(txSig,snrdB,'measured');
        
        % Demodulate the noisy signal
        rxSym = pskdemod(rxSig,M);
        
        % Convert received symbols to bits
        dataOut = int2bit(rxSym,K);
        % rxDecoded = decode(dataOut,n,k,'hamming');
        
        % Calculate the number of bit errors
        nErrors = biterr(dataIn,dataOut);
        
        % Increment the error and bit counters
        numErrs = numErrs + nErrors;
        numBits = numBits + N;    
    end
    
    % Estimate the BER
    
    berEst(i) = numErrs/numBits;
    
end
berTheory = berawgn(EbNoVec,'psk',M, 'nondiff');
semilogy(EbNoVec,berEst,'*')
hold on
semilogy(EbNoVec,berTheory)
grid
legend('Estimated BER','Theoretical BER')
xlabel('Eb/No (dB)')
ylabel('Bit Error Rate')

For further understanding, you can refer the following documentation:

• berawgn: provides the BER of an AWGN channel https://www.mathworks.com/help/comm/ref/berawgn.html

I hope this resolves your query.