Documentation

Initialization

Initialize the system variables and create System objects for modulation, demodulation, AWGN channel, and error rate operations. Since the comm.AWGNChannel System object™ and the randi function use the default random stream, set the random number generator seed to ensure repeatable results. The state of the random number generator is stored before setting the random stream seed, and restored at the end of the example.

M = 8; % Modulation order for 8-PSK
SamplesPerFrame = 10000;
maxNumErrs=100;
maxNumBits=1e8;

prevState = rng;
rng(529558);

Create PSK modulator and demodulator System objects to map the binary input data to 8-PSK Gray- and binary-coded constellations.

pskmod = comm.PSKModulator('ModulationOrder',M,'SymbolMapping','Gray', ...
    'PhaseOffset',0,'BitInput',true);
pskdemod = comm.PSKDemodulator('ModulationOrder',M,'SymbolMapping','Gray', ...
    'PhaseOffset',0,'BitOutput',true,'OutputDataType','uint8', ...
    'DecisionMethod','Hard decision');
pskmodb = comm.PSKModulator('ModulationOrder',M,'SymbolMapping','Binary', ...
    'PhaseOffset',0,'BitInput',true);
pskdemodb = comm.PSKDemodulator('ModulationOrder',M,'SymbolMapping','Binary', ...
    'PhaseOffset',0,'BitOutput',true,'OutputDataType','uint8', ...
    'DecisionMethod','Hard decision');

Create an AWGN channel System object to add noise to the modulated signal. The noise method is configured to ${\mathit{E}}_{\mathrm{b}}/{\mathit{N}}_0$ for the processing loop. The PSK modulator generates symbols with 1 W of power, so the signal power property of the AWGN channel object is set to 1 W also.

awgnchan = comm.AWGNChannel('NoiseMethod','Signal to noise ratio (Eb/No)', ...
    'BitsPerSymbol',log2(M),'SignalPower',1);

Create a symbol error rate and bit error rate calculator System objects to compare the demodulated integer and bit data with the original source data. This comparison yields symbol error and bit error statistics. The output of the error rate calculator System object is a three-element vector containing the calculated error rate, the number of errors observed, and the amount of data processed. The simulation uses the three-element vector to determine when to stop the simulation.

symerror = comm.ErrorRate;
biterror = comm.ErrorRate;
biterrorb = comm.ErrorRate;

Frame Processing Loop

Configure a frame processing loop where data is coded, modulated, and demodulated using 8-PSK modulation. The loop simulates the communications system for ${\mathit{E}}_{\mathrm{b}}/{\mathit{N}}_0$ values in the range 0 dB to 12 dB in steps of 2 dB.

For each ${\mathit{E}}_{\mathrm{b}}/{\mathit{N}}_0$ value, the simulation stops when either the maximum number of errors (maxNumErrs) or the maximum number of bits (maxNumBits) processed by the bit error rate calculator System object is reached for the Gray coded bits.

EbNoVec = 0:2:12; % Eb/No values to simulate
SERVec = zeros(size(EbNoVec)); % Initialize SER history
BERVec = zeros(size(EbNoVec)); % Initialize BER history for Gray ordered
BERVecb = zeros(size(EbNoVec)); % Initialize BER history for binary ordered
for p = 1:length(EbNoVec)
  % Reset System objects
  reset(symerror);
  reset(biterror);
  reset(biterrorb);
  awgnchan.EbNo = EbNoVec(p);
  % Reset SER / BER for the current Eb/No value
  SER = zeros(3,1);
  BER = zeros(3,1);
  while (BER(2)<maxNumErrs) && (BER(3)<maxNumBits)
    % Generate random data
    txSym = randi([0 M-1],SamplesPerFrame,1,'uint8');  
    txBits = reshape(de2bi(txSym,log2(M),'left-msb')',[],1); % Convert symbols to bits
    
    tx = pskmod(txBits);
    txb = pskmodb(txBits);
    rx = awgnchan(tx);
    rxb = awgnchan(txb);
    rxBits = pskdemod(rx);
    rxBitsb = pskdemodb(rxb);
    rxSym = bi2de(reshape(rxBits,log2(M),[])','left-msb');
    
    SER = symerror(txSym,rxSym); % Symbol error rate for Gray-coded data 
    BER = biterror(txBits,rxBits); % Bit error rate for Gray-coded data
    BERb = biterrorb(txBits,rxBitsb); % Bit error rate for natural binary-coded data
  end
  % Save history of SER and BER values
  SERVec(p) = SER(1);
  BERVec(p) = BER(1);
  BERVecb(p) = BERb(1);
end

Restore the default stream.

rng(prevState)

Results Analysis

Analyze the data from the example and compare theoretical performance with simulation performance. The theoretical symbol error probability of MPSK is

where erfc is the complementary error function, ${\mathit{E}}_{\mathit{S}}/{\mathit{N}}_0$ is the ratio of energy in a symbol to noise power spectral density, and M is the number of symbols.

To determine the bit error probability, convert the symbol error probability, ${\mathit{P}}_{\mathrm{E}}$, to its bit error equivalent. There is no general formula for the symbol to bit error conversion. Nevertheless, upper and lower limits are easy to establish. The actual bit error probability, ${\mathit{P}}_{\mathit{b}}$, can be shown to be bounded by

The lower limit corresponds to the case where the symbols have undergone Gray coding. The upper limit corresponds to the case of pure binary coding.

Calculate theoretical error probabilities by using the use the berawgn function. Plot the simulated symbol error rate for Gray coding, bit error rate for Gray and natural binary coding, and the theoretical symbol error and bit error probabilities for Gray coding.

[theorBER,theorSER] = berawgn(EbNoVec,'psk',M,'nondiff');

figure;
semilogy(EbNoVec,SERVec,'o',EbNoVec,BERVecb,'x',EbNoVec,BERVec,'*', ...
         EbNoVec,theorSER,'-',EbNoVec,theorBER,'-');
legend('Symbol error rate','Bit error rate (Binary)','Bit error rate (Gray)', ...
          'Theoretical Symbol error rate','Theoretical Bit error rate', ...
          'Location','SouthWest');
xlabel('Eb/No (dB)'); ylabel('Error Probability');
title('Symbol and Bit Error Probability');
grid on;