Matlab: Help in understanding if the example for Maximum Likelihood Equalizer is properly functioning

Question

I am stuck in the part where I need to apply the MLSE Equalizer with Viterbi code. The equalizer is an optimal. I am using the Communication Toolbox http://www.mathworks.com/help/comm/ref/comm.mlseequalizer-class.html#properties

and cannot understand pretty much if the simulation is working properly or not. Basically, the equalizer is an FIR model and the channel is a multipath (not Rayleigh). Following the example in the documentation, I cannot understand

(1) What this statement does? Why is it calling the ML Equalizer here?

  modSignal = step(hMod, data);

(2) In the example code, there is no statement for additive Additive White Gaussian noise. The example given in the documentation does not clearly mention if the additive white Gaussian noise has been incorporated into the transmitted signal. It only says Introduce channel distortion . What kind of distortion is unclear.

(3) Is the example properly functioning because the simulation BER = 0 for all SNR levels. I am sceptical if I am doing it properly.

I shall be really obliged if some help is provided in how I should properly use the MLSE equalizer. Thank you.

UPDATE : The code throws error :

Error using comm.MLSEEqualizer/step
Complexity mismatch with input 1; expected complex, got real.

Error in MLSEEqualization_test (line 47)
      eqSignal = step(hMLSEE,chanOutput);

I wanted to see if I can apply MLSE for real signal also such as BPSK and to minimize the use of built-in functions for data generation, modulation and demodulation. This would help me to understand how the theoretical concepts can be put together in the code. That is why I am not using the communication toolbox for data generation and other steps; I am using only the MLSE EQualizer module provided in the toolbox. How can I go about this error message? The channel coefficients are real, noise is real and the data is also real. Thank you.

N = 10^3 % number of bits or symbols
rand('state',100); % initializing the rand() function
randn('state',200); % initializing the randn() function


n = 1/sqrt(2)*[randn(1,N) + j*randn(1,N)]; % white gaussian noise, 0dB variance 
Eb_N0_dB = [0:10]; % multiple Eb/N0 values
h=[1 0.6 0.3]';

L = length(h);
for aa = 1:length(Eb_N0_dB)

    % Transmitter
ip = rand(1,N)>0.0; % generating 0,1 with equal probability
data = 2*ip-1; % BPSK modulation 0 -> -1; 1 -> 1 

   chanOutput = filter(h,1,data');



    %% ----- ADDED
    EbN0 = Eb_N0_dB(aa);
    SNR_linear  = 10^(EbN0/10)*log2(M);
    SigPWR      = 1; %unit circle symbols
    NoisePWR    = SigPWR/SNR_linear;
    NoiseFactor = sqrt(NoisePWR/2); %half power complex half imaginary

    hMod = comm.QPSKModulator(0,'SymbolMapping','Binary');
  %  hDemod = comm.QPSKDemodulator(0,'SymbolMapping','Binary');
    const = step(hMod,(0:M-1)'); % BPSK constellation

            %% ----- ADD NOISE !!      
  %    noiseVec   = NoiseFactor*randn(N,1)+1j*NoiseFactor*randn(N,1);
   noiseVec   = NoiseFactor*randn(N,1);
      chanOutput = chanOutput+noiseVec;

    hMLSEE = comm.MLSEEqualizer('TracebackDepth',10,...
                      'Channel',h, 'Constellation',const);

    hError = comm.ErrorRate;

 % Equalize the channel output and demodulate
      eqSignal = step(hMLSEE,chanOutput);
      demodData = real(eqSignal) > 0;
   %   demodData = step(hDemod,eqSignal);
 % Compute BER
      errorStats = step(hError, data, demodData);

    fprintf('Error rate = %f\nNumber of errors = %d\n', ...
      errorStats(1), errorStats(2))

    %assume 1 symbol error = 1 bit error (needs Gray coding in real system for this to be true)
    BER(aa) = errorStats(1); % SER ~= BER with undelrying Gray coding
end

%% changed, this is already divided by N*N2
simBer = BER; % simulated ber


theoryBer = 0.5*erfc(sqrt(10.^(Eb_N0_dB/10))); % theoretical ber
close all
figure
semilogy(Eb_N0_dB,theoryBer,'b.-');
hold on
semilogy(Eb_N0_dB,simBer,'mx-');
axis([0 10 10^-5 0.5])
grid on
legend('theory', 'simulation');
xlabel('Eb/No, dB');
ylabel('Bit Error Rate');
title('Bit error probability curve for BPSK modulation');

Answer 1

I had a look at your code, and I know why you get 0 BER.

The thing is that you do not add any noise, you should use Eb/N0 to calculate the noise variance, and add a complex noise on your channel output.

The distortion added by "chanOutput = filter(chCoeffs,1,modSignal);" merely adds linear ISI, by convolving the channel impulse response with the input signal. This linear distortion is exactly known to the equalizer since you pass it the exact channel coefficients. This is never the case in real applications.

Even assuming you do know the channel coefficients, you still need to add noise immediately after the channel distortion and before the equalizer, to get a meaningful result of BER over Eb/N0.

Noise can be generated with randn having unit variance, then you need to multiply this noise by a constant.

I did not spend too much time understanding what you want to verify, but i did correct the script to include noise.

clear
N = 10^4; % number of symbols
N2= 10;   %repeat with same Eb/N0

% Eb_N0_dB = [0:25]; % multiple Eb/N0 values
Eb_N0_dB = (0:10); % multiple Eb/N0 values
nTx = 1;  %SISO 
nRx = 1;

M = 4; %QPSK

for ii = 1:length(Eb_N0_dB)

    %% ----- ADDED
    EbN0 = Eb_N0_dB(ii);
    SNR_linear  = 10^(EbN0/10)*log2(M);
    SigPWR      = 1; %unit circle symbols
    NoisePWR    = SigPWR/SNR_linear;
    NoiseFactor = sqrt(NoisePWR/2); %half power complex half imaginary

    hMod = comm.QPSKModulator(0,'SymbolMapping','Binary');
    hDemod = comm.QPSKDemodulator(0,'SymbolMapping','Binary');
 % Channel coefficients
    chCoeffs = [.986; .845; .237; .12345+.31i]; 
    hMLSEE = comm.MLSEEqualizer('TracebackDepth',10,...
                      'Channel',chCoeffs, 'Constellation',[1 1i -1 -1i]);
 % Create an error rate calculator
    hError = comm.ErrorRate;
    for n = 1:N2
      data= randi([0 3],N,1);
      modSignal = step(hMod, data);

 % Introduce channel distortion.
      chanOutput = filter(chCoeffs,1,modSignal); 

      %% ----- ADD NOISE !!      
      noiseVec   = NoiseFactor*randn(N,1)+1j*NoiseFactor*randn(N,1);
      chanOutput = chanOutput+noiseVec;


 % Equalize the channel output and demodulate
      eqSignal = step(hMLSEE,chanOutput);
      demodData = step(hDemod,eqSignal);
 % Compute BER
      errorStats = step(hError, data, demodData);
%       nErr=errorStats(1) ;
    end
    fprintf('Error rate = %f\nNumber of errors = %d\n', ...
      errorStats(1), errorStats(2))

    %assume 1 symbol error = 1 bit error (needs Gray coding in real system for this to be true)
    SER(ii) = errorStats(1); % SER ~= BER with undelrying Gray coding
end

%% changed, this is already divided by N*N2
simBer = SER; % simulated ber


EbN0Lin = 10.^(Eb_N0_dB/10);
theoryBer_nRx1 = 0.5.*(1-1*(1+1./EbN0Lin).^(-0.5)); 
p = 1/2 - 1/2*(1+1./EbN0Lin).^(-1/2);
theoryBerMRC_nRx2 = p.^2.*(1+2*(1-p)); 

close all
figure
semilogy(Eb_N0_dB,theoryBer_nRx1,'bp-','LineWidth',2);
hold on
semilogy(Eb_N0_dB,theoryBerMRC_nRx2,'kd-','LineWidth',2);
semilogy(Eb_N0_dB,simBer,'mo-','LineWidth',2);
axis([0 25 10^-5 0.5])
grid on
legend('theory (nTx=1,nRx=1)', 'theory (nTx=1,nRx=2, MRC)', 'sim (nTx=2, nRx=2, ML)');
xlabel('Average Eb/No,dB');
ylabel('Bit Error Rate');
title('BER for BPSK modulation with ML equalizer');