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High Rate Convolutional Codes for Turbo Coding

Concatenated convolutional codes offer high reliability and have gained in prominence and usage as turbo codes. The comm.TurboEncoder and comm.TurboDecoder System objects support rate 1/n convolutional codes only. This example shows the parallel concatenation of two rate 2/3 convolutional codes to achieve an effective rate 1/3 turbo code by using comm.ConvolutionalEncoder and comm.APPDecoder System objects.

System parameters

blkLength = 1024;          % Block length
EbNo = 0:5;                % Eb/No values to loop over
numIter = 3;               % Number of decoding iterations
maxNumBlks = 1e2;          % maximum number of blocks per Eb/No value

Convolutional Encoder/Decoder Parameters

trellis = poly2trellis([5 4],[23 35 0; 0 5 13]);
k = log2(trellis.numInputSymbols);      % number of input bits
n = log2(trellis.numOutputSymbols);     % number of output bits
intrIndices = randperm(blkLength/k)';   % Random interleaving
decAlg = 'True App';                    % Decoding algorithm
modOrder = 2;                           % PSK-modulation order

Initialize System Objects

Initialize Systems object for convolutional encoding, APP Decoding, BPSK modulation and demodulation, AGWN channel, and error rate computation. The demodulation output soft bits using a log-likelihood ratio method.

cEnc1 = comm.ConvolutionalEncoder('TrellisStructure',...
        trellis,'TerminationMethod','Truncated');
cEnc2 = comm.ConvolutionalEncoder('TrellisStructure',...
        trellis,'TerminationMethod','Truncated');
cAPPDec1 = comm.APPDecoder('TrellisStructure',trellis,...
            'TerminationMethod','Truncated','Algorithm',decAlg);
cAPPDec2 = comm.APPDecoder('TrellisStructure',trellis,...
            'TerminationMethod','Truncated','Algorithm',decAlg);

bpskMod = comm.BPSKModulator;
bpskDemod = comm.BPSKDemodulator('DecisionMethod','Log-likelihood ratio', ...
    'VarianceSource','Input port');

awgnChan = comm.AWGNChannel('NoiseMethod','Variance', ...
    'VarianceSource','Input port');

bitError = comm.ErrorRate; % BER measurement

Frame Processing Loop

Loop through a range of Eb/N0 values to generate results for BER performance. The helperTurboEnc and helperTurboDec helper functions perform the turbo encoding and decoding.

ber = zeros(length(EbNo),1); 
bitsPerSymbol = log2(modOrder);
turboEncRate = k/(2*n);

for ebNoIdx = 1:length(EbNo)
    % Calculate the noise variance from EbNo
    EsNo = EbNo(ebNoIdx) + 10*log10(bitsPerSymbol);
    SNRdB = EsNo + 10*log10(turboEncRate); % Account for code rate
    noiseVar = 10^(-SNRdB/10);

    for  numBlks = 1:maxNumBlks 
        % Generate binary data
        data = randi([0 1],blkLength,1);

        % Turbo encode the data
        [encodedData,outIndices] = helperTurboEnc(data,cEnc1,cEnc2, ...
            trellis,blkLength,intrIndices);

        % Modulate the encoded data
        modSignal = bpskMod(encodedData);

        % Pass the modulated signal through an AWGN channel
        receivedSignal = awgnChan(modSignal,noiseVar);

        % Demodulate the noisy signal using LLR to output soft bits
        demodSignal = bpskDemod(receivedSignal,noiseVar);

        % Turbo decode the demodulated data
        receivedBits = helperTurboDec(-demodSignal,cAPPDec1,cAPPDec2, ...
            trellis,blkLength,intrIndices,outIndices,numIter); 
        
        % Calculate the error statistics
        errorStats = bitError(data,receivedBits);        
    end
    
    ber(ebNoIdx) = errorStats(1);
    reset(bitError);
end

Display Results

While the practical wireless systems, such as LTE and CCSDS, specify base rate-1/n convolutional codes for turbo codes, the results show use of higher rate convolutional codes as turbo codes is viable.

figure; 
semilogy(EbNo, ber, '*-');
grid on; 
xlabel('E_b/N_0 (dB)'); 
ylabel('BER'); 
title('High Rate Convolutional Codes for Turbo Coding'); 
legend(['N = ' num2str(blkLength) ', ' num2str(numIter) ' iterations']);

Figure contains an axes. The axes with title High Rate Convolutional Codes for Turbo Coding contains an object of type line. This object represents N = 1024, 3 iterations.

Helper Functions

function [yEnc,outIndices] = helperTurboEnc(data,hCEnc1,hCEnc2,trellis,blkLength,intrIndices)
% Turbo encoding using two parallel convolutional encoders.
% No tail bits handling and assumes no output stream puncturing.

    % Trellis parameters
    k = log2(trellis.numInputSymbols);
    n = log2(trellis.numOutputSymbols);
    cLen = blkLength*n/k;

    punctrVec = [0;0;0;0;0;0];      % assumes all streams are output
    N = length(find(punctrVec==0));

    % Encode random data bits
    y1 = step(hCEnc1, data);
    y2 = step(hCEnc2, reshape(intrlv(reshape(data, k, [])',intrIndices)', [], 1));
    y1D = reshape(y1(1:cLen), n, []);
    y2D = reshape(y2(1:cLen), n, []);
    yDTemp = [y1D; y2D];
    y = yDTemp(:);

    % Generate output indices vector using puncturing vector
    idx = 0 : 2*n : (blkLength - 1)*2*(n/k);
    punctrVecIdx = find(punctrVec==0);
    dIdx = repmat(idx, N, 1) + punctrVecIdx;
    outIndices = dIdx(:);
    yEnc = y(outIndices);
end

function yDec = helperTurboDec(yEnc,cAPPDec1,cAPPDec2,trellis,blkLength,intrIndices,inIndices,numIter)
% Turbo decoding using two a-posteriori probability (APP) decoders

    % Trellis parameters
    k = log2(trellis.numInputSymbols);
    n = log2(trellis.numOutputSymbols);
    rCodLen = 2*(n/k)*blkLength;
    typeyEnc = class(yEnc);

    % Re-order encoded bits according to outIndices
    x = zeros(rCodLen, 1);
    x(inIndices) = yEnc;

    % Generate output of first encoder
    yD = reshape(x(1:rCodLen), 2*n, []);
    lc1D = yD(1:n, :);
    Lc1_in = lc1D(:);

    % Generate output of second encoder
    lc2D   = yD(n+1:2*n, :);
    Lc2_in = lc2D(:);

    % Initialize unencoded data input
    Lu1_in = zeros(blkLength, 1, typeyEnc);

    % Turbo Decode
    out1 = zeros(blkLength/k, k, typeyEnc);
    for iterIdx = 1 : numIter
        [Lu1_out, ~] = step(cAPPDec1, Lu1_in, Lc1_in);
        tmp = Lu1_out(1:blkLength);
        Lu2_in = reshape(tmp, k, [])';
        [Lu2_out, ~] = step(cAPPDec2, ...
            reshape(Lu2_in(intrIndices, :)', [], 1), Lc2_in);
        out1(intrIndices, :) = reshape(Lu2_out(1:blkLength), k, [])';
        Lu1_in = reshape(out1', [], 1);
    end
    % Calculate llr and decoded bits for the final iteration
    llr = reshape(out1', [], 1) + Lu1_out(1:blkLength);
    yDec = cast((llr>=0), typeyEnc);
end