Establish Simulation Framework

In this example, to achieve a more accurate BER estimate, the number of bits to process is increased from the value used in the Use Pulse Shaping on 16-QAM Signal example. Other simulation variables match the settings in that example.

Define simulation parameters for a 16-QAM modulation scheme with raised cosine filtering and an AWGN channel.

M = 16; % Modulation order
k = log2(M); % Number of bits per symbol
numBits = 1e6; % Number of bits to process
sps = 4; % Number of samples per symbol (oversampling factor)
filtlen = 10; % Filter length in symbols
rolloff = 0.25; % Filter rolloff factor

Generate Random Data

Set the rng function to its default state, or any static seed value, so that the example produces repeatable results. Then, use the randi function to generate random binary data.

rng default;                     % Use default random number generator
dataIn = randi([0 1],numBits,1); % Generate vector of binary data

Apply Convolutional Encoding

To correct errors arising from the noisy channel, apply convolutional coding to the data before transmission and Viterbi decoding to the received data. The decoder uses a hard decision algorithm, which means each received data bit is interpreted as either 0 or 1.

Define a convolutional coding trellis for a rate 2/3 code by using the poly2trellis function. The defined trellis represents the convolutional code that the convenc function uses for encoding the binary vector, dataIn.

constrlen = [5 4]; % Code constraint length
genpoly = [23 35 0; 0 5 13] % Generator polynomials

genpoly = 2×3

    23    35     0
     0     5    13

tPoly = poly2trellis(constrlen,genpoly);
codeRate = 2/3;

Encode the input data by using the tPoly trellis.

dataEnc = convenc(dataIn,tPoly);

Modulate Data

Reshape the input vector into a matrix of 4-bit binary data. Then, use the bi2de function to convert the encoded binary data to an integer format.

dataEncMatrix = reshape(dataEnc, ...
    length(dataEnc)/k,k); % Reshape data into binary 4-tuples
dataSymbolsIn = bi2de(dataEncMatrix); % Convert to integers

Use the qammod function to apply 16-QAM modulation.

dataMod = qammod(dataSymbolsIn,M);

Apply Raised Cosine Filtering

Use the rcosdesign function to create an RRC filter.

rrcFilter = rcosdesign(rolloff,filtlen,sps);

Use the upfirdn function to upsample the signal by the oversampling factor and apply the RRC filter. The upfirdn function pads the upsampled signal with zeros at the end to flush the filter. Then, the function applies the filter.

txSignal = upfirdn(dataMod,rrcFilter,sps,1);

Apply AWGN Channel

Using the number of bits per symbol (k) and the number of samples per symbol (sps), convert the ratio of energy per bit to noise power spectral density (EbNo) to an SNR value for use by the awgn function. When converting the ${\mathit{E}}_{\mathrm{b}}/{\mathit{N}}_0$ to SNR, you must account for the number of information bits per symbol. With no FEC applied, each symbol corresponded to k bits. With FEC applied, each symbol corresponds to (k $\times$ codeRate) information bits. For the 2/3 code rate and 16-QAM transmissions used in this example, three symbols correspond to 12 coded bits and 8 uncoded (information) bits.

EbNo = 10;
snr = EbNo+10*log10(k*codeRate)-10*log10(sps);

Pass the filtered signal through an AWGN channel.

rxSignal = awgn(txSignal,snr,'measured');

Receive and Demodulate Signal

Filter the received signal by using the RRC filter. Remove a portion of the signal to account for the filter delay.

rxFiltSignal = upfirdn(rxSignal,rrcFilter,1,sps); % Downsample and filter
rxFiltSignal = rxFiltSignal(filtlen + 1:end - filtlen); % Account for delay

Use the qamdemod function to demodulate the received filtered signal.

dataSymbolsOut = qamdemod(rxFiltSignal,M);

Apply Viterbi Decoding

Use the de2bi function to convert the recovered integer symbols into binary data.

dataOutMatrix = de2bi(dataSymbolsOut,k);
codedDataOut = dataOutMatrix(:); % Return data in column vector

Use the vitdec function, configured for hard decisions and continuous operation mode, to decode the convolutionally encoded data. The continuous operation mode maintains the internal state when the decoder is repeatedly invoked, such as when receiving frames of data operating in a loop. The continuous operation mode also adds delay to the system. Although this example does not use a loop, the 'cont' mode is used for the purpose of illustrating how to compensate for the delay in this decoding operation.

traceBack = 16; % Traceback length for decoding
numCodeWords = floor(length(codedDataOut)*2/3); % Number of complete codewords
dataOut = vitdec(codedDataOut(1:numCodeWords*3/2), ...
           tPoly,traceBack,'cont','hard'); % Decode data

Compute System BER

Use the biterr function to compute the number of errors and the BER by comparing dataIn and dataOut. The delay introduced by the transmit and receive RRC filters is already accounted for in the recovered data, but the decoder delay is not accounted for yet. The continuous operation mode of the Viterbi decoder incurs a delay with a duration in bits equal to the traceback length, traceBack, times the number of input streams at the encoder. For the 2/3 code rate used in this example, the encoder has two input streams, so the delay is 2×traceBack bits. As a result, the first 2×traceBack bits in the decoded vector, dataOut, are zeros. When computing the BER, discard the first 2×traceBack bits in dataOut and the last 2×traceBack bits in the original vector, dataIn.

decDelay = 2*traceBack; % Decoder delay, in bits
[numErrors,ber] = ...
   biterr(dataIn(1:end - decDelay),dataOut(decDelay + 1:end));       
fprintf('\nThe bit error rate is %5.2e, based on %d errors.\n', ...
    ber,numErrors)

The bit error rate is 1.00e-04, based on 100 errors.

For the same ${\mathit{E}}_{\mathrm{b}}/{\mathit{N}}_0$ of 10 dB, less errors occur when using FEC and the BER improves from approximately 2×10-3 to 1×10-4.

More About Delays

The decoding operation in this example incurs a delay that causes the output of the decoder to lag the input. Timing information does not appear explicitly in the example, and the length of the delay depends on the specific operations being performed. Delays occur in various communications system operations, including convolutional decoding, convolutional interleaving and deinterleaving, equalization, and filtering. To find out the duration of the delay caused by specific functions or operations, see the specific documentation for those functions or operations. For more information on delays, see Delays of Convolutional Interleavers and Fading Channels.