This section builds an example step-by-step to give you a first look
at the Communications Toolbox™ software. This section also shows how Communications Toolbox functionalities build upon the computational and
visualization tools in the underlying MATLAB^{®} environment.

This example shows how to process a binary data stream using a communication system that consists of a baseband modulator, channel, and demodulator. The system's bit error rate (BER) is computed and the transmitted and received signals are displayed in a constellation diagram.

The following table summarizes the basic operations used, along with relevant Communications Toolbox and MATLAB functions. The example uses baseband 16-QAM (quadrature amplitude modulation) as the modulation scheme and AWGN (additive white Gaussian noise) as the channel model.

Task | Function |
---|---|

Generate a Random Binary Data Stream | `randi` |

Convert the Binary Signal to an Integer-Valued Signal | `bi2de` |

Modulate using 16-QAM | `qammod` |

Add White Gaussian Noise | `awgn` |

Create a Constellation Diagram | `scatterplot` |

Demodulate using 16-QAM | `qamdemod` |

Convert the Integer-Valued Signal to a Binary Signal | `de2bi` |

Compute the System BER | `biterr` |

The conventional format for representing a signal in
MATLAB is a vector or matrix. This example uses the
`randi`

function to create a
column vector that contains the values of a binary data
stream. The length of the binary data stream (that is, the
number of rows in the column vector) is arbitrarily set to
30,000.

The code below also creates a stem plot of a portion of the
data stream, showing the binary values. Notice the use of
the colon (`:`

) operator in MATLAB to select a portion of the vector.

Define parameters.

M = 16; % Size of signal constellation k = log2(M); % Number of bits per symbol n = 30000; % Number of bits to process numSamplesPerSymbol = 1; % Oversampling factor

Create a binary data stream as a column vector.

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

Plot the first 40 bits in a stem plot.

stem(dataIn(1:40),'filled'); title('Random Bits'); xlabel('Bit Index'); ylabel('Binary Value');

The `qammod`

function implements a
rectangular, M-ary QAM modulator, M being 16 in this
example. The default configuration is such that the object
receives integers between 0 and 15 rather than 4-tuples of
bits. In this example, we preprocess the binary data stream
`dataIn`

before using the
`qammod`

function. In particular,
the `bi2de`

function is used to convert
each 4-tuple to a corresponding integer.

Perform a bit-to-symbol mapping.

dataInMatrix = reshape(dataIn,length(dataIn)/k,k); % Reshape data into binary k-tuples, k = log2(M) dataSymbolsIn = bi2de(dataInMatrix); % Convert to integers

Plot the first 10 symbols in a stem plot.

figure; % Create new figure window. stem(dataSymbolsIn(1:10)); title('Random Symbols'); xlabel('Symbol Index'); ylabel('Integer Value');

Having generated the `dataSymbolsIn`

column
vector, use the `qammod`

function to apply
16-QAM modulation for both binary and Gray coded
bit-to-symbol mappings. Recall that `M`

is
16, the alphabet size.

Apply modulation.

dataMod = qammod(dataSymbolsIn,M,'bin'); % Binary coding, phase offset = 0 dataModG = qammod(dataSymbolsIn,M); % Gray coding, phase offset = 0

The results are complex column vectors whose values are elements of the 16-QAM signal constellation. A later step in this example will plot the constellation diagram.

To learn more about modulation functions, see Digital Modulation. Also, note that the `qammod`

function
does not apply pulse shaping. To extend this example to use
pulse shaping, see Pulse Shaping Using a Raised Cosine Filter.
For an example that uses Gray coding with PSK modulation,
see Gray Coded
8-PSK.

The ratio of bit energy to noise power spectral density,
*E _{b}/N_{0}*,
is arbitrarily set to 10 dB. From that value, the
signal-to-noise ratio (SNR) can be determined. Given the
SNR, the modulated signal,

`dataMod`

, is
passed through the channel by using the
`awgn`

function.The `numSamplesPerSymbol`

variable is
not significant in this example but will make it
easier to extend the example later to use pulse
shaping.

Calculate the SNR when the channel has an
*E _{b}/N_{0}*
= 10 dB.

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

Pass the signal through the AWGN channel for both the binary and Gray coded symbol mappings.

receivedSignal = awgn(dataMod,snr,'measured'); receivedSignalG = awgn(dataModG,snr,'measured');

The `scatterplot`

function is used to display
the in-phase and quadrature components of the modulated
signal, `dataMod`

, and its received, noisy
version, `receivedSignal`

. By looking at
the resultant diagram, the effects of AWGN are readily
observable.

Use the `scatterplot`

function to show the
constellation diagram.

sPlotFig = scatterplot(receivedSignal,1,0,'g.'); hold on scatterplot(dataMod,1,0,'k*',sPlotFig)

The `qamdemod`

function is used to demodulate
the received data and output integer-valued data
symbols.

Demodulate the received signals using the
`qamdemod`

function.

```
dataSymbolsOut = qamdemod(receivedSignal,M,'bin');
dataSymbolsOutG = qamdemod(receivedSignalG,M);
```

The `de2bi`

function is used to convert the
data symbols from the QAM demodulator,
`dataSymbolsOut`

, into a binary
matrix, `dataOutMatrix`

with dimensions of
N_{sym}-by-N_{bits/sym},
where N_{sym} is the total number of QAM
symbols and N_{bits/sym} is the number
of bits per symbol, four in this case. The matrix is then
converted into a column vector whose length is equal to the
number of input bits, 30,000. The process is repeated for
the Gray coded data symbols,
`dataSymbolsOutG`

.

Reverse the bit-to-symbol mapping performed earlier.

dataOutMatrix = de2bi(dataSymbolsOut,k); dataOut = dataOutMatrix(:); % Return data in column vector dataOutMatrixG = de2bi(dataSymbolsOutG,k); dataOutG = dataOutMatrixG(:); % Return data in column vector

The function `biterr`

is used to calculate
the bit error statistics from the original binary data
stream, `dataIn`

, and the received data
streams, `dataOut`

and
`dataOutG`

.

Use the error rate function to compute the error statistics
and use `fprintf`

to display the
results.

[numErrors,ber] = biterr(dataIn,dataOut); fprintf('\nThe binary coding bit error rate = %5.2e, based on %d errors\n', ... ber,numErrors)

The binary coding bit error rate = 2.40e-03, based on 72 errors

[numErrorsG,berG] = biterr(dataIn,dataOutG); fprintf('\nThe Gray coding bit error rate = %5.2e, based on %d errors\n', ... berG,numErrorsG)

The Gray coding bit error rate = 1.33e-03, based on 40 errors

Observe that Gray coding significantly reduces the bit error rate.

The example in the previous section, Modulate a Random Signal, created a scatter plot from the modulated signal. Although the plot showed the points in the QAM constellation, the plot did not indicate which integers of the modulator are mapped to a given constellation point. This section illustrates two possible mappings: 1) binary coding, and 2) Gray coding. It was previously demonstrated that Gray coding provides superior bit error rate performance.

Apply 16-QAM modulation to all possible input values using the default symbol mapping, binary.

M = 16; % Modulation order x = (0:15)'; % Integer input y1 = qammod(x,16,'bin'); % 16-QAM output

Use the `scatterplot`

function to plot the
constellation diagram and annotate it with binary
representations of the constellation points.

```
scatterplot(y1)
text(real(y1)+0.1, imag(y1), dec2bin(x))
title('16-QAM, Binary Symbol Mapping')
axis([-4 4 -4 4])
```

Apply 16-QAM modulation to all possible input values using Gray-coded symbol mapping.

y2 = qammod(x,16,'gray'); % 16-QAM output, Gray-coded

Use the `scatterplot`

function to plot the
constellation diagram and annotate it with binary
representations of the constellation points.

```
scatterplot(y2)
text(real(y2)+0.1, imag(y2), dec2bin(x))
title('16-QAM, Gray-coded Symbol Mapping')
axis([-4 4 -4 4])
```

In the binary mapping plot, notice that symbols 1 (```
0
0 0 1
```

) and 2 (`0 0 1 0`

)
correspond to adjacent constellation points on the left side
of the diagram. The binary representations of these integers
differ by two bits unlike the Gray-coded signal
constellation in which each point differs by only one bit
from its direct neighbors, which leads to better BER
performance.

The Modulate a Random Signal example was modified to employ a
pair of square-root raised cosine (RRC) filters to perform pulse
shaping and matched filtering. The filters are created by the
`rcosdesign`

function. In Error Correction using a Convolutional Code, this example is extended by
introducing forward error correction (FEC) to improve BER
performance.

To create a BER simulation, a modulator, demodulator, communication channel, and error counter functions must be used and certain key parameters must be specified. In this case, 16-QAM modulation is used in an AWGN channel.

Set the simulation parameters.

M = 16; % Size of signal constellation k = log2(M); % Number of bits per symbol numBits = 3e5; % Number of bits to process numSamplesPerSymbol = 4; % Oversampling factor

Set the square-root, raised cosine filter parameters.

span = 10; % Filter span in symbols rolloff = 0.25; % Roloff factor of filter

Create a square-root, raised cosine filter using the
`rcosdesign`

function.

rrcFilter = rcosdesign(rolloff, span, numSamplesPerSymbol);

Display the RRC filter impulse response using the
`fvtool`

function.

fvtool(rrcFilter,'Analysis','Impulse')

Use the `randi`

function to generate random
binary data. The `rng`

function should be
set to its default state so that the example produces
repeatable results.

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

Reshape the input vector into a matrix of 4-bit binary data, which is then converted into integer symbols.

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

Apply 16-QAM modulation using
`qammod`

.

dataMod = qammod(dataSymbolsIn, M);

Using the `upfirdn`

function, upsample and
apply the square-root, raised cosine filter.

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

The `upfirdn`

function upsamples the
modulated signal, `dataMod`

, by a factor of
`numSamplesPerSymbol`

, pads the
upsampled signal with zeros at the end to flush the filter
and then applies the filter.

Set the
*E _{b}/N_{0}*
to 10 dB and convert the SNR given the number of bits per
symbol,

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

Pass the filtered signal through an AWGN channel.

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

Filter the received signal using the square-root, raised cosine filter and remove a portion of the signal to account for the filter delay in order to make a meaningful BER comparison.

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

These functions apply the same square-root raised cosine
filter that the transmitter used earlier, and then
downsample the result by a factor of
`nSamplesPerSymbol`

. The last
command removes the first `Span`

symbols
and the last `Span`

symbols in the
decimated signal because they represent the cumulative delay
of the two filtering operations. Now
`rxFiltSignal`

, which is the input
to the demodulator, and `dataSymbolsOut`

,
which is the output from the modulator, have the same vector
size. In the part of the example that computes the bit error
rate, it is required to compare vectors that have the same
size.

Apply 16-QAM demodulation to the received, filtered signal.

dataSymbolsOut = qamdemod(rxFiltSignal, M);

Using the `de2bi`

function, convert the
incoming integer symbols into binary data.

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

Apply the `biterr`

function to determine
the number of errors and the associated BER.

[numErrors, ber] = biterr(dataIn, dataOut); fprintf('\nThe bit error rate = %5.2e, based on %d errors\n', ... ber, numErrors)

The bit error rate = 1.83e-03, based on 550 errors

Create an eye diagram for a portion of the filtered signal.

eyediagram(txSignal(1:2000),numSamplesPerSymbol*2);

The `eyediagram`

function creates an eye
diagram for part of the filtered noiseless signal. This
diagram illustrates the effect of the pulse shaping. Note
that the signal shows significant intersymbol interference
(ISI) because the filter is a square-root raised cosine
filter, not a full raised cosine filter.

Created a scatter plot of the received signal before and after filtering.

Notice that the first `scatterplot`

command
scales `rxSignal`

by
`sqrt(numSamplesPerSymbol)`

when
plotting. This is because the filtering operation changes
the signal's power.

Building upon the Pulse Shaping Using a Raised Cosine Filter example, this example shows how bit error rate performance improves with the addition of forward error correction, FEC, coding.

To create the simulation, a modulator, demodulator, raised cosine filter pair, communication channel, and error counter functions are used and certain key parameters are specified. In this case, a 16-QAM modulation scheme with raised cosine filtering is used in an AWGN channel. With the exception of the number of bits, the specified parameters are identical to those used in the previous example.

Set the simulation variables. The number of bits is increased from the previous example so that the bit error rate may be estimated more accurately.

M = 16; % Size of signal constellation k = log2(M); % Number of bits per symbol numBits = 100000; % Number of bits to process numSamplesPerSymbol = 4; % Oversampling factor

Use the `randi`

function to generate
random, binary data once the `rng`

function has been called. When set to its default value, the
`rng`

function ensures that the
results from this example are repeatable.

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

The performance of the Pulse Shaping Using a Raised Cosine Filter example can be significantly improved upon by employing forward error correction. In this example, convolutional coding is applied to the transmitted bit stream in order to correct errors arising from the noisy channel. Because it is often implemented in real systems, the Viterbi algorithm is used to decode the received data. A hard decision algorithm is used, which means that the decoder interprets each input as either a “0” or a “1”.

Define a convolutional coding trellis for a rate 2/3 code. The
`poly2trellis`

function defines
the trellis that represents the convolutional code that
`convenc`

uses for encoding the
binary vector, `dataIn`

. The two input
arguments of the `poly2trellis`

function
indicate the code’s constraint length and generator
polynomials, respectively.

tPoly = poly2trellis([5 4],[23 35 0; 0 5 13]); codeRate = 2/3;

Encode the input data using the previously created trellis.

dataEnc = convenc(dataIn, tPoly);

The encoded binary data is converted into an integer format so that 16-QAM modulation can be applied.

Reshape the input vector into a matrix of 4-bit binary data, which is then converted into integer symbols.

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

Apply 16-QAM modulation.

dataMod = qammod(dataSymbolsIn, M);

As in the Pulse Shaping Using a Raised Cosine Filter example, RRC filtering
is applied to the modulated signal before transmission. The
example makes use of the `rcosdesign`

function to create the filter and the
`upfirdn`

function to filter the
data.

Specify the filter span and rolloff factor for the square-root, raised cosine filter.

span = 10; % Filter span in symbols rolloff = 0.25; % Roloff factor of filter

Create the filter using the `rcosdesign`

function.

rrcFilter = rcosdesign(rolloff, span, numSamplesPerSymbol);

Using the `upfirdn`

function, upsample and
apply the square-root, raised cosine filter.

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

Calculate the signal-to-noise ratio, SNR, based on the input
*E _{b}/N_{0}*,
the number of samples per symbol, and the code rate.
Converting from

`codeRate`

information bits. More concretely, three symbols correspond
to 12 coded bits in 16-QAM, which correspond to 8 uncoded
(information) bits.EbNo = 10; snr = EbNo + 10*log10(k*codeRate)-10*log10(numSamplesPerSymbol);

Pass the filtered signal through an AWGN channel.

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

Filter the received signal using the RRC filter and remove a portion of the signal to account for the filter delay in order to make a meaningful BER comparison.

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

Demodulate the received, filtered signal using the
`qamdemod`

function.

dataSymbolsOut = qamdemod(rxFiltSignal, M);

Use the `de2bi`

function to convert the
incoming integer symbols into bits.

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

Decode the convolutionally encoded data with a Viterbi
decoder. The syntax for the `vitdec`

function instructs it to use hard decisions. The 'cont'
argument instructs it to use a mode designed for maintaining
continuity when the function is repeatedly invoked (as in a
loop). 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), ... % Decode data tPoly,traceBack,'cont','hard');

Using the `biterr`

function, compare
`dataIn`

and
`dataOut`

to obtain the number of
errors and the bit error rate while taking the decoding
delay into account. The continuous operation mode of the
Viterbi decoder incurs a delay whose duration in bits equals
the traceback length, `traceBack`

, times
the number of input streams at the encoder. For this rate
2/3 code, 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 bit error rate, the first
2×`traceBack`

bits in
`dataOut`

and the last
2×`traceBack`

bits in the
original vector, `dataIn`

, are discarded.
Without delay compensation, the BER computation is
meaningless.

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

The bit error rate = 6.00e-05, based on 6 errors

It can be seen that for the same
E_{b}/N_{0}
of 10 dB, the number of errors when using FEC is reduced as
the BER is improves from
2.0×10^{-3} to
6.9×10^{-4}.

The decoding operation in this example incurs a delay, which means that the output of the decoder lags the input. Timing information does not appear explicitly in the example, and the duration of the delay depends on the specific operations being performed. Delays occur in various communications-related operations, including convolutional decoding, convolutional interleaving/deinterleaving, equalization, and filtering. To find out the duration of the delay caused by specific functions or operations, refer to the specific documentation for those functions or operations. For example:

The

`vitdec`

reference page