comm.RSEncoder

Encode data using Reed-Solomon encoder

Description

The comm.RSEncoder System object™ creates a Reed-Solomon (RS) code with message and codeword lengths that you specify. For more information, see Algorithms.

To encode data using an RS-coding scheme:

Create the comm.RSEncoder object and set its properties.
Call the object with arguments, as if it were a function.

To learn more about how System objects work, see What Are System Objects?

Creation

Syntax

enc = comm.RSEncoder

enc = comm.RSEncoder(N,K)

enc = comm.RSEncoder(N,K,GP)

enc = comm.RSEncoder(N,K,GP,S)

enc = comm.RSEncoder(___,Name=Value)

Description

enc = comm.RSEncoder creates an RS encoder System object, enc, that applies RS encoding.

example

enc = comm.RSEncoder(N,K) sets CodewordLength to N and MessageLength to K. For example, comm.RSEncoder(7,3) creates a (7,3) RS encoder object.

enc = comm.RSEncoder(N,K,GP) additionally sets GeneratorPolynomial to GP and GeneratorPolynomialSource to 'Property'.

enc = comm.RSEncoder(N,K,GP,S) additionally sets ShortMessageLength to S and ShortMessageLengthSource to 'Property'.

example

enc = comm.RSEncoder(___,Name=Value) specifies options using one or more name-value arguments in addition to the input arguments in previous syntaxes. For example, comm.RSEncoder(BitInput="true") creates an RS encoder object configured to encode an input column vector of bits.

example

Properties

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Unless otherwise indicated, properties are nontunable, which means you cannot change their values after calling the object. Objects lock when you call them, and the release function unlocks them.

If a property is tunable, you can change its value at any time.

For more information on changing property values, see System Design in MATLAB Using System Objects.

`BitInput` — Option to input bits
`false` or `0` (default) | `true` or `1`

Option to input bits, specified as a logical 0 (false) or 1 (true).

When you set this property to false, the input X must be a numeric, column vector of integers with a length equal to an integer multiple of MessageLength. Each group of MessageLength input elements represents one message word that the object encodes. The object outputs encoded data, Y, as a column vector of integers whose length is an integer multiple of (CodewordLength – N_punctures). Each symbol that forms the input message and output codeword is an integer in the range [0, (2^M – 1)]. These integers correspond to an element of the finite Galois field gf(2^M).
When you set this property to true, X must be a column vector of bits with an integer multiple of (MessageLength×M) bits. A group of M bits represents an integer in the range [0, (2^M – 1)] that belongs to the finite Galois field gf(2^M). The data type of input X must be a numeric, logical, or unsigned fixed point of word length 1 (fi (Fixed-Point Designer) object). Each group of (MessageLength×M) input bits represents one message word that the object encodes. The object outputs encoded data, Y, as a column vector of integers whose length is an integer multiple of ((CodewordLength – N_punctures)×M).

N_punctures is the number of punctures. You specify the number of punctures with PuncturePatternSource and PuncturePattern. M is the degree of the primitive polynomial that you specify with PrimitivePolynomialSource and PrimitivePolynomial.

For more information, see Input and Output Signal Lengths in BCH and RS System Objects.

Data Types: logical

`CodewordLength` — Codeword length of RS code
7 (default) | positive integer

Codeword length of the RS code in symbols, specified as a positive integer.

If you set PrimitivePolynomialSource to "Auto", CodewordLength must be in the range (3, 2¹⁶ – 1].
If you set PrimitivePolynomialSource to "Property", CodewordLength must be in the range [3, 2^M – 1], where M is the degree of the primitive polynomial that you specify with PrimitivePolynomialSource and PrimitivePolynomial. M must be in the range [3, 16]. The difference (CodewordLength – MessageLength) must be an even integer.

For a full-length RS code, CodewordLength must be (2^M – 1). If CodewordLength is less than (2^M – 1), the object assumes a shortened RS code.

For more information, see Input and Output Signal Lengths in BCH and RS System Objects.

`MessageLength` — Message length
`3` (default) | positive integer

Message length in symbols, specified as a positive integer.

`ShortMessageLengthSource` — Source of short message length
`"Auto"` (default) | `"Property"`

Source of the short message length, specified as "Auto" or "Property".

When you set this property to "Auto", the object derives the shortening from CodewordLength and MessageLength. The object defines the RS code based on the CodewordLength, MessageLength, GeneratorPolynomial, and PrimitivePolynomial properties.
When you set this property to "Property", the object uses ShortMessageLength.

`ShortMessageLength` — Shortened message length
`3` (default) | positive integer

Shortened message length in symbols, specified as a positive integer whose value must be less than or equal to MessageLength. When ShortMessageLength < MessageLength, the object shortens the RS code.

Dependencies

This property applies when you set ShortMessageLengthSource to "Property".

`GeneratorPolynomialSource` — Source of generator polynomial
`"Auto"` (default) | `"Property"`

Source of the generator polynomial, specified as "Auto" or "Property".

When you set this property to "Auto", the object calculates the generator polynomial based on the value of PrimitivePolynomialSource and PrimitivePolynomial.
- When you set PrimitivePolynomialSource to "Auto", the object computes the generator polynomial as rsgenpoly(CodewordLength + SL, MessageLength + SL).
- When you set PrimitivePolynomialSource to "Property", the object computes the generator polynomial as rsgenpoly(CodewordLength + SL, MessageLength + SL, PrimitivePolynomial).
- In both cases, SL = (2^M – 1) – CodewordLength is the shortening length and M is the degree of the primitive polynomial you specify with PrimitivePolynomialSource and PrimitivePolynomial.
When you set this property to "Property", the object uses GeneratorPolynomial to define the generator polynomial.

`GeneratorPolynomial` — Generator polynomial
`[1 3 1 2 3]` (default) | integer row vector | `gf` row vector | character vector | string scalar

Generator polynomial, specified as a polynomial string, integer row vector, or Galois field row vector. The length of the generator polynomial must be (CodewordLength – MessageLength + 1).

Galois field row vector entries must be in the range [0, 2^M – 1] and represent a generator polynomial in descending order of powers. Each coefficient must be an element of the Galois field gf(2^M), represented in integer format.

Express the polynomial as one of these options:

Character vector or string scalar of a polynomial whose constant term is 1. For more information, see Representation of Polynomials in Communications Toolbox.
Binary-valued row vector that represents the coefficients of the polynomial in order of descending powers. The length of this vector must be M + 1, where M is the degree of the polynomial. The first and last entries must be 1, indicating the leading term with degree M and a constant term of 1.
Integer-valued row vector of elements that represent the exponents for the nonzero terms of the polynomial in order of descending powers. The last entry must be 0, indicating a constant term of 1.

The default is the result of rsgenpoly(7,3,[],[],'double'), which corresponds to [1 3 1 2 3].

Dependencies

This property applies when you set GeneratorPolynomialSource to "Property".

Data Types: double | string | char

`CheckGeneratorPolynomial` — Option to check generator polynomial
`true` or `1` (default) | `false` or `0`

Option to check the generator polynomial, specified as a logical 1 (true) or 0 (false). Set this property to true to confirm that X^N + 1 is divisible by the generator polynomial you specify in GeneratorPolynomial. N is CodewordLength.

A valid generator polynomial is given by (X – α^B)×(X – α^{(B +
1)})×...×(X – α^{(B + N –
K – 1)}). Where α is a root of the primitive polynomial, B is an integer, N is CodewordLength, and K is MessageLength. If the value of B is 1, then you can set this property to false. For more information about B, see the rsgenpoly.

Tips

For codes with generator polynomials of high degree, disabling the check speeds up processing. As a best practice, perform the check at least once before setting this property to false.

Dependencies

This property applies when GeneratorPolynomialSource is set to "Property".

Data Types: logical

`PrimitivePolynomialSource` — Source of primitive polynomial
`"Auto"` (default) | `"Property"`

Source of the primitive polynomial, specified as "Auto" or "Property".

When you set this property to "Auto", the object uses a primitive polynomial of degree M = ceil(log2(CodewordLength + 1)).
When you set this property to "Property", the object uses PrimitivePolynomial to define the primitive polynomial.

`PrimitivePolynomial` — Primitive polynomial
`[1 0 1 1]` (default) | binary row vector | character vector | string scalar

Primitive polynomial, specified as a binary row vector that defines the finite field gf(2^M) in descending order of powers corresponding to the integers that form messages and codewords.

Express the polynomial as one of these options:

Character vector or string scalar of a polynomial whose constant term is 1. For more information, see Representation of Polynomials in Communications Toolbox.
Binary-valued row vector that represents the coefficients of the polynomial in order of descending powers. The length of this vector must be M + 1, where M is the degree of the polynomial. The first and last entries must be 1, indicating the leading term with degree M and a constant term of 1.
Integer-valued row vector of elements that represent the exponents for the nonzero terms of the polynomial in order of descending powers. The last entry must be 0, indicating a constant term of 1.

The default is the result of int2bit(primpoly(3),4)', which is [1 0 1 1] or the polynomial x³+x+1.

Dependencies

This property applies when you set PrimitivePolynomialSource to "Property".

`PuncturePatternSource` — Source of puncture pattern
`"None"` (default) | `"Property"`

Source of the puncture pattern, specified as "None" or "Property".

When you set this property to "None", the object does not apply puncturing to the code.
When you set this property to "Property", the object punctures the code using the puncture pattern vector defined by PuncturePattern.

`PuncturePattern` — Puncture pattern
`[ones(2,1); zeros(2,1)]` (default) | binary column vector

Puncture pattern, specified as a binary column vector with a length of (N – K), where N is CodewordLength and K is MessageLength. Zeros in the puncture pattern vector indicate the position of the parity symbols that are punctured or excluded from each codeword.

Dependencies

This property applies when you set PuncturePatternSource to "Property".

`OutputDataType` — Output data type
`"Same as input"` (default) | `"double"` | `"logical"`

Output data type, specified as "Same as input", "double", or "logical".

Dependencies

This property applies when you set BitInput to true.

Usage

Syntax

Y = enc(X)

Description

Y = enc(X) encodes the input message using a (,CodewordLength,MessageLength) RS encoder. For information about the input and output lengths when you run the object, see Input and Output Signal Lengths in BCH and RS System Objects.

example

Input Arguments

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`X` — Message to encode
column vector

Message to encode, specified as a column vector of integers or bits based on the setting of BitInput. A group of MessageLength input elements represents one message word to be encoded.

Output Arguments

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`Y` — RS-encoded codeword
binary-valued column vector

RS-encoded codeword, returned as a binary-valued column vector. The output vector length and data type depends on BitInput and OutputDataType.

Data Types: single | logical | fi object

Object Functions

To use an object function, specify the System object as the first input argument. For example, to release system resources of a System object named obj, use this syntax:

release(obj)

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Common to All System Objects

`step`	Run System object algorithm
`release`	Release resources and allow changes to System object property values and input characteristics
`reset`	Reset internal states of System object

Examples

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Transmit RS-Encoded, 8-DPSK-Modulated Symbol Stream

Open Live Script

Transmit an RS-encoded, 8-DPSK-modulated symbol stream through an AWGN channel. After channel filtering demodulate and decode the received signal, and then count errors.

M = 8;
rsEnc = comm.RSEncoder;
rsDec = comm.RSDecoder;
errorRate = comm.ErrorRate(ComputationDelay=3);

for counter = 1:20
    data = randi([0 7], 30, 1);
    encodedData = rsEnc(data);
    modSignal = dpskmod(encodedData,M);
    receivedSignal = awgn(modSignal,10);
    demodSignal = dpskdemod(receivedSignal,M);
    receivedSymbols = rsDec(demodSignal);
    errorStats = errorRate(data,receivedSymbols);
end

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

Error rate = 0.128978
Number of errors = 77

Estimate BER of QPSK in AWGN with Reed-Solomon Coding

Open Live Script

Transmit Reed-Solomon encoded data using QPSK over an AWGN channel. Demodulate and decode the received signal and collect error statistics. Compute the theoretical bit error rate (BER) for coded and noncoded data. Plot the BER results to compare performance.

Define the example parameters.

rng(1993);     % Seed random number generator for repeatable results
M = 4;         % Modulation order
bps = log2(M); % Bits per symbol
N = 7;         % RS codeword length
K = 5;         % RS message length

Create AWGN channel and error rate objects.

awgnChannel = comm.AWGNChannel( ...
    BitsPerSymbol=bps);
errorRate = comm.ErrorRate;

Create a (7,5) Reed-Solomon encoder and decoder pair that accepts bit inputs.

rsEncoder = comm.RSEncoder( ...
    BitInput=true, ...
    CodewordLength=N, ...
    MessageLength=K);
rsDecoder = comm.RSDecoder( ...
    BitInput=true, ...
    CodewordLength=N, ...
    MessageLength=K);

Set the range of $E_{b} / N_{0}$ values and account for RS coding gain. Initialize the error statistics matrix.

ebnoVec = (3:0.5:8)';
ebnoVecCodingGain = ...
    ebnoVec + 10*log10(K/N); % Account for RS coding gain
errorStats = zeros(length(ebnoVec),3);

Estimate the bit error rate for each $E_{b} / N_{0}$ value. The simulation runs until either 100 errors or $1 0^{7}$ bits is encountered. The simulation loop encodes and modulates random data, filters the signal through an AWGN channel, demodulate and decodes the signal, and then computes error statistics.

for i = 1:length(ebnoVec)
    awgnChannel.EbNo = ebnoVecCodingGain(i);
    reset(errorRate)
    while errorStats(i,2) < 100 && errorStats(i,3) < 1e7
        data = randi([0 1],1500,1);
        encData = rsEncoder(data);
        modData = pskmod(encData,M,InputType='bit');
        rxSig = awgnChannel(modData);
        rxData = pskdemod(rxSig,M,OutputType='bit');
        decData = rsDecoder(rxData);
        errorStats(i,:) = errorRate(data,decData);
    end
end

Fit a curve to the BER data using berfit. Generate an estimate of QPSK performance with and without coding by using the bercoding and berawgn functions.

berCurveFit = berfit(ebnoVecCodingGain,errorStats(:,1));
berwCoding = bercoding(ebnoVec,'RS','hard',N,K,'psk',M,'nondiff');
berNoCoding = berawgn(ebnoVec,'psk',M,'nondiff');

Plot the RS-coded BER data, curve fit of the BER data, theoretical performance with RS coding, and theoretical performance without RS coding. The (7,5) RS code improves the $E_{b} / N_{0}$ required to achieve a $1 0^{- 2}$ bit error rate by approximately 1.2 dB.

semilogy(ebnoVecCodingGain,errorStats(:,1),'b*', ...
    ebnoVecCodingGain,berCurveFit,'c-', ...
    ebnoVecCodingGain,berwCoding,'r', ...
    ebnoVec,berNoCoding)
ylabel('BER')
xlabel('Eb/No (dB)')
legend( ...
    'RS-coded BER','Curve fit', ...
    'Theory with coding','Theory no coding')
grid

Figure contains an axes object. The axes object with xlabel Eb/No (dB), ylabel BER contains 4 objects of type line. One or more of the lines displays its values using only markers These objects represent RS-coded BER, Curve fit, Theory with coding, Theory no coding.

Transmit Shortened, RS-Encoded, 256-QAM-Modulated Symbol Stream

Open Live Script

Transmit a shortened, RS-encoded, 256-QAM-modulated symbol stream through an AWGN channel. Then demodulate, decode, and count errors.

Set the parameters for the Reed-Solomon code, where N is the codeword length, K is the nominal message length, and S is the shortened message length. Set the modulation order, M, and the number of frames, L.

N = 255;
K = 239;
S = 188;
M = 256;
L = 50;
bps = log2(M);

Create an AWGN channel System object and an error rate System object.

awgnChan = comm.AWGNChannel( ...
    NoiseMethod='Signal to noise ratio (Eb/No)', ...
    EbNo=15, ...
    BitsPerSymbol=bps);
errorRate = comm.ErrorRate(ComputationDelay=3);

Create the Reed-Solomon generator polynomial from the DVB-T standard.

gp = rsgenpoly(N,K,[],0);

Create a Reed-Solomon encoder and decoder pair using the shortened message length, S, and the DVB-T generator polynomial, gp.

enc = comm.RSEncoder(N,K,gp,S);
dec = comm.RSDecoder(N,K,gp,S);

Generate random symbol frames whose length equals one message block. Encode, modulate, apply AWGN, demodulate, decode, and collect statistics.

for counter = 1:L
    data = randi([0 1],S*bps,1);
    encodedData = enc(bit2int(data,bps));
    modSignal = qammod(encodedData,M,UnitAveragePower=true);
    rxSignal = awgnChan(modSignal);
    demodSignal = qamdemod(rxSignal,M,UnitAveragePower=true);
    rxBits = dec(demodSignal);
    dataOut = int2bit(rxBits,bps);
    errorStats = errorRate(data(:),dataOut(:));
end

Display the error rate and number of errors.

fprintf('Error rate = %5.2e\nNumber of errors = %d\n', ...
    errorStats(1), errorStats(2))

Error rate = 2.01e-02
Number of errors = 1514

Reed-Solomon Coding with Erasures

Open Live Script

This example shows how to configure the comm.RSEncoder and comm.RSDecoder System objects to perform Reed-Solomon (RS) block coding with erasures when simulating a communications system. A codeword with erasures can have those erasures in either message symbols or parity symbols. A punctured codeword has only parity symbols removed, and a shortened codeword has only message symbols removed.

RS decoders can correct both errors and erasures. A receiver that identifies the most unreliable symbols in a given codeword can generate symbol erasures and replace those symbols with zeros. When sending a symbol to the decoder, the receiver passes a flag to the decoder to indicate if the symbol is an erasure rather than a valid code symbol. In a similar fashion, the encoder can apply a puncture pattern that always removes specific parity symbols from its output. Given the puncture pattern, the decoder inserts zeros in the puncture positions, and treats those symbols the same way it treats erasures. Punctured codes offer code rate flexibility, but at the expense of some error correction capability. For the same demodulator input energy per bit to noise power spectral density ratio ( $E_{b} / N_{0}$ ), shortened codes, however, achieve the same code rate flexibility without degrading the error correction performance.

In this example, you simulate a communications system that consists of a random source, an RS encoder, a 64-QAM modulator, an AWGN channel, a 64-QAM demodulator, and an RS decoder. You analyze the RS coding with erasures by comparing the channel bit error rate (BER) performance versus the coded BER performance. You obtain the channel BER by comparing inputs for the QAM modulator to outputs from the QAM demodulator and the coded BER by comparing inputs for the RS encoder to outputs from the RS decoder.

Initialization

The helperRSCodingConfig.m helper function initializes the simulation parameters and configures the comm.AWGNChannel and comm.ErrorRate System objects that simulate the communications system. The helper function also sets the uncoded $E_{b} / N_{0}$ ratio (EbNoUncoded) to 15 dB and the simulation stop criteria for 500 errors or a maximum $5 \times 10^{6}$ bits are transmitted.

helperRSCodingConfig;

Configure RS Encoder/Decoder

This example uses a (63,53) RS code operating with a 64-QAM modulation scheme. This code can correct (63 – 53)/2 = 5 errors, or it can alternatively correct (63 – 53) = 10 erasures. For each codeword at the output of the 64-QAM demodulator, the receiver determines the six least reliable symbols using the helperRSCodingGetErasures.m helper function. The indices that point to the location of these unreliable symbols are an input to the RS decoder. The RS decoder treats these symbols as erasures resulting in an error correction capability of (10 – 6)/2 = 2 errors per codeword.

Create a comm.RSEncoder System object and set the BitInput property to false so that the encoder inputs and outputs are integer symbols.

N = 63;  % Codeword length
K = 53;  % Message length
rsEncoder = comm.RSEncoder(N,K,BitInput=false);
numErasures = 6;

Create a comm.RSDecoder System object that matches the configuration of the comm.RSEncoder object.

rsDecoder = comm.RSDecoder(N,K,BitInput=false);

Set the ErasuresInputPort property to true to specify erasures as an input to the decoder object.

rsDecoder.ErasuresInputPort = true;

Set the NumCorrectedErrorsOutputPort property to true so that the decoder outputs the number of corrected errors. A nonnegative value in the error output denotes the number of corrected errors in the input codeword. A value of –1 in the error output indicates a decoding error. A decoding error occurs when the input codeword has more errors than the error correction capability of the RS code.

rsDecoder.NumCorrectedErrorsOutputPort = true;

Stream Processing Loop

Simulate the communications system for an uncoded $E_{b} / N_{0}$ ratio of 15 dB. The uncoded $E_{b} / N_{0}$ is the ratio that would be measured at the input of the channel if there were no coding in the system.

EbNoCoded = EbNoUncoded + 10*log10(K/N);
channel.EbNo = EbNoCoded;

Loop the simulation until it reaches the target number of errors or the maximum number of transmissions.

chanErrorStats = zeros(3,1);
codedErrorStats = zeros(3,1);
correctedErrors = 0;
while (codedErrorStats(2) < targetErrors) && ...
        (codedErrorStats(3) < maxNumTransmissions)

The data symbols transmit one message word at a time. Each message word has K symbols in the range [0, N].

    data = randi([0 N],K,1);

Encode the message word. The encoded word, encData, is (N – numPunc) symbols long.

    encData = rsEncoder(data);

Modulate encoded data and add noise. Then demodulate channel output.

    modData = qammod(encData,M);
    chanOutput = channel(modData);
    demodData = qamdemod(chanOutput,M);

Use the helperRSCodingGetErasures.m helper function to find the six least reliable symbols and generate an erasures vector. The length of the erasures vector must be equal to the number of symbols in the demodulated codeword. A one in the ith element of the vector erases the ith symbol in the codeword. Zeros in the vector indicate no erasures.

    erasuresVec = helperRSCodingGetErasures(chanOutput,numErasures);

Decode the data. Log the number of errors corrected by the RS decoder.

    [estData,errs] = rsDecoder(demodData,erasuresVec);
    if (errs >= 0)
        correctedErrors = correctedErrors + errs;
    end

When computing the channel and coded BERs, convert integers to bits.

    chanErrorStats(:,1) = ...
        chanBERCalc(int2bit(encData,log2(M)), ...
            int2bit(demodData,log2(M)));
    codedErrorStats(:,1) = ...
        codedBERCalc(int2bit(data,log2(M)), ...
            int2bit(estData,log2(M)));
end

The error rate measurement objects, chanBERCalc and codedBERCalc, output 3-by-1 vectors containing BER measurement updates, the number of errors, and the total number of bit transmissions. Display the channel BER, the coded BER and the total number of errors corrected by the RS decoder.

chanBitErrorRate = chanErrorStats(1)

chanBitErrorRate = 
0.0017

codedBitErrorRate = codedErrorStats(1)

codedBitErrorRate = 
0

totalCorrectedErrors = correctedErrors

totalCorrectedErrors = 
882

To run the simulation for a set of $E_{b} / N_{0}$ values, you can add a for-loop around the processing loop above. For a simulation with uncoded $E_{b} / N_{0}$ values in the range 4:15 dB, a target number of errors equal to 5000, and maximum number of transmissions equal to $50 \times 10^{6}$ , the results in the figure show that the channel BER is worse than the theoretical 64-QAM BER because $E_{b} / N_{0}$ is reduced by the code rate.

Helper Functions

Reed-Solomon Coding with Erasures and Punctures

Open Live Script

This example shows how to configure the comm.RSEncoder and comm.RSDecoder System objects to perform Reed-Solomon (RS) block coding with erasures and puncture codes when simulating a communications system. An encoder can generate punctures to remove specific parity symbols from its output. Given the puncture pattern, the decoder inserts zeros in the puncture positions and treats those symbols as erasures. The decoder treats encoder-generated punctures and receiver-generated erasures in exactly the same way when it decodes. Puncturing provides code rate flexibility, but at the expense of some error correction capability.

In this example, you simulate a communications system that consists of a random source, an RS encoder, a 64-QAM modulator, an AWGN channel, a 64-QAM demodulator, and an RS decoder. You analyze the RS coding with erasures and puncturing by comparing the channel bit error rate (BER) performance versus the coded BER performance. You obtain the channel BER by comparing inputs for the QAM modulator to outputs from the QAM demodulator and the coded BER by comparing inputs for the RS encoder to outputs from the RS decoder.

Initialization

helperRSCodingConfig;

Configure RS Encoder/Decoder

This example uses the same (63,53) RS code operating with a 64-QAM modulation scheme that is configured for erasures and code puncturing. The RS algorithm decodes receiver-generated erasures and corrects encoder-generated punctures. For each codeword, the sum of the punctures and erasures cannot exceed twice the error-correcting capability of the code.

Create a comm.RSEncoder System object and set the BitInput property to false so that the encoder inputs and outputs are integer symbols.

N = 63;  % Codeword length
K = 53;  % Message length
rsEncoder = comm.RSEncoder(N,K,BitInput=false);
numErasures = 6;

Create a comm.RSDecoder System object that matches the configuration of the comm.RSEncoder object. Then set the ErasuresInputPort property to true to specify erasures as an input to the decoder object.

rsDecoder = comm.RSDecoder(N,K,BitInput=false);
rsDecoder.ErasuresInputPort = true;

To enable code puncturing, set the PuncturePatternSource property to 'Property' and set the PuncturePattern property to the desired puncture pattern vector. The same puncture vector must be specified in both the encoder and decoder. This example punctures two symbols from each codeword. Values of 1 in the puncture pattern vector indicate nonpunctured symbols, and values of 0 indicate punctured symbols.

numPuncs = 2;
rsEnc.PuncturePatternSource = 'Property';
rsEnc.PuncturePattern = [ones(N-K-numPuncs,1); zeros(numPuncs,1)];

rsDec.PuncturePatternSource = 'Property';
rsDec.PuncturePattern = rsEnc.PuncturePattern;

Stream Processing Loop

Since the signal going into the AWGN channel is the encoded signal, you must convert the uncoded $E_{b} / N_{0}$ values so that they correspond to the energy ratio at the encoder output. This ratio is the coded $E_{b} / N_{0}$ ratio. If you input K symbols to the encoder and obtain N output symbols, then the energy relation is given by the K/N rate. Since the length of the codewords generated by the RS encoder is reduced by the number of punctures specified in the puncture pattern vector, the value of the coded $E_{b} / N_{0}$ ratio needs to be adjusted to account for these punctures. In this example, the number of output symbols is (N – numPuncs) and the uncoded $E_{b} / N_{0}$ ratio relates to the coded $E_{b} / N_{0}$ as shown below. Set the EbNo property of the AWGN channel object to the computed coded $E_{b} / N_{0}$ value.

EbNoCoded = EbNoUncoded + 10*log10(K/(N - numPuncs));
channel.EbNo = EbNoCoded;

Loop the simulation until it reaches the target number of errors or the maximum number of transmissions.

chanErrorStats = zeros(3,1);
codedErrorStats = zeros(3,1);
correctedErrors = 0;
while (codedErrorStats(2) < targetErrors) && ...
        (codedErrorStats(3) < maxNumTransmissions)

The data symbols transmit one message word at a time. Each message word has K symbols in the range [0, N].

    data = randi([0 N],K,1);

Encode the message word. The encoded word, encData, is (N – numPunc) symbols long.

    encData = rsEncoder(data);

Modulate encoded data and add noise. Then demodulate channel output.

    modData = qammod(encData,M);
    chanOutput = channel(modData);
    demodData = qamdemod(chanOutput,M);

Use the helperRSCodingGetErasures.m helper function to find the six least reliable symbols and generate an erasures vector. The length of the erasures vector must be equal to the number of symbols in the demodulated codeword. A one in the ith element of the vector erases the ith symbol in the codeword. Zeros in the vector indicate no erasures.

    erasuresVec = helperRSCodingGetErasures(chanOutput,numErasures);

Decode the data. Log the number of errors corrected by the RS decoder.

    [estData,errs] = rsDecoder(demodData,erasuresVec);
    if (errs >= 0)
        correctedErrors = correctedErrors+errs;
    end

When computing the channel and coded BERs, convert integers to bits.

    chanErrorStats(:,1) = ...
        chanBERCalc(int2bit(encData,log2(M)), ...
            int2bit(demodData,log2(M)));
    codedErrorStats(:,1) = ...
        codedBERCalc(int2bit(data,log2(M)), ...
            int2bit(estData,log2(M)));
end

chanBitErrorRate = chanErrorStats(1)

chanBitErrorRate = 
0.0015

codedBitErrorRate = codedErrorStats(1)

codedBitErrorRate = 
0

totalCorrectedErrors = correctedErrors

totalCorrectedErrors = 
632

RS coding with only erasures
RS coding with erasures and puncturing
Theoretical BER for 64-QAM

The coded $E_{b} / N_{0}$ is slightly higher than the channel $E_{b} / N_{0}$ , so the channel BER is slightly better in the punctured case. On the other hand, the coded BER is worse in the punctured case because the two punctures reduce the error correcting capability of the code by one, leaving it able to correct only (10 – 6 – 2) / 2 = 1 error per codeword.

Help Functions

Reed-Solomon Coding with Erasures, Punctures, and Shortening

Open Live Script

This example shows how to configure the comm.RSEncoder and comm.RSDecoder System objects to perform Reed-Solomon (RS) block coding that shortens a (63,53) code to a (28,18) code. You simulate a communications system that consists of a random source, an RS encoder, a 64-QAM modulator, an AWGN channel, a 64-QAM demodulator, and an RS decoder.

In this example, you analyze the effects of RS coding with erasures, puncturing, and shortening by comparing the channel bit error rate (BER) performance versus the coded BER performance. You obtain the channel BER by comparing inputs for the QAM modulator to outputs from the QAM demodulator and the coded BER by comparing inputs for the RS encoder to outputs from the RS decoder. Puncturing is the removal of parity symbols from a codeword, and shortening is the removal of message symbols from a codeword. Puncturing has the benefit of making the code rate more flexible, at the expense of some error correction capability. Shortened codes achieve the same code rate flexibility without degrading the error correction performance for the same demodulator input $E_{b} / N_{0}$ .

Initialization

helperRSCodingConfig;

Configure RS Encoder/Decoder

This example uses a (63,53) RS code operating with a 64-QAM modulation scheme. The RS coding operation includes erasures, puncturing, and code shortening. This example shows how to shorten the (63,53) code to a (28,18) code.

To shorten a (63,53) code by 10 symbols to a (53,43) code, you can simply enter 53 and 43 for the CodewordLength and MessageLength properties, respectively (since $2 ⌈ \log_{2} (53 + 1) ⌉ - 1 = 63$ ). However, to shorten it by 35 symbols to a (28,18) code, you must explicitly specify that the symbols belong to the Galois field GF(26). Otherwise, the RS System objects assume that the code is shortened from a (31,21) code (since $2 ⌈ \log_{2} (28 + 1) ⌉ - 1 = 31$ ).

Create a pair of comm.RSEncoder and comm.RSDecoder System objects so that they perform block coding with a (28,18) code shortened from a (63,53) code that is configured to input and output integer symbols. Configure the decoder to accept an erasure input and two punctures. For each codeword at the output of the 64-QAM demodulator, the receiver determines the six least reliable symbols by using the helperRSCodingGetErasures.m helper function. The indices that point to the location of these unreliable symbols are an input to the RS decoder.

N = 63;  % Codeword length
K = 53;  % Message length
S = 18;  % Shortenened message length
numErasures = 6;
numPuncs = 2;
rsEncoder = comm.RSEncoder(N,K, ...
    BitInput=false, ...
    PuncturePatternSource="Property", ...
    PuncturePattern = [ones(N-K-numPuncs,1); zeros(numPuncs,1)]);
rsDecoder = comm.RSDecoder(N,K, ...
    BitInput=false, ...
    ErasuresInputPort=true, ...
    PuncturePatternSource="Property", ...
    PuncturePattern = rsEncoder.PuncturePattern);

Set the shortened codeword length and message length values.

rsEncoder.ShortMessageLength = S;
rsDecoder.ShortMessageLength = S;

Specify the field of ${GF (2}^{6})$ in the RS encoder/decoder System objects, by setting the PrimitivePolynomialSource property to 'Property' and the PrimitivePolynomial property to a sixth degree primitive polynomial.

primPolyDegree = 6;
rsEncoder.PrimitivePolynomialSource = 'Property';
rsEncoder.PrimitivePolynomial = ...
    int2bit(primpoly(primPolyDegree,'nodisplay'),7)';

rsDecoder.PrimitivePolynomialSource = 'Property';
rsDecoder.PrimitivePolynomial = ...
    int2bit(primpoly(primPolyDegree,'nodisplay'),7)';

Stream Processing Loop

Since the signal going into the AWGN channel is the encoded signal, you must convert the uncoded $E_{b} / N_{0}$ values so that they correspond to the energy ratio at the encoder output. This ratio is the coded $E_{b} / N_{0}$ ratio. If you input K symbols to the encoder and obtain N output symbols, then the energy relation is given by the K/N rate. Since the length of the codewords generated by the RS encoder is shortened and reduced by the number of punctures specified in the puncture pattern vector, the value of the coded $E_{b} / N_{0}$ ratio needs to be adjusted to account for the shortened code and the punctures. In this example, the number of output symbols is (N – numPuncs – S) and the uncoded $E_{b} / N_{0}$ ratio relates to the coded $E_{b} / N_{0}$ as shown below. Set the EbNo property of the AWGN channel object to the computed coded $E_{b} / N_{0}$ value.

EbNoCoded = EbNoUncoded + 10*log10(S/(N - numPuncs - K + S));
channel.EbNo = EbNoCoded;

Loop the simulation until it reaches the target number of errors or the maximum number of transmissions.

chanErrorStats = zeros(3,1);
codedErrorStats = zeros(3,1);
correctedErrors = 0;
while (codedErrorStats(2) < targetErrors) && ...
        (codedErrorStats(3) < maxNumTransmissions)

The data symbols transmit one message word at a time. Each message word has (K – S) symbols in the range [0, (2^primPolyDegree) – 1].

    data = randi([0 2^primPolyDegree-1],S,1);

Encode the shortened message word. The encoded word encData is (N – numPuncs – S) symbols long.

    encData = rsEncoder(data);

Modulate encoded data and add noise. Then demodulate channel output.

    modData = qammod(encData,M);
    chanOutput = channel(modData);
    demodData = qamdemod(chanOutput,M);

    erasuresVec = helperRSCodingGetErasures(chanOutput,numErasures);

Decode the data. Log the number of errors corrected by the RS decoder.

    [estData,errs] = rsDecoder(demodData,erasuresVec);
    if (errs >= 0)
        correctedErrors = correctedErrors + errs;
    end

When computing the channel and coded BERs, convert integers to bits.

    chanErrorStats(:,1) = ...
        chanBERCalc(int2bit(encData,log2(M)), ...
            int2bit(demodData,log2(M)));
    codedErrorStats(:,1) = ...
        codedBERCalc(int2bit(data,log2(M)), ...
            int2bit(estData,log2(M)));
end

The error rate measurement objects, chanBERCalc and codedBERCalc, output 3-by-1 vectors containing BER measurement updates, the number of errors, and the total number of bit transmissions. Display the channel BER, the coded BER, and the total number of errors corrected by the RS decoder.

chanBitErrorRate = chanErrorStats(1)

chanBitErrorRate = 
0.0036

codedBitErrorRate = codedErrorStats(1)

codedBitErrorRate = 
4.6399e-05

totalCorrectedErrors = correctedErrors

totalCorrectedErrors = 
1416

RS coding with only erasures
RS coding with erasures and puncturing
RS coding with erasures, puncturing, and shortening
Theoretical BER for 64-QAM

The BER out of the 64-QAM demodulator is worse with shortening than it is without shortening. The code rate of the shortened code is much lower than the code rate of the nonshortened code. Therefore, the coded $E_{b} / N_{0}$ into the demodulator is worse with shortening. A shortened code has the same error correcting capability as nonshortened code for the same $E_{b} / N_{0}$ , but the reduction in $E_{b} / N_{0}$ manifests in the form of a higher BER out of the RS decoder with shortening than without.

The coded $E_{b} / N_{0}$ is slightly higher than the channel $E_{b} / N_{0}$ , so the channel BER is slightly better than the coded BER in the shortened case. The degraded coded $E_{b} / N_{0}$ occurs because the code rate of the shortened code is lower than that of the nonshortened code. Shortening results in degraded coded BER, most noticeably at lower $E_{b} / N_{0}$ values.

Helper functions

Algorithms

expand all

This object implements Reed-Solomon (RS) coding as described in [1] and Algorithms for BCH and RS Errors-only Decoding.

Input and Output Signal Lengths in BCH and RS System Objects

The notation y = c * x denotes that y is an integer multiple of x.

The number of punctures equals the number of zeros in the puncture vector.

M is the degree of the primitive polynomial. Each group of M bits represents an integer in the range [0, (2^M–1)] that belongs to the finite Galois field gf(2^M).

ShortMessageLengthSource "Auto" "Property"

`ShortMessageLengthSource`	`"Auto"`	`"Property"`
`comm.BCHEncoder` and `comm.RSEncoder(BitInput=false)`	Input Length: `c * MessageLength` Output Length: `c * (CodewordLength – number of punctures)`	Input Length: `c * ShortMessageLength` Output Length: `c * (CodewordLength - MessageLength + ShortMessageLength - number of punctures)`
`comm.BCHDecoder` and `comm.RSDecoder(BitInput=false)`	Input Length: `c * (CodewordLength – number of punctures)` Output Length: `c * MessageLength` Erasures Length: `c * (CodewordLength – number of punctures)`	Input Length: `c * (CodewordLength - MessageLength + ShortMessageLength - number of punctures)` Output Length: `c * ShortMessageLength` Erasures Length: `c * (CodewordLength - MessageLength + ShortMessageLength - number of punctures)`
`comm.RSEncoder(BitInput=true)`	Input Length: `c * (MessageLength * M)` Output Length: `c * ((CodewordLength – number of punctures) * M)`	Input Length: `c * (ShortMessageLength * M)` Output Length: `c * ((CodewordLength - MessageLength + ShortMessageLength - number of punctures) * M)`
`comm.RSDecoder(BitInput=true)`	Input Length: `c * ((CodewordLength – number of punctures) * M)` Output Length: `c * (MessageLength * M)` Erasures Length: `c * (CodewordLength – number of punctures)`	Input Length: `c * ((CodewordLength - MessageLength + ShortMessageLength - number of punctures) * M)` Output Length: `c * (ShortMessageLength * M)` Erasures Length: `c * (CodewordLength - MessageLength + ShortMessageLength - number of punctures)`

comm.BCHEncoder

and

comm.RSEncoder(BitInput=false)

Input Length:

c * MessageLength

Output Length:

c * (CodewordLength – number of punctures)

Input Length:

c * ShortMessageLength

Output Length:

c * (CodewordLength - MessageLength + ShortMessageLength - number of punctures)

comm.BCHDecoder

and

comm.RSDecoder(BitInput=false)

Input Length:

c * (CodewordLength – number of punctures)

Output Length:

c * MessageLength

Erasures Length:

c * (CodewordLength – number of punctures)

Input Length:

c * (CodewordLength - MessageLength + ShortMessageLength - number of punctures)

Output Length:

c * ShortMessageLength

Erasures Length:

c * (CodewordLength - MessageLength + ShortMessageLength - number of punctures)

comm.RSEncoder(BitInput=true)

Input Length:

c * (MessageLength * M)

Output Length:

c * ((CodewordLength – number of punctures) * M)

Input Length:

c * (ShortMessageLength * M)

Output Length:

c * ((CodewordLength - MessageLength + ShortMessageLength - number of punctures) * M)

comm.RSDecoder(BitInput=true)

Input Length:

c * ((CodewordLength – number of punctures) * M)

Output Length:

c * (MessageLength * M)

Erasures Length:

c * (CodewordLength – number of punctures)

Input Length:

c * ((CodewordLength - MessageLength + ShortMessageLength - number of punctures) * M)

Output Length:

c * (ShortMessageLength * M)

Erasures Length:

c * (CodewordLength - MessageLength + ShortMessageLength - number of punctures)

References

[1] Clark, George C., and J. Bibb Cain. Error-Correction Coding for Digital Communications. Applications of Communications Theory. New York: Plenum Press, 1981.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

See System Objects in MATLAB Code Generation (MATLAB Coder).

When you use this object to generate code, you must set the generator polynomial to a double-precision integer row vector.

Version History

Introduced in R2012a

comm.RSEncoder

Description

Creation

Syntax

Description

Properties

BitInput — Option to input bits false or 0 (default) | true or 1

CodewordLength — Codeword length of RS code 7 (default) | positive integer

MessageLength — Message length 3 (default) | positive integer

ShortMessageLengthSource — Source of short message length "Auto" (default) | "Property"

ShortMessageLength — Shortened message length 3 (default) | positive integer

Dependencies

GeneratorPolynomialSource — Source of generator polynomial "Auto" (default) | "Property"

GeneratorPolynomial — Generator polynomial [1 3 1 2 3] (default) | integer row vector | gf row vector | character vector | string scalar

Dependencies

CheckGeneratorPolynomial — Option to check generator polynomial true or 1 (default) | false or 0

Tips

Dependencies

PrimitivePolynomialSource — Source of primitive polynomial "Auto" (default) | "Property"

PrimitivePolynomial — Primitive polynomial [1 0 1 1] (default) | binary row vector | character vector | string scalar

Dependencies

PuncturePatternSource — Source of puncture pattern "None" (default) | "Property"

PuncturePattern — Puncture pattern [ones(2,1); zeros(2,1)] (default) | binary column vector

Dependencies

OutputDataType — Output data type "Same as input" (default) | "double" | "logical"

Dependencies

Usage

Syntax

Description

Input Arguments

X — Message to encode column vector

Output Arguments

Y — RS-encoded codeword binary-valued column vector

Object Functions

Common to All System Objects

Examples

Transmit RS-Encoded, 8-DPSK-Modulated Symbol Stream

Estimate BER of QPSK in AWGN with Reed-Solomon Coding

Transmit Shortened, RS-Encoded, 256-QAM-Modulated Symbol Stream

Reed-Solomon Coding with Erasures

Reed-Solomon Coding with Erasures and Punctures

Reed-Solomon Coding with Erasures, Punctures, and Shortening

Algorithms

Input and Output Signal Lengths in BCH and RS System Objects

References

Extended Capabilities

C/C++ Code Generation Generate C and C++ code using MATLAB® Coder™.

Version History

See Also

Objects

Functions

Topics

`BitInput` — Option to input bits
`false` or `0` (default) | `true` or `1`

`CodewordLength` — Codeword length of RS code
7 (default) | positive integer

`MessageLength` — Message length
`3` (default) | positive integer

`ShortMessageLengthSource` — Source of short message length
`"Auto"` (default) | `"Property"`

`ShortMessageLength` — Shortened message length
`3` (default) | positive integer

`GeneratorPolynomialSource` — Source of generator polynomial
`"Auto"` (default) | `"Property"`

`GeneratorPolynomial` — Generator polynomial
`[1 3 1 2 3]` (default) | integer row vector | `gf` row vector | character vector | string scalar

`CheckGeneratorPolynomial` — Option to check generator polynomial
`true` or `1` (default) | `false` or `0`

`PrimitivePolynomialSource` — Source of primitive polynomial
`"Auto"` (default) | `"Property"`

`PrimitivePolynomial` — Primitive polynomial
`[1 0 1 1]` (default) | binary row vector | character vector | string scalar

`PuncturePatternSource` — Source of puncture pattern
`"None"` (default) | `"Property"`

`PuncturePattern` — Puncture pattern
`[ones(2,1); zeros(2,1)]` (default) | binary column vector

`OutputDataType` — Output data type
`"Same as input"` (default) | `"double"` | `"logical"`

`X` — Message to encode
column vector

`Y` — RS-encoded codeword
binary-valued column vector

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.