Integer-Output RS Decoder

Decode Reed-Solomon code to recover integer vector data

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Libraries:
Communications Toolbox / Error Detection and Correction / Block

Description

The Integer-Output RS Decoder block recovers a message vector from a Reed-Solomon codeword vector. For proper decoding, the parameter values in this block must match those in the corresponding Integer-Input RS Encoder block.

The Reed-Solomon code has message length K, and codeword length N – number of punctures. You specify N and K directly in the block dialog. The symbols for the code are integers in the range [0, 2^M-1], which represent elements of the finite field GF(2^M). Restrictions on M and N are described in Restrictions on the M and the Codeword Length N below.

This icon shows optional ports.

The input and output are integer-valued signals that represent codewords and messages, respectively. For more information, see Input and Output Signal Length in RS Blocks. The block inherits the output data type from the input data type. For information about the data types each block port supports, see Supported Data Types.

For more information on representing data for Reed-Solomon codes, see the section Integer Format (Reed-Solomon Only).

If the decoder is processing multiple codewords per frame, then the same puncture pattern holds for all codewords.

The default value of M is ceil(log2(N+1)), that is, the smallest integer greater than or equal to log2(N+1). You can change the value of M from the default by specifying the primitive polynomial for GF(2^M), as described in Specify the Primitive Polynomial below.

You can also specify the generator polynomial for the Reed-Solomon code, as described in Specify the Generator Polynomial.

An (N, K) Reed-Solomon code can correct up to floor((N-K)/2) symbol errors (not bit errors) in each codeword.

If decoding fails, the message portion of the decoder input is returned unchanged as the decoder output.

The sample times of the input and output signals are equal.

Examples

Reed-Solomon Coding with Erasures, Punctures, and Shortening in Simulink

Configure Reed-Solomon (RS) codes to perform block coding with erasures, punctures, and shortening.

Ports

Input

In — Reed-Solomon codeword

integer column vector

Reed-Solomon codeword, specified as an (N_C×(N – K + S – P)-by-1 integer column vector. N_C is the number of codewords, N is the Codeword length N, K is the Message length K, S is the Shortened message length S, P is the number of punctures per codeword.

For more information, see Input and Output Signal Length in RS Blocks.

Data Types: single | double | integer

Era — Erasure vector

binary column vector

Erasure vector, specified as a binary column vector input signal with the same size as the input Reed-Solomon codeword.

Erasure values of 1 correspond to erased bits in the same position in the codeword. Values of 0 correspond to bits that are not erased. For more information, see Puncturing and Erasures.

Dependencies

To enable this port, select Enable erasures input port.

Data Types: double | Boolean

Output

Out — Decoded message

integer column vector

Decoded message, returned as one of the following:

• When there is no message shortening, a (N_C×K)-by-1 integer column vector.

• When there is message shortening, a (N_C×S)-by-1 integer column vector.

N_C is the number of message words, K is the Message length K (symbols), and S is the Shortened message length S (symbols).

Note

The number of decoded message words equals the number of codewords.

For more information, see Input and Output Signal Length in RS Blocks.

Err — Decoding errors

integer vector

Symbol decoding errors, returned as an integer vector with N_C elements, where N_C is the number of codewords. This port indicates the number of symbol errors detected during decoding of each codeword. A negative integer indicates that the block detected more errors than it could correct by using the specified coding scheme.

Note

An (N,K) Reed-Solomon code can correct up to floor((N-K)/2) symbol errors (not bit errors) in each codeword. When a received codeword contains more than (N-K)/2 symbol errors, a decoding failure occurs.

Dependencies

To enable this port, select Output number of corrected symbol errors.

Data Types: double

For more information, see Supported Data Types.

Parameters

To edit block parameters interactively, use the Property Inspector. From the Simulink^® Toolstrip, on the Simulation tab, in the Prepare gallery, select Property Inspector.

Codeword length N — Codeword length

7 (default) | integer

Codeword length, specified as an integer.

For more information, see Restrictions on the M and the Codeword Length N and Input and Output Signal Length in RS Blocks.

Message length K — Message word length

3 (default) | integer

Message word length, specified as an integer in the range [1, N–2], where N is the codeword length.

Shortened message length S — Shortened message word length

3 (default) | integer

Shortened message word length, specified as an integer, such that S ≤ K. When Shortened message length S < Message length K, the Reed-Solomon code is shortened.

You still specify N and K values for the full-length (N, K) code but the decoding is shortened to an (N–K+S, S) code.

Dependencies

To enable this parameter, select Specify shortened message length.

Generator polynomial — Generator polynomial

rsgenpoly(7, 3, [], [], 'double') (default) | polynomial character vector | binary row vector | binary Galois row vector

Generator polynomial with values in the range [0, 2^M–1], in order of descending power, specified as one of the following:

• A polynomial character vector. For more information, see Representation of Polynomials in Communications Toolbox.

• An integer row vector that represents the coefficients of the generator polynomial in order of descending power.

• An integer Galois row vector that represents the coefficients of the generator polynomial in order of descending power.

Each coefficient is an element of the Galois field defined by the primitive polynomial. For more information, see Specify the Generator Polynomial.

Example: [1 3 1 2 3], which is equivalent to rsgenpoly(7,3)

Dependencies

To enable this parameter, select Specify generator polynomial.

Primitive polynomial — Primitive polynomial

'X^3 + X + 1' (default) | polynomial character vector | binary row vector

Primitive polynomial in order of descending power. This polynomial is of order M and defines the finite Galois field GF(2^M) corresponding to the integers that form message words and codewords. Specify the primitive polynomial as one of the following:

• A polynomial character vector. For more information, see Representation of Polynomials in Communications Toolbox.

• A binary row vector that represents the coefficients of the generator polynomial.

For more information, see Specify the Primitive Polynomial.

Example: 'X^3 + X + 1', which is the primitive polynomial used for a (7,3) code, ppoly = primpoly(3,'nodisplay'); int2bit(ppoly,ceil(log2(max(ppoly))))'

Dependencies

To enable this parameter, select Specify primitive polynomial.

Puncture vector — Puncture vector

[ones(2,1); zeros(2,1)] (default) | binary column vector

Puncture vector, specified as an (N–K)-by-1 binary column vector. Element indices with 1s represent data symbol indices that pass through the block unaltered. Element indices with 0s represent data symbol indices that get punctured, or removed, from the data stream. For more information, see Puncturing and Erasures.

Note

If the encoder is processing multiple codewords per frame, then the same puncture pattern holds for all codewords.

Dependencies

To enable this parameter, select Puncture code.

Enable erasures input port — Enable erasures input port

off (default) | on

Selecting this check box enables the erasures port, Era. For more information, see Puncturing and Erasures.

Output number of corrected symbol errors — Enable port to output number of corrected symbol errors

off (default) | on

Selecting this check box enables an additional output port, Err, which indicates the number of symbol errors the block corrected in the input codeword.

Block Characteristics

Data Types	double | integer | single
Multidimensional Signals	no
Variable-Size Signals	no

More About

Input and Output Signal Length in RS Blocks

The Reed-Solomon code has a message word length, K, or shortened message word length, S. The codeword length is N – K + S – P, where N is the full codeword length and P is the number of punctures per codeword. When there is no message shortening, the codeword length expression reduces to N – P, because K = S. If the decoder is processing multiple codewords per frame, then the same puncture pattern holds for all codewords.

This table provides expressions for the input and output signal lengths for the Reed-Solomon encoder and decoder.

The notation y = N_C × x denotes that y is an integer multiple of x.

	Input, Erasure, and Output Vector Lengths
RS Block Coder	No Message Shortening Used	Message Shortening Used
Integer-Input RS Encoder	Input Length (symbols): NC × K Output Length (symbols): NC × (N–P)	Input Length (symbols): NC × S Output Length (symbols): NC × (N–K+S–P)
Integer-Output RS Decoder	Input Length (symbols): NC × (N–P) Erasures Length (symbols): NC × (N–P) Output Length (symbols): NC × K	Input Length (symbols): NC × (N–K+S–P) Erasures Length (symbols): NC × (N–K+S–P) Output Length (symbols): NC × S

• N is the codeword length.

• K is the message word length.

• S is the shortened message word length.

• N_C is the number of codewords (and message words).

• P is the number of punctures, and is equal to the number of zeros in the puncture vector.

• M is the degree of the primitive polynomial. Each group of M bits represents an integer between 0 and 2M–1 that belongs to the finite Galois field GF(2M).

For more information on representing data for Reed-Solomon codes, see Integer Format (Reed-Solomon Only).

Restrictions on the M and the Codeword Length N

• If you do not select Specify primitive polynomial, valid values for the codeword length, N, are from 7 to 65535. In this case, the block uses the default primitive polynomial of degree M = ceil(log2(N+1)). You can display the default primitive polynomial by running primpoly(ceil(log2(N+1))).

• If you select Specify primitive polynomial, valid values for the primitive polynomial degree, M, are from 3 to 16. The valid values for N in this case are from 7 to 2^M–1. Selecting Specify primitive polynomial enables you to specify the primitive polynomial that defines the finite field GF(2^M), which corresponds to the values that form message words and codewords.

Specify the Primitive Polynomial

You can specify the primitive polynomial that defines the finite field GF(2^M), corresponding to the integers that form messages and codewords. To do so, first select Specify primitive polynomial. Then, in the Primitive polynomial text box, enter a binary row vector that represents a primitive polynomial over GF(2^M), in descending order of powers. For example, to specify the polynomial x³+x+1, enter the vector [1 0 1 1].

If you do not select Specify primitive polynomial, the block uses the default primitive polynomial of degree M = ceil(log2(N+1)). You can display the default polynomial by entering primpoly(ceil(log2(N+1))) at the MATLAB^® prompt.

Specify the Generator Polynomial

Select Specify generator polynomial to enable the Generator polynomial parameter for specifying the generator polynomial of the Reed-Solomon code. Enter an integer row vector with element values from 0 to 2^M-1. The vector represents a polynomial, in descending order of powers, whose coefficients are elements of GF(2^M) represented in integer format. For more information about integer and binary format, see Integer Format (Reed-Solomon Only). The generator polynomial must be equal to a polynomial with this factored form:

g(x) = (x+α^b)(x+α^b+1)(x+α^b+2)...(x+α^b+N-K-1)

α is the primitive element of the Galois field over which the input message is defined, and b is an integer.

If you do not select Specify generator polynomial, the block uses the default generator polynomial, corresponding to b=1, for Reed-Solomon encoding. You can display the default generator polynomial by running rsgenpoly.

• If you are using the default primitive polynomial (Specify primitive polynomial is not selected), the default generator polynomial is rsgenpoly(N,K), where N = 2M-1.

• If you are not using the default primitive polynomial (Specify primitive polynomial is selected) and you specify the primitive polynomial as poly, the generator polynomial is rsgenpoly(N,K,poly).

Note

The degree of the generator polynomial is N − K, where N is the codeword length and K is the message word length.

Puncturing and Erasures

1s and 0s have precisely opposite meanings for the puncture and erasure vectors.

In a puncture vector,

• 1 means that the data symbol is passed through the block unaltered.

• 0 means that the data symbol is to be punctured, or removed, from the data stream.

In an erasure vector,

• 1 means that the data symbol is to be replaced with an erasure symbol.

• 0 means that the data symbol is passed through the block unaltered.

These conventions apply to both the encoder and the decoder. For more information, see Shortening, Puncturing, and Erasures.

Supported Data Types

Port	Supported Data Types
In	Double-precision floating point Single-precision floating point 8-, 16-, and 32-bit signed integers 8-, 16-, and 32-bit unsigned integers
Out	Double-precision floating point Single-precision floating point 8-, 16-, and 32-bit signed integers 8-, 16-, and 32-bit unsigned integers
Era	Double-precision floating point Boolean
Err	Double-precision floating point Single-precision floating point 8-, 16-, and 32-bit signed integers If the input is uint8, uint16, or uint32, then the number of errors output datatype is int8, int16, or int32, respectively.

Algorithms

This block uses the Berlekamp-Massey decoding algorithm. For information about this algorithm, see Algorithms for BCH and RS Errors-only Decoding.

References

[1] Blahut, Richard E. Algebraic Codes for Data Transmission. Cambridge University Press, 2003.

[2] Wicker, Stephen B. Error Control Systems for Digital Communication and Storage. Upper Saddle River, NJ: Prentice Hall, 1995.

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

[4] Berlekamp, Elwyn R., Algebraic Coding Theory, New York, McGraw-Hill, 1968.

Extended Capabilities

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

Version History

Introduced before R2006a