LDPC Decoder

Decode binary low-density parity-check (LDPC) code

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Libraries:
Communications Toolbox / Error Detection and Correction / Block

Description

The LDPC Decoder block uses the belief propagation algorithm to decode a binary LDPC code, which is input to the block as the soft-decision output (log-likelihood ratio of received bits) from demodulation. The block decodes generic binary LDPC codes where no patterns in the parity-check matrix are assumed. For more information, see Belief Propagation Decoding.

The input and output are discrete-time signals. The ratio of the output sample time to the input sample time is:

• N/K when only the information-part of the codeword is decoded

• 1 when the entire codeword is decoded

N is the length of the received signal and must be in the range (0, 2³¹). K is the length of the uncoded message and must be less than N.

This icon shows all ports, including optional ports, for the LDPC Decoder block.

Examples

LDPC Encode and Decode QPSK-Modulated Signal in Simulink

Transmit an LDPC-encoded, QPSK-modulated bit stream through an AWGN channel. Demodulate and decode the received signal. Compute the error statistics.

The snr variable is initialized using the InitFcn callback in Model Properties>Callbacks. The SNR dB parameter in the AWGN Channel block and the Variance parameter in the QPSK Demodulator Baseband block initialize settings using the snr variable.

The simulation is configured to process one input frame of data. The Error Rate Calculation block compares the transmitted binary data stream with the information data bits output by the LDPC Decoder block.

For SNR = 1 dB, the error rate is 0.

Extended Examples

DVB-S.2 Link, Including LDPC Coding in Simulink

The state-of-the-art channel coding scheme used in the second generation Digital Video Broadcasting standard (DVB-S.2), which is deployed by DIRECTV in the United States. The coding scheme is based on concatenation of LDPC (Low-Density Parity-Check) and BCH codes. LDPC codes, invented by Gallager in his seminal doctoral thesis in 1960, can achieve extremely low error rates near channel capacity by using a low-complexity iterative decoding algorithm. The outer BCH codes are used to correct sporadic errors made by the LDPC decoder.

• Open Model

Ports

Input

In — Log-likelihood ratios

column vector

Log-likelihood ratios, specified as an N-by-1 column vector containing the soft-decision output from demodulation. N is the number of bits in the LDPC codeword before modulation. Each element is the log-likelihood ratio for a received bit and the value is more likely to be 0 if the log-likelihood ratio is positive. The first K elements correspond to the information-part of the input message.

Data Types: double

Output

Out — Decoded data

column vector

Decoded data, returned as a column vector. The Decision type parameter specifies whether the block outputs hard decisions or soft decisions (log-likelihood ratios).

• If the Output format parameter is set to Information part, the output includes only the information-part of the received codeword.

• If the Output format parameter is set to Whole codeword, the output includes the whole log-likelihood ratio vector.

Data Types: double | Boolean

Iter — Number of executed decoding iterations

positive integer

Number of executed decoding iterations, returned as a positive integer.

Dependencies

To enable this port, select the Output number of iterations executed parameter.

Data Types: double

ParChk — Final parity checks

column vector

Final parity checks after decoding the input LDPC code, returned as an (N-K)-by-1 column vector. N is the number of bits in the LDPC codeword before modulation. K is the length of the uncoded message.

Dependencies

To enable this port, select the Output final parity checks parameter.

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.

Parity-check matrix (sparse binary (N-K)-by-N matrix) — Parity-check matrix

ldpcDecoderConfig().ParityCheckMatrix (default) | sparse binary matrix | nonsparse index matrix

Parity-check matrix, specified as a sparse (N – K)-by-N binary-valued matrix. N is the length of the received signal and must be in the range (0, 2³¹). K is the length of the uncoded message and must be less than N. The last (N – K) columns in the parity-check matrix must be an invertible matrix in the Galois field of order 2, gf(2).

You can also specify the parity-check matrix as a two-column nonsparse index matrix, I, that defines the row and column indices of the 1s in the parity-check matrix such that sparse(I(:,1),I(:,2),1).

This parameter accepts numeric data types. When you set this parameter to a sparse binary matrix, this parameter also accepts the Boolean data type.

Example: ldpcDecoderConfig().ParityCheckMatrix configures the binary matrix as a 108-by-648 sparse logical matrix.

Output format — Output value format

Information part (default) | Whole codeword

Output value format, specified as one of these values:

• Information part — The block outputs a K-by-1 column vector containing only the information-part of the received log-likelihood ratio vector. K is the length of the uncoded message.

• Whole codeword — The block outputs an N-by-1 column vector containing the whole log-likelihood ratio vector. N is the length of the received signal.

  N and K must align with the dimension of the (N–K)-by-K parity-check matrix.

Decision type — Decision method

Hard decision (default) | Soft decision

Decision method used for decoding, specified as one of these values:

• Hard decision — The block outputs decoded data of data type double or boolean. Specify this data type using the Output data type parameter.

• Soft decision — The block outputs log-likelihood ratios of data type double.

Output data type — Output value data type

double (default) | boolean

Output value data type, specified as double or boolean.

Dependencies

To enable this parameter, set the Decision type parameter to Hard decision.

Number of iterations — Maximum number of decoding iterations

50 (default) | positive integer

Maximum number of decoding iterations, specified as a positive integer.

Stop iterating when all parity-checks are satisfied — Condition for iteration termination

off (default) | on

Select this parameter to terminate decoding after all parity checks are satisfied. If not all parity checks are satisfied, decoding terminates after the number of iterations specified by the Number of iterations parameter.

Output number of iterations executed — Output number of iterations executed

off (default) | on

Select this parameter to enable the Iter output port.

Output final parity-checks — Output number of iterations executed

off (default) | on

Select this parameter to enable the ParChk output port.

Block Characteristics

Data Types	Boolean | double
Multidimensional Signals	no
Variable-Size Signals	no

Algorithms

This block performs LDPC decoding using the belief propagation algorithm, also known as a message-passing algorithm.

Belief Propagation Decoding

The implementation of the belief propagation algorithm is based on the decoding algorithm presented by Gallager [2].

For transmitted LDPC-encoded codeword c = c₀, c₁, …, c_n-1, the input to the LDPC decoder is the log-likelihood ratio (LLR) value $$L(c_i)=\log \left(\frac{\mathrm{Pr}(c_i=0|\text{channel output for }{c}_i)}{\mathrm{Pr}(c_i=1|\text{channel output for }{c}_i)}\right)$$.

In each iteration, the key components of the algorithm are updated based on these equations:

$$L(r_{ji})=2\text{atanh}\left({\displaystyle \prod _{i^{\prime }\in V_j\backslash i}\tanh \left(\frac{1}{2}L(q_{i^{\prime }j})\right)}\right)$$,

$$L(q_{ij})=L(c_i)+{\displaystyle \sum _{j^{\prime }\in C_i\backslash j}L(r_{j^{\prime }i})}$$, initialized as $$L(q_{ij})=L(c_i)$$ before the first iteration, and

$$L(Q_i)=L(c_i)+{\displaystyle \sum _{j^{\prime }\in C_i}L(r_{j^{\prime }i})}$$.

At the end of each iteration, L(Q_i) contains the updated estimate of the LLR value for transmitted bit c_i. The value L(Q_i) is the soft-decision output for c_i. If L(Q_i) ≤ 0, the hard-decision output for c_i is 1. Otherwise, the hard-decision output for c_i is 0.

If decoding is configured to stop when all of the parity checks are satisfied, the algorithm verifies the parity-check equation (H c' = 0) at the end of each iteration. When all of the parity checks are satisfied, or if the maximum number of iterations is reached, decoding stops.

Index sets $$C_i\backslash j$$ and $$V_j\backslash i$$ are based on the parity-check matrix (PCM). Index sets C_i and V_j correspond to all nonzero elements in column i and row j of the PCM, respectively.

This figure shows the computation of these index sets in a given PCM for i = 5 and j = 3.

To avoid infinite numbers in the algorithm equations, atanh(1) and atanh(–1) are set to 19.07 and –19.07, respectively. Due to finite precision, MATLAB^® returns 1 for tanh(19.07) and –1 for tanh(-19.07).

References

[1] Gallager, Robert G. Low-Density Parity-Check Codes. Cambridge, MA: MIT Press, 1963.

Extended Capabilities

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

Version History

Introduced in R2007a

R2024b: Change default parity-check matrix

With this update, the default parity-check matrix for the LDPC Decoder block is now ldpcDecoderConfig().ParityCheckMatrix instead of dvbs2ldpc(1/2).