LDPC Decoder

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

Library:
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.

Ports

Input

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`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

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`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

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`Parity-check matrix (sparse binary (N-K)-by-N matrix)` — Parity-check matrix
`dvbs2ldpc(1/2)` (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.

The default value uses the dvbs2ldpc function to configure a sparse parity-check matrix for half-rate LDPC coding, as specified in the DVB-S.2 standard.

Example: dvbs2ldpc(R,'indices') configures the index matrix for the DVB-S.2 standard, where R is the code rate, and 'indices' specifies the output format of dvbs2ldpc as a two-column double-precision matrix that defines the row and column indices of the 1s in the parity-check matrix.

Data Types: double | Boolean

`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.

Model 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.

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.

Block Characteristics

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

Algorithms

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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.

For transmitted LDPC-encoded codeword c = c₀, c₁, …, c_n_-1, the input to the LDPC decoder is the log-likelihood ratio (LLR) value $L (c_{i}) = \log (\frac{\Pr (c_{i} = 0 | channel output for c_{i})}{\Pr (c_{i} = 1 | channel output for c_{i})})$ .

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

$L (r_{j i}) = 2 atanh (\prod_{i^{'} \in V_{j} \ i} \tanh (\frac{1}{2} L (q_{i^{'} j})))$ ,

$L (q_{i j}) = L (c_{i}) + \sum_{j^{'} \in C_{i} \ j} L (r_{j^{'} i})$ , initialized as $L (q_{i j}) = L (c_{i})$ before the first iteration, and

$L (Q_{i}) = L (c_{i}) + \sum_{j^{'} \in C_{i}} L (r_{j^{'} i})$ .

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

Index sets $C_{i} \ j$ and $V_{j} \ i$ 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 highlights 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.

Documentation

LDPC Decoder

Description