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LDPC (low density parity check)

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Low-density parity-check (LDPC) codes are a type of error-correcting code that is widely used in modern digital communication systems. These codes have been the subject of intense research over the past few decades due to their excellent error-correcting performance and low-complexity decoding algorithms. In this article, we will provide an in-depth explanation of LDPC codes, their properties, and their applications.

Background

LDPC codes were first introduced by Robert Gallager in his PhD thesis in 1963, but it was not until the 1990s that their potential for practical applications was recognized. LDPC codes belong to the class of linear block codes, which means that they encode a block of input data into a longer code word by adding redundant bits to the data. These redundant bits are used to detect and correct errors that may occur during transmission.

LDPC codes are characterized by their sparse parity-check matrices, which means that the matrix has a small number of non-zero elements. This sparsity property is what makes LDPC codes particularly attractive, as it enables efficient decoding algorithms to be developed.

Encoding

The LDPC encoding process involves multiplying the input data vector by a sparse parity-check matrix to produce a code word. The parity-check matrix is typically constructed using a bipartite graph, where the input data vector corresponds to one set of nodes, and the parity-check matrix corresponds to the other set of nodes.

To illustrate this, consider the example of a (3,6) LDPC code, where the input data vector has three bits and the code word has six bits. The parity-check matrix for this code can be represented by the bipartite graph shown below:Copy code  ┌───┬───┬───┬───┬───┬───┐ ┌─┤ 1 │   │   │ 1 │   │ 1 │ │ ├───┼───┼───┼───┼───┼───┤ │ │   │ 1 │   │ 1 │ 1 │   │ └─┤   │   │ 1 │   │ 1 │ 1 │ └───┴───┴───┴───┴───┴───┘

The rows of the parity-check matrix correspond to the check equations that are used to verify the correctness of the code word. The columns of the matrix correspond to the bits of the code word. The non-zero entries in the matrix indicate which bits are involved in each check equation. For example, the first check equation involves bits 1, 4, and 6.

To encode a message using this code, we simply multiply the message vector by the parity-check matrix modulo 2. For example, if the message vector is (1, 0, 1), then the corresponding code word is obtained by multiplying the message vector by the parity-check matrix as follows:csharpCopy code[1 0 1] ⨂ ┌───┬───┬───┬───┬───┬───┐ │ 1 │ 0 │ 0 │ 1 │ 0 │ 1 │ └───┴───┴───┴───┴───┴───┘ = [0 1 1 1 1 0]

The resulting code word is (0, 1, 1, 1, 1, 0), which is twice as long as the original message vector.

Decoding

The LDPC decoding process is the process of recovering the original message vector from a received code word that may have errors due to noise or other transmission impairments. LDPC codes are decoded using iterative algorithms, which means that the decoder makes multiple passes over the received code word, updating the estimated values of the message bits based on the parity-check equations.

The most commonly used decoding algorithm for LDPC codes is the belief propagation algorithm, also known as the sum-product algorithm. This algorithm exploits the sparsity of the parity-check matrix to perform efficient message passing between the variable nodes (bits) and check nodes (parity checks) in the bipartite graph.

The belief propagation algorithm starts by initializing the estimates of the message bits to their received values. Then, it performs several rounds of message passing between the variable nodes and check nodes, using the following update rules:

For each variable node i that is connected to check nodes j, compute the log-likelihood ratio (LLR) of the bit using the received code word and the current estimates of the other bits:

LLR(i) = log[Pr(xi=1|y)/Pr(xi=0|y)]

where y is the received code word and Pr(xi=1|y) and Pr(xi=0|y) are the conditional probabilities of the bit xi being 1 or 0 given the received code word y.

For each check node j that is connected to variable nodes i, compute the LLR of the check node using the LLRs of the connected variable nodes:

LLR(j) = 2tanh⁡(1/2∑i∈N(j)\i LLR(i))

where N(j)\i denotes the set of variable nodes connected to check node j except for variable node i.

For each variable node i, compute the updated estimate of the bit using the LLRs of the connected check nodes:

xi = sign[∏j∈M(i) LLR(j)]

where M(i) denotes the set of check nodes connected to variable node i.

The belief propagation algorithm continues to iterate between the variable nodes and check nodes until a stopping criterion is met, such as a maximum number of iterations or a minimum distance between the estimated message vector and the received code word.

Properties

LDPC codes have several important properties that make them attractive for practical applications:

  • Excellent error-correcting performance: LDPC codes can achieve very low error rates close to the Shannon limit, which is the theoretical limit of error correction for a given channel capacity.
  • Low-complexity decoding: The belief propagation algorithm is computationally efficient and can be implemented using parallel processing techniques, making it well-suited for hardware and software implementations.
  • Scalability: LDPC codes can be designed for arbitrary code lengths and rates by adjusting the size and sparsity of the parity-check matrix.
  • Robustness to transmission impairments: LDPC codes can correct errors due to a variety of channel impairments, including additive white Gaussian noise, multipath fading, and phase distortion.
  • Compatibility with modern communication systems: LDPC codes are used in several wireless communication standards, such as Wi-Fi, WiMAX, and 5G, and are also used in high-speed wired communication systems, such as Ethernet and optical communication.

Applications

LDPC codes have a wide range of applications in modern digital communication systems, including:

  • Wireless communication systems: LDPC codes are used in several wireless communication standards, such as Wi-Fi, WiMAX, and 5G, to provide reliable and high-speed data transmission over wireless channels.
  • Wired communication systems: LDPC codes are used in high-speed wired communication systems, such as Ethernet and optical communication, to provide error-free transmission over long distances.
  • Storage systems: LDPC codes are used in modern storage systems, such as hard disk drives and flash memory, to provide error correction and ensure data integrity.
  • Satellite communication systems: LDPC codes are used in satellite communication systems to provide reliable and high-speed data transmission over long distances and through atmospheric interference.
  • Digital broadcasting: LDPC codes are used in digital broadcasting systems, such as DVB-S2 and ATSC 3.0, to provide error correction and ensure high-quality video and audio transmission.
  • Quantum error correction: LDPC codes are also used in quantum error correction schemes to protect quantum information from errors due to decoherence and other quantum noise sources.

Design and Optimization

The design of LDPC codes involves selecting the size and sparsity of the parity-check matrix to achieve a desired code rate and error-correction performance. The sparsity of the parity-check matrix is usually specified by the check node degree distribution, which determines the number of variable nodes connected to each check node.

The optimization of LDPC codes involves finding the best check node degree distribution that maximizes the code performance for a given code length and rate. This optimization problem can be formulated as a constrained optimization problem, where the objective function is the code performance, such as the bit error rate or the frame error rate, and the constraints are the code length, rate, and sparsity of the parity-check matrix.

Several optimization techniques have been proposed for LDPC codes, including genetic algorithms, particle swarm optimization, and simulated annealing. These techniques can be used to find the optimal check node degree distribution for a given code length and rate, or to search for the best LDPC code among a large pool of candidate codes.

Conclusion

LDPC codes are a class of error-correcting codes that have gained significant attention in recent years due to their excellent error-correction performance, low-complexity decoding algorithms, and scalability to arbitrary code lengths and rates. LDPC codes have become a key component in modern digital communication systems, including wireless and wired communication systems, storage systems, satellite communication systems, and digital broadcasting. The design and optimization of LDPC codes is an active area of research, with several optimization techniques available to find the optimal check node degree distribution for a given code length and rate.