CN (Check Node)

Introduction

Check nodes (CNs) are a key component of low-density parity-check (LDPC) codes, which are error-correcting codes used in a variety of communication systems. In this article, we will discuss the basic principles of CNs, their role in the decoding process, and some practical considerations for designing and implementing LDPC codes.

LDPC Codes

LDPC codes are a class of linear error-correcting codes that are based on sparse parity-check matrices. They were first introduced by Robert Gallager in the 1960s, but it was not until the 1990s that they were rediscovered and became popular due to their excellent performance and low decoding complexity.

The fundamental idea behind LDPC codes is to represent a message as a sequence of binary digits and encode it by multiplying it with a sparse parity-check matrix. The resulting codeword has the property that each row and column of the matrix has an equal number of ones and zeros, which makes it easy to detect and correct errors.

Decoding LDPC Codes

The decoding of LDPC codes is a complex process that involves iteratively updating the probabilities of the bits in the codeword until a satisfactory solution is reached. This process is called message passing, and it involves two types of nodes: variable nodes (VNs) and check nodes (CNs).

VNs represent the bits in the codeword and are connected to CNs via edges, which represent the parity-check equations of the code. CNs, on the other hand, represent the parity-check equations of the code and are connected to VNs via edges that represent the corresponding bits.

Check Nodes (CNs)

CNs are responsible for determining the reliability of the information received from the VNs. In other words, they check whether the parity-check equations are satisfied or not. If a parity-check equation is satisfied, the CN sends a message to the corresponding VN indicating that the bit is likely to be correct. If a parity-check equation is not satisfied, the CN sends a message to the corresponding VNs indicating that the bits involved in the equation are likely to be in error.

The messages sent by CNs to VNs are called check-to-variable (C2V) messages, and they represent the reliability of the corresponding bits in the codeword. The messages are computed using the following formula:

C2V(xi) = prod_j (1 - 2yjpij(xi))

where xi is the bit corresponding to the VN, yj is the value of the jth neighbor of the CN, and pij(xi) is the probability that xi takes the value j given the current state of the other bits.

The messages sent by VNs to CNs are called variable-to-check (V2C) messages, and they represent the reliability of the parity-check equations. The messages are computed using the following formula:

V2C(yj) = tanh(prod_i (1 - 2xipji(yj)/C2V(xi)))

where yj is the value of the CN, xi is the value of the ith neighbor of the CN, pji(yj) is the probability that the CN is satisfied given that yj takes the value i, and C2V(xi) is the message received by the VN corresponding to xi.

The decoding process continues until a satisfactory solution is reached or a maximum number of iterations is reached. A satisfactory solution is one where the probability of error is below a certain threshold.

Designing LDPC Codes

The design of LDPC codes involves selecting an appropriate parity-check matrix that satisfies certain constraints, such as being sparse, having a large girth, and having a low error rate. The girth of a graph is the length of the shortest cycle in the graph, and a large girth ensures that the decoding process converges quickly and avoids trapping sets, which are groups of bits that are difficult to decode.

One popular method for designing LDPC codes is to use the protograph construction, which involves designing a small graph called a protograph and using it to construct a larger parity-check matrix. The protograph is a template that is repeated to form the larger matrix, and it is chosen to satisfy certain properties such as having a large girth and a low error rate.

Another method for designing LDPC codes is to use the density evolution algorithm, which involves simulating the message passing process and computing the probability of error as a function of the density of the parity-check matrix. The algorithm can be used to optimize the parameters of the code such as the degree distribution and the number of iterations.

Practical Considerations

In practice, LDPC codes are widely used in communication systems such as satellite communications, wireless networks, and optical communications. They offer excellent performance in terms of error correction and can be efficiently implemented using hardware or software.

One important consideration when implementing LDPC codes is the choice of the decoding algorithm. The most common algorithm is the belief propagation algorithm, which is based on the message passing principle. However, other algorithms such as the min-sum algorithm and the sum-product algorithm can also be used.

Another consideration is the choice of the parity-check matrix. The parity-check matrix can be designed to satisfy different constraints depending on the application, such as having a high code rate or a low error rate. The choice of the matrix also affects the decoding complexity, as a more complex matrix may require more iterations to converge.

Conclusion

Check nodes (CNs) are a key component of low-density parity-check (LDPC) codes, which are error-correcting codes used in a variety of communication systems. CNs are responsible for determining the reliability of the information received from the variable nodes (VNs) and sending messages to the VNs indicating the likelihood of error. The decoding process involves iteratively updating the probabilities of the bits in the codeword until a satisfactory solution is reached. The design of LDPC codes involves selecting an appropriate parity-check matrix that satisfies certain constraints such as being sparse and having a large girth. In practice, LDPC codes offer excellent performance in terms of error correction and can be efficiently implemented using hardware or software.