# Analysis of TDMP Algorithm of LDPC Codes Based on Density Evolution and Gaussian Approximation
## 1 Introduction  
Low-density parity-check (LDPC) codes were first proposed by [PERSON] [1] in 1962. In 1996, [PERSON] and [PERSON] [2] re-researched LDPC codes and discovered that irregular LDPC performs better than regular LDPC [34], the irregular LDPC codes can perform close to the Shannon limit error performance. Based on this, [PERSON] [5] proposed an extended method to convert the regular LDPC codes into the irregular LDPC codes. On the basis of the Slepian-Wolf coding problem, a coding theorem of finite-substitution linear coding was proposed. The research shows that there exist a series of linear coders for any set of finite-correlation discrete memoryless sources [6]. In Reference [7], rate-compatible LDPC coding was implemented by graphics expansion to ensure coding performance and to simplify the structure of the encoder. Based on the probabilistic iterative decoding algorithm, [PERSON] [8] proposed the belief propagation (BP) algorithm, which is currently of great academic interest. In BP algorithm decoding, information is passed along the edge of Tanner graph until the decoding end [9]. The BP decoding schedules are designed according to the reliability of check nodes, which can improve the error-correcting performance of BP decoding [10]. In addition, only one subset of check nodes is updated in each iteration, which reduces the resource consumption in the decoding process. Based on [PERSON]'s [1] thought, [PERSON] [11] introduced the concept of density evolution, which can calculate the capacity of LDPC codes under message-passing decoding. The density evolution algorithm is also applied to the multi-edge type framework of LDPC [12]. [PERSON] used the density evolution to analyze the convergence of BP iterative decoding and proposed an iterative decoding threshold analysis method for LDPC convolutional codes [13]. In the 5G communication standard [14], [PERSON] uses the density evolution algorithm to analyze LDPC codes of a sparse-code multiple-access [15] system and designs a multi-user de-noising algorithm. For the binary-input additive white Gaussian noise (BIAWGN) channel, we can replace density evolution (DE) with Gaussian approximation [16] to simplify the analysis of the decoding algorithm. In the performance analysis of low-density parity check-bit interleaved coded modulation system in [17], the probability density function (PDF) of the log-likelihood ratio (LLR) messages transmitted between variables and check nodes is calculated using a Gaussian mixture approximation method for reducing the complexity of the density evolution process. [PERSON] analyzed the function \(\phi(x)\) involved in Gaussian approximation on the basis in Reference [16] and obtained a new approximation function [18]. The threshold of the new approximation function is closer to that of density evolution compared with the approximation function in Reference [16]. Though the BP algorithm, which is the basic algorithm of LDPC codes, can obtain near Shannon limit error performance, the check nodes need to deal with many non-linear operations, the complexity of the BP algorithm is higher. With further research on the decoding algorithm, [PERSON] and [PERSON] proposed the Turbo Decoding Message Passing (TDMP) algorithm [19; 20], which can obtain high throughput, and at the same time increase the convergence speed of iterative decoding. Under ideal conditions (in an ideal situation), the number of iterations of the TDMP algorithm is half of that of the BP algorithm. [PERSON] analyzed the PDF calculation of the Tanner graph edge transfer [21]. [PERSON] used an Extrinsic Information Transfer Chart (EXIT) to analyze the irregular LDPC codes and proved that there was a similar relationship between regular LDPC and irregular LDPC's EXIT [22]. In Reference [23], the mixed method of density evolution initialization node and Gaussian approximation calculation could reduce the amount of calculation and facilitate code design.