# Rate Matching for LDPC Codes

Rate matching is a technique used in Low Density Parity Check (LDPC) codes to adjust the code rate of the LDPC code to match the desired transmission rate of the communication system. LDPC codes are widely used in many communication systems due to their high coding performance and low decoding complexity. However, the code rate of an LDPC code is fixed, and it may not match the transmission rate of the communication system. This can result in wasted bandwidth or a loss in coding performance. Rate matching is the process of adjusting the code rate of an LDPC code to match the transmission rate of the communication system.

In this article, we will discuss rate matching for LDPC codes in detail, including the motivation for rate matching, the rate matching process, different rate matching schemes, and their advantages and disadvantages.

## Motivation for Rate Matching

The motivation for rate matching in LDPC codes is to adjust the code rate of the LDPC code to match the transmission rate of the communication system. In a communication system, the transmission rate is determined by the channel bandwidth and the modulation scheme used. The transmission rate is the number of bits per second that can be transmitted over the channel. The code rate of an LDPC code is the ratio of the number of information bits to the total number of bits transmitted. The code rate is typically fixed for an LDPC code, and it may not match the transmission rate of the communication system.

When the code rate of an LDPC code is higher than the transmission rate of the communication system, it results in wasted bandwidth. For example, if the transmission rate of the communication system is 1 Mbps, and the code rate of the LDPC code is 2/3, then the maximum number of information bits that can be transmitted is 666,666 bits per second. However, the LDPC code will transmit 1 million bits per second, resulting in wasted bandwidth.

When the code rate of an LDPC code is lower than the transmission rate of the communication system, it results in a loss in coding performance. For example, if the transmission rate of the communication system is 1 Mbps, and the code rate of the LDPC code is 1/2, then the maximum number of information bits that can be transmitted is 500,000 bits per second. However, the LDPC code will transmit only 500,000 bits per second, resulting in a loss in coding performance.

Therefore, rate matching is essential to adjust the code rate of an LDPC code to match the transmission rate of the communication system, and to achieve maximum coding performance without wasting bandwidth.

## Rate Matching Process

The rate matching process for LDPC codes involves two main steps: puncturing and shortening. Puncturing involves removing some of the parity check equations of the LDPC code to reduce the number of parity bits transmitted, which increases the code rate. Shortening involves removing some of the information bits of the LDPC code to reduce the number of information bits transmitted, which decreases the code rate.

The LDPC encoder takes the input data and generates a codeword by adding parity check equations to the information bits. The puncturing process involves removing some of the parity check equations from the codeword to reduce the number of parity bits transmitted. The shortening process involves removing some of the information bits from the codeword to reduce the number of information bits transmitted.

After the rate matching process, the resulting codeword has a code rate that matches the transmission rate of the communication system. The LDPC decoder then receives the codeword and uses the parity check equations to decode the information bits.

The rate matching process can be applied in two ways: systematic and non-systematic. In systematic rate matching, the information bits are preserved in the resulting codeword. In non-systematic rate matching, the information bits are not preserved in the resulting codeword.

## Puncturing

Puncturing is a process used in rate matching for LDPC codes to increase the code rate by removing some of the parity check equations from the codeword. The puncturing process involves selecting a subset of the parity check equations of the LDPC code to be transmitted and removing the rest.

The selection of the parity check equations can be based on different criteria, such as the parity check degree and the parity check weight. The parity check degree is the number of ones in a parity check equation, and the parity check weight is the sum of the absolute values of the ones in a parity check equation.

One common puncturing scheme is the Progressive Edge-Growth (PEG) algorithm, which selects the parity check equations based on their parity check degree. The PEG algorithm starts with an empty set of parity check equations and adds parity check equations with the lowest degree until the desired code rate is achieved. The PEG algorithm has a low complexity and is widely used in practical LDPC systems.

Another puncturing scheme is the Random Puncturing (RP) algorithm, which selects the parity check equations randomly. The RP algorithm has a higher complexity than the PEG algorithm but can achieve better coding performance.

## Shortening

Shortening is a process used in rate matching for LDPC codes to decrease the code rate by removing some of the information bits from the codeword. The shortening process involves selecting a subset of the information bits of the LDPC code to be transmitted and removing the rest.

The selection of the information bits can be based on different criteria, such as the information bit reliability and the information bit weight. The information bit reliability is the likelihood of an information bit being transmitted correctly, and the information bit weight is the number of ones in an information bit.

One common shortening scheme is the Min-Sum Shortening (MSS) algorithm, which selects the information bits with the lowest reliability and removes them from the codeword. The MSS algorithm has a low complexity and is widely used in practical LDPC systems.

Another shortening scheme is the Random Shortening (RS) algorithm, which selects the information bits randomly. The RS algorithm has a higher complexity than the MSS algorithm but can achieve better coding performance.

## Advantages and Disadvantages of Rate Matching Schemes

Different rate matching schemes have different advantages and disadvantages in terms of coding performance, complexity, and flexibility.

Systematic rate matching schemes preserve the information bits in the resulting codeword, which makes decoding easier and faster. However, systematic rate matching schemes have a lower coding performance than non-systematic rate matching schemes.

Non-systematic rate matching schemes do not preserve the information bits in the resulting codeword, which makes decoding harder and slower. However, non-systematic rate matching schemes have a higher coding performance than systematic rate matching schemes.

Puncturing schemes based on parity check degree, such as the PEG algorithm, have a lower complexity than puncturing schemes based on parity check weight or random selection, such as the RP algorithm. However, puncturing schemes based on parity check degree may result in a suboptimal coding performance.

Shortening schemes based on information bit reliability, such as the MSS algorithm, have a lower complexity than shortening schemes based on information bit weight or random selection, such as the RS algorithm. However, shortening schemes based on information bit reliability may result in a suboptimal coding performance.

In practice, the choice of rate matching scheme depends on the specific requirements of the communication system, such as the required code rate, coding performance, and complexity.

## Conclusion

Rate matching is a critical process in LDPC coding systems that adapts the code rate of the LDPC code to match the transmission rate of the communication system. Rate matching involves puncturing and shortening the LDPC codeword to increase or decrease the code rate, respectively.

Different rate matching schemes have different advantages and disadvantages in terms of coding performance, complexity, and flexibility. Systematic rate matching schemes preserve the information bits in the resulting codeword, while non-systematic rate matching schemes do not. Puncturing schemes based on parity check degree have a lower complexity than puncturing schemes based on parity check weight or random selection, while shortening schemes based on information bit reliability have a lower complexity than shortening schemes based on information bit weight or random selection.

The choice of rate matching scheme depends on the specific requirements of the communication system, such as the required code rate, coding performance, and complexity. In practice, the PEG algorithm for puncturing and the MSS algorithm for shortening are widely used due to their low complexity and good coding performance.