Log-likelihood ratio

Definition

The log-likelihood ratio is a statistical measure that quantifies the relative likelihood of a particular hypothesis given some observed data. In coding theory, it plays a vital role in decoding processes for error-correcting codes, particularly in the context of Turbo codes and Low-Density Parity-Check (LDPC) codes, where it is used to assess the reliability of received signals and make decisions about the transmitted bits.

5 Must Know Facts For Your Next Test

1. The log-likelihood ratio is computed as the logarithm of the ratio of the probabilities of two competing hypotheses, often expressed as $$ ext{LLR} = ext{log} \left( \frac{P(data | H_1)}{P(data | H_0)} \right)$$.
2. In Turbo and LDPC decoding algorithms, the log-likelihood ratio is used to provide soft information, which helps improve the accuracy of decisions made during decoding.
3. The LLR can take both positive and negative values, indicating confidence in either hypothesis, where a positive value suggests that the bit is likely a '1', and a negative value suggests it is likely a '0'.
4. Efficient computation of LLRs is crucial for the performance of decoding algorithms; high-performance implementations often utilize hardware acceleration to achieve real-time processing.
5. Log-likelihood ratios are essential for iterative decoding schemes, as they allow for updates based on previously received messages, leading to improved error correction capabilities.

Review Questions

• How does the log-likelihood ratio function in Turbo codes during the decoding process?
  • In Turbo codes, the log-likelihood ratio serves as a way to represent soft information about the received bits. During decoding, it helps in assessing the likelihood of each bit being either a '0' or a '1' based on the received signals. The iterative nature of Turbo decoding utilizes these LLRs to update beliefs about the bits and improve decision-making in each iteration, ultimately enhancing error correction performance.
• Compare and contrast how log-likelihood ratios are used in both Turbo codes and LDPC codes.
  • Both Turbo codes and LDPC codes leverage log-likelihood ratios to improve decoding accuracy, but they do so through different mechanisms. Turbo codes typically involve parallel concatenation and iterative decoding between two or more encoders, while LDPC codes rely on sparse parity-check matrices and belief propagation algorithms. In both cases, LLRs facilitate the communication of soft information that leads to better-informed decisions about transmitted bits, yet their specific implementation strategies differ significantly.
• Evaluate the impact of log-likelihood ratios on the overall performance of error-correcting codes like Turbo and LDPC codes.
  • The use of log-likelihood ratios significantly enhances the performance of error-correcting codes such as Turbo and LDPC codes by enabling iterative refinement of decoded outputs. As LLRs are updated through each iteration, they provide progressively more accurate information about the transmitted bits. This iterative approach allows these codes to approach Shannon limits more closely, resulting in improved error rates under challenging conditions. The ability to convey soft information through LLRs not only increases robustness against noise but also makes these coding schemes highly effective in modern communication systems.

Related terms

Likelihood: The probability of observing the given data under a specific model or hypothesis.

Turbo Codes: A type of error-correcting code that uses iterative decoding and parallel concatenation of two or more convolutional codes.

LDPC Codes: Low-Density Parity-Check codes are a class of linear error-correcting codes characterized by sparse parity-check matrices.