Glossary

11.3 BCJR Algorithm

3 min read•august 7, 2024

The is a powerful decoding method for . It uses forward and backward recursions to calculate probabilities of states and transitions in a , enabling accurate soft-decision decoding of received sequences.

This algorithm plays a crucial role in turbo codes and iterative decoding. By computing log-likelihood ratios for each bit, it provides soft information that can be used in iterative decoding schemes, improving overall error correction performance.

Recursion and Metrics

Forward and Backward Recursion

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  • calculates the probability of reaching each state in the trellis at time kk based on the probabilities of the states at time k−1k-1 and the branch metrics connecting them
  • Denoted as αk(s)\alpha_k(s), where ss is the state at time kk
  • calculates the probability of reaching each state in the trellis at time kk based on the probabilities of the states at time k+1k+1 and the branch metrics connecting them
  • Denoted as βk(s)\beta_k(s), where ss is the state at time kk
  • Recursions are initialized with known probabilities at the start and end states of the trellis (α0(s0)=1\alpha_0(s_0) = 1, βN(sN)=1\beta_N(s_N) = 1)

Branch and State Metrics

  • Branch metrics represent the probability of transitioning from one state to another along a specific branch in the trellis
  • Calculated using the channel output and the expected output for each possible (Hamming distance or Euclidean distance)
  • State metrics combine the branch metrics and the probabilities from the forward and backward recursions to determine the likelihood of being in a particular state at time kk
  • Denoted as γk(s′,s)=p(yk,sk=s∣sk−1=s′)\gamma_k(s',s) = p(y_k, s_k=s | s_{k-1}=s'), where s′s' is the previous state and ss is the current state

Probabilities and Ratios

A Posteriori Probabilities (APP)

  • APP represents the probability of a particular bit being 0 or 1 given the entire received sequence
  • Calculated by summing the state metrics for all states at time kk where the bit is 0 or 1
  • APP(uk=0)=∑(s′,s):uk=0αk−1(s′)γk(s′,s)βk(s)APP(u_k=0) = \sum_{(s',s):u_k=0} \alpha_{k-1}(s') \gamma_k(s',s) \beta_k(s)
  • APP(uk=1)=∑(s′,s):uk=1αk−1(s′)γk(s′,s)βk(s)APP(u_k=1) = \sum_{(s',s):u_k=1} \alpha_{k-1}(s') \gamma_k(s',s) \beta_k(s)

Log-Likelihood Ratios (LLRs) and Max-Log-MAP Approximation

  • LLRs represent the logarithm of the ratio of the APP for a bit being 1 to the APP for a bit being 0
  • LLR(uk)=log⁡APP(uk=1)APP(uk=0)LLR(u_k) = \log \frac{APP(u_k=1)}{APP(u_k=0)}
  • Max-log-MAP approximation simplifies the LLR calculation by using the maximum state metrics instead of summing over all states
  • LLR(uk)≈max⁡(s′,s):uk=1(αk−1(s′)+γk(s′,s)+βk(s))−max⁡(s′,s):uk=0(αk−1(s′)+γk(s′,s)+βk(s))LLR(u_k) \approx \max_{(s',s):u_k=1} (\alpha_{k-1}(s') + \gamma_k(s',s) + \beta_k(s)) - \max_{(s',s):u_k=0} (\alpha_{k-1}(s') + \gamma_k(s',s) + \beta_k(s))
  • Reduces at the cost of some performance loss

Trellis Representation

Trellis Diagram

  • Trellis diagram is a graphical representation of the encoding process and the possible state transitions
  • Each node represents a state at a specific time step, and each branch represents a possible transition between states
  • Input bits determine the path through the trellis (upper branch for 0, lower branch for 1)
  • Output bits are associated with each branch, determined by the generator polynomials of the convolutional code
  • Trellis structure depends on the encoder's memory and generator polynomials (rate 1/2 encoder with memory 2 has 4 states and 8 branches per time step)
  • BCJR algorithm operates on the trellis to calculate forward and backward recursions, branch metrics, and state metrics for decoding

Key Terms to Review (20)

Backward recursion: Backward recursion is a method used in dynamic programming and algorithms, where the solution to a problem is built by recursively solving smaller subproblems in reverse order. This technique is particularly useful for optimizing the calculation of probabilities and metrics in certain algorithms, allowing for an efficient traversal of data structures like trellis diagrams. In the context of specific algorithms, backward recursion can help to efficiently compute values based on previously calculated results, streamlining the process.
Bahl: Bahl refers to a significant figure in coding theory, particularly known for the development of algorithms used for decoding convolutional codes. This term is closely linked with the BCJR algorithm, which is named after its creators, including Bahl, and is crucial for efficient decoding processes in communication systems. Bahl's contributions have enabled improved performance in error correction, making it a foundational concept in coding theory.
BCJR Algorithm: The BCJR algorithm, named after its developers Bahl, Cocke, Jelinek, and Raviv, is a soft-decision decoding technique used for error correction in convolutional codes. This algorithm uses a forward-backward approach to calculate the posterior probabilities of the state transitions and symbols, providing a powerful method to improve decoding performance. It plays a significant role in soft-decision decoding by leveraging the received signals' likelihood to enhance the accuracy of the decoded information, which is essential in applications involving noisy communication channels.
Bit error rate: Bit error rate (BER) is a metric that quantifies the number of bit errors in a digital transmission system, expressed as a ratio of the number of erroneous bits to the total number of transmitted bits. This measurement is critical for assessing the performance and reliability of communication systems, particularly in the presence of noise and interference. A lower BER indicates a more reliable system and is essential in designing effective error correction techniques.
Block codes: Block codes are a type of error-correcting code that encodes data in fixed-size blocks, allowing for the detection and correction of errors that may occur during data transmission or storage. These codes are defined by their length and dimension, providing a structured method to represent information, which connects to various coding techniques and mathematical properties.
Cocke: Cocke refers to a method developed by John Cocke for decoding convolutional codes using the BCJR algorithm. This approach significantly enhances the efficiency of error correction in digital communications by utilizing a recursive structure that efficiently computes the probabilities of different states in a trellis diagram. The algorithm is crucial for improving the reliability of data transmission over noisy channels.
Computational Complexity: Computational complexity is a field in computer science that studies the resources required for solving computational problems, specifically focusing on time and space. It examines how the difficulty of a problem relates to the size of the input and classifies problems based on their inherent difficulty, often using complexity classes such as P, NP, and NP-complete. This understanding is critical when applying algorithms in areas like decoding, where efficiency can impact performance significantly.
Convolutional Codes: Convolutional codes are a type of error-correcting code that are generated by passing data sequences through a linear finite state machine, producing encoded output as a function of the current input and previous inputs. This coding technique is essential for ensuring data integrity in communication systems and is deeply connected to several aspects of coding theory, including the use of generator and parity check matrices, systematic encoding techniques, and various decoding algorithms.
Forward recursion: Forward recursion is a computational technique used to process sequences in a systematic manner, particularly in algorithms involving dynamic programming and probabilistic models. In the context of decoding, this method is integral for calculating the probabilities of different states at each time step, moving sequentially from the start to the end of a sequence. It lays the groundwork for backward recursion, allowing for efficient computation of metrics essential in algorithms like the BCJR.
Frame error rate: Frame error rate refers to the percentage of incorrectly received data frames in a communication system. It's crucial for assessing the reliability and performance of various decoding techniques, impacting how well data can be retrieved from transmitted signals under various conditions, including noise and interference.
Input sequence: An input sequence is a series of symbols or bits that are fed into a system for processing, particularly in the context of decoding and encoding information. In coding theory, the input sequence is essential for algorithms that analyze and reconstruct data, as it directly affects the performance and reliability of communication systems. Understanding the characteristics and implications of the input sequence helps in designing better codes and improving error correction methods.
Jelinek: Jelinek refers to a specific decoding algorithm used in the context of coding theory, particularly within the framework of trellis-coded modulation. It is designed to efficiently decode signals transmitted over noisy channels, optimizing the trade-off between computational complexity and performance in error correction.
Log-likelihood ratio: The log-likelihood ratio is a statistical measure used to quantify the support for one hypothesis over another based on observed data. It compares the likelihood of the data under two competing hypotheses, often expressed as the logarithm of the ratio of their probabilities. This concept plays a crucial role in decision-making processes, especially in decoding schemes that leverage probabilities to enhance accuracy, particularly in methods that handle uncertain or noisy information.
Maximum a posteriori probability: Maximum a posteriori probability (MAP) is a statistical estimation technique that identifies the most probable value of a parameter given observed data and prior knowledge. This approach combines likelihood with prior distributions to yield a posterior distribution, from which the mode is selected as the MAP estimate. MAP is crucial in contexts where one seeks to infer hidden states or parameters in systems like communication channels, especially when utilizing algorithms to optimize decisions.
Output Sequence: An output sequence is a series of symbols produced by a coding system, typically representing the results of encoding input data through a specific algorithm or model. This sequence is crucial in understanding how data is transmitted and decoded in communication systems, particularly in relation to the reliability and accuracy of information transfer.
Raviv: Raviv refers to a specific algorithm utilized for decoding convolutional codes, particularly in the context of the BCJR algorithm. This technique is crucial for efficiently estimating hidden states in communication systems, enabling the recovery of transmitted information even in the presence of noise and errors.
Soft decision decoding: Soft decision decoding is a technique used in error correction coding where the decoder considers not just the received bits but also the confidence level associated with each bit. This approach allows for more nuanced interpretations of the received signals, which can lead to better error correction performance compared to hard decision decoding, where bits are simply treated as either 0 or 1. By leveraging probabilistic information from the received signal, soft decision decoding is crucial for improving the efficiency and reliability of various coding schemes, especially in convolutional codes and turbo codes.
Time Complexity: Time complexity is a computational concept that measures the amount of time an algorithm takes to complete as a function of the length of the input. It helps in understanding how the runtime of an algorithm grows with the size of the input, which is crucial when analyzing algorithms like the BCJR algorithm, particularly in applications like error correction in coding theory.
Trellis diagram: A trellis diagram is a graphical representation used to visualize the state transitions of a finite-state machine, particularly in coding theory and digital communications. It helps in understanding the possible sequences of states and the transitions between them, especially during decoding processes. Trellis diagrams are crucial for algorithms that rely on the systematic exploration of paths, such as those used in decoding convolutional codes and evaluating different sequences in the presence of noise.
Viterbi Algorithm: The Viterbi Algorithm is a dynamic programming algorithm used for decoding convolutional codes by finding the most likely sequence of hidden states, known as the Viterbi path, given a sequence of observed events. This algorithm is critical in error correction for digital communications, as it efficiently determines the optimal path through a trellis structure that represents all possible states and transitions. By applying the Viterbi Algorithm, one can achieve maximum likelihood decoding, making it essential for reliable data transmission.
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