10.3 Viterbi Algorithm
3 min read•august 7, 2024
The is a key technique for decoding convolutional codes. It uses to find the most likely sequence of hidden states in a , efficiently determining the original message from a noisy received signal.
This algorithm employs soft-decision or , with soft-decision typically offering better performance. It uses operations at each trellis stage and a to find the most likely path, balancing accuracy and computational complexity.
Viterbi Algorithm Basics
Decoding Methods
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- decoding determines the most likely transmitted sequence by comparing received sequence with all possible transmitted sequences
- uses received signal values as input to the decoder, providing more information than hard-decision decoding
- Utilizes analog information about the received signal
- Generally achieves better performance compared to hard-decision decoding
- Hard-decision decoding uses binary values (0 or 1) as input to the decoder, discarding analog information about the received signal
- Simplifies the decoding process but may result in suboptimal performance
- refers to the number of trellis stages considered in the decoding process
- Longer decoding depth improves error correction capability but increases complexity
- Typical values range from 3 to 7 times the constraint length of the
Key Concepts
- Viterbi algorithm is a dynamic programming approach to finding the most likely sequence of hidden states in a
- Widely used in convolutional code decoding for its efficiency and optimality
- Trellis diagram represents the possible state transitions of the encoder over time
- Each stage corresponds to a time step, and each node represents a possible encoder state
- Branches between nodes represent state transitions based on input bits
- measures the likelihood or distance of a particular path through the trellis
- Calculated using the received sequence and the expected output for each state transition
- Lower path metric indicates a more likely path (for distance metrics)
Viterbi Algorithm Operations
Core Steps
- Add-Compare-Select (ACS) operation is performed at each trellis stage to update path metrics and survivor paths
- Add: Calculate branch metrics for each possible state transition
- Compare: Compare path metrics of arriving branches at each node
- Select: Choose the branch with the best path metric as the
- Traceback operation determines the most likely state sequence by tracing the survivor paths backward through the trellis
- Starts from the node with the best path metric at the final stage
- Follows the survivor path backward to the initial stage
- stores the survivor paths at each trellis stage
- Typically implemented as a register exchange or trace back memory
- Size depends on the decoding depth and the number of states in the trellis
Algorithm Variations
- Viterbi algorithm can be applied to both terminated and unterminated convolutional codes
- have a known initial and final state (usually all zeros)
- require a sufficiently long decoding depth to achieve good performance
- reduces memory requirements by processing the trellis in fixed-size windows
- Only the most recent window of survivor paths is stored
- Trade-off between memory usage and decoding latency
Distance Metrics in Viterbi Algorithm
Commonly Used Metrics
- counts the number of bit differences between the received sequence and the expected output sequence
- Suitable for hard-decision decoding, where the inputs are binary values
- Computationally simple but may not achieve optimal performance
- calculates the squared difference between the received signal values and the expected output values
- Suitable for soft-decision decoding, where the inputs are analog signal values
- Provides better performance by incorporating more information about the received signal
- Requires more computational complexity compared to Hamming distance
Metric Calculation
- represents the distance or likelihood of a particular state transition
- Calculated for each possible state transition at each trellis stage
- Depends on the chosen distance metric (Hamming or Euclidean)
- Path metric accumulates the branch metrics along a particular path through the trellis
- Updated at each trellis stage using the ACS operation
- Represents the total distance or likelihood of a path up to the current stage
- Normalization techniques can be applied to prevent path metric overflow
- Ensures numerical stability for long decoding depths
- Examples include rescaling or subtracting a common value from all path metrics
Key Terms to Review (20)
Add-compare-select: Add-compare-select is a key operation in decoding algorithms, particularly used in the Viterbi Algorithm. This operation efficiently combines the processes of adding and comparing metrics of different paths in a trellis structure, allowing for the selection of the most likely path through a state space. It plays a crucial role in optimizing performance by minimizing the complexity of decoding processes.
Branch metric: A branch metric is a quantitative measure used in decoding algorithms, particularly in the context of convolutional codes. It represents the cost or distance associated with transitioning between different states in a state machine model, and is crucial for evaluating the most likely path through a trellis diagram. The branch metric plays an essential role in optimizing the decoding process by helping to identify the most probable sequence of transmitted symbols.
Convolutional code: A convolutional code is a type of error-correcting code used in digital communication that encodes data by passing it through a series of shift registers and applying a set of mathematical operations. This coding technique helps ensure reliable data transmission by adding redundancy to the data stream, which can be used to detect and correct errors that occur during transmission. Convolutional codes are particularly important in improving the performance of communication systems, especially when combined with algorithms for decoding like the Viterbi Algorithm.
Decoding depth: Decoding depth refers to the number of stages or steps required to successfully decode a received signal in communication systems, particularly when using decoding algorithms. It is crucial because it directly impacts the efficiency and accuracy of decoding, especially in scenarios with noise and errors. Understanding decoding depth helps in evaluating the performance of various decoding methods, ensuring that the most effective approach is employed for reliable data recovery.
Dynamic Programming: Dynamic programming is a method for solving complex problems by breaking them down into simpler subproblems, storing the results of these subproblems to avoid redundant calculations. This technique is particularly useful in optimization problems and is applied in various algorithms to efficiently compute solutions, significantly reducing the time complexity compared to naive approaches.
Euclidean Distance Metric: The Euclidean distance metric is a mathematical measure used to determine the straight-line distance between two points in a multidimensional space. It is widely applied in various fields, including coding theory, for assessing the similarity or dissimilarity between data points. This metric is crucial when evaluating error-correcting codes, as it helps to quantify how different encoded messages are from each other.
Hamming Distance Metric: The Hamming distance metric is a measure of the difference between two strings of equal length, calculated by counting the number of positions at which the corresponding symbols differ. This concept is crucial in coding theory as it helps determine error detection and correction capabilities of codes, particularly in the context of comparing received and transmitted data sequences.
Hard-decision decoding: Hard-decision decoding is a method used in error correction where the decoder makes a binary decision on each received signal, determining whether it is a '0' or '1'. This approach simplifies the decoding process by using a definitive value rather than considering the likelihood of errors, which makes it efficient but can lead to suboptimal performance when errors are present in the received signal.
Hidden Markov Model: A Hidden Markov Model (HMM) is a statistical model that represents systems that are assumed to be a Markov process with unobservable (hidden) states. HMMs are widely used in various fields such as speech recognition, natural language processing, and bioinformatics, as they provide a way to model temporal sequences where the states of the system are not directly visible but can be inferred through observed data.
Maximum likelihood: Maximum likelihood is a statistical method used to estimate the parameters of a statistical model, where the goal is to find the parameter values that make the observed data most probable. This concept is crucial for decoding techniques that optimize performance by selecting the most likely transmitted symbols based on received signals. It connects closely to methods that use soft-decision decoding and algorithms like the Viterbi algorithm, which seeks to maximize the likelihood of correctly interpreting sequences in noisy environments.
Path Memory: Path memory refers to the ability of a decoding algorithm, particularly in the context of trellis-based codes, to retain information about previously traversed paths in order to make optimal decisions on the current state. This feature is essential for efficiently decoding convolutional codes, as it allows the algorithm to consider multiple potential sequences and select the one that is most likely to have been transmitted. By keeping track of these paths, the decoder can improve its accuracy and reduce errors during the decoding process.
Path Metric: A path metric is a numerical value assigned to a path through a graph, representing the cost associated with traversing that path. In the context of coding theory, particularly in algorithms like the Viterbi Algorithm, the path metric helps determine the most likely sequence of states (or paths) that a system has taken based on received information, thereby allowing for efficient decoding of data.
Sliding Window Viterbi Algorithm: The Sliding Window Viterbi Algorithm is a variation of the Viterbi algorithm that improves efficiency in decoding convolutional codes by limiting the size of the path memory to a defined window. This method enables the algorithm to process input sequences in smaller segments, thereby reducing computational complexity and memory usage, which is particularly beneficial for long sequences. By maintaining only a subset of possible states, it streamlines the decision-making process while still achieving near-optimal decoding performance.
Soft-decision decoding: Soft-decision decoding is a method in coding theory that takes into account not just the received bits, but also the reliability or confidence level of each bit during the decoding process. This technique improves error correction capabilities by using probabilities rather than making a hard yes-or-no decision for each bit, allowing for more nuanced interpretations of the received signal. By leveraging additional information from the received signals, soft-decision decoding can lead to better performance in various decoding algorithms, including those used in sequential decoding and convolutional codes.
Survivor Path: A survivor path refers to the most likely sequence of states in a hidden Markov model that can be traced back through the Viterbi algorithm, based on observed outputs. This path is crucial for decoding sequences in communication systems, allowing for optimal error correction by identifying the state transitions that produce the observed data while minimizing the probability of error.
Terminated codes: Terminated codes are a type of error-correcting code that ensures proper signaling when data transmission ends. They are essential in coding theory, especially in contexts where messages must be clearly demarcated to prevent ambiguity at the receiver's end. These codes enhance the reliability of communication systems by providing a clear indication of when the data stream has concluded.
Traceback: In the context of coding theory, traceback refers to the process of reconstructing the most likely sequence of states in a hidden Markov model after performing an inference algorithm like the Viterbi algorithm. This process allows for determining the optimal path that leads to a given output sequence by retracing the steps taken during the computation. It’s crucial because it helps in decoding sequences efficiently and accurately.
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.
Unterminated codes: Unterminated codes are a type of coding scheme in which the encoded data does not have a distinct end marker or termination sequence. This absence of an explicit termination allows for a continuous stream of data, which can be beneficial for certain applications, particularly in decoding processes where the Viterbi Algorithm is employed to find the most likely sequence of hidden states from observed events.
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.