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