are powerful used in digital communications. and are essential tools for understanding and working with these codes, providing visual ways to depict and decode received messages.
These diagrams help us grasp the inner workings of convolutional encoders and decoders. By exploring state transitions, , and , we can see how these codes protect data from errors and how receivers can recover the original message.
State and Trellis Diagrams
Representing Convolutional Codes with State Diagrams
State diagrams visually represent the behavior of a convolutional encoder
Consist of nodes representing and edges representing state transitions
Each edge is labeled with the and corresponding
Provide a compact way to describe the encoding process
Trellis Diagrams: Unfolding the State Diagram
Trellis diagrams are an alternative representation of convolutional codes
Obtained by unfolding the state diagram over time
Each stage in the trellis corresponds to a time step in the encoding process
Nodes at each stage represent the possible encoder states at that time
Edges connecting nodes represent state transitions and are labeled with input/output bits
Exploring Encoder States and Transitions
The number of states in a convolutional encoder depends on the (K)
For a rate 1/n encoder with constraint length K, there are 2K−1 states
Each state corresponds to a unique sequence of the past K−1 input bits
State transitions occur based on the current state and the incoming input bit
The output bits for each transition are determined by the
Trellis Sections and Encoding
A trellis section represents one time step in the encoding process
It consists of nodes representing the current states and edges representing transitions
The input bit determines which transition is taken from each state
The output bits associated with each transition form the encoded sequence
By traversing the trellis section for each input bit, the entire message is encoded
Metrics and Paths
Branch Metrics: Measuring Transition Likelihood
Branch metrics quantify the likelihood or "distance" of a particular
Calculated by comparing the received bits with the expected output bits for each transition
Common branch metric is the (number of bit differences)
Branch metrics are assigned to each edge in the trellis diagram
Lower branch metrics indicate a closer match between received and expected bits
Path Metrics: Accumulating Branch Metrics
Path metrics represent the accumulated likelihood or "distance" of a sequence of state transitions
Calculated by summing the branch metrics along a path through the trellis
Each path corresponds to a possible transmitted sequence
Path metrics are updated at each stage of the trellis based on the incoming branch metrics
The path with the lowest metric is considered the most likely transmitted sequence
Survivor Paths and Decoding Decisions
At each stage in the trellis, multiple paths may converge at a given state
The path with the lowest metric among the converging paths is called the survivor path
are retained for each state, while the other paths are discarded
are made by tracing back the survivor paths through the trellis
The traced-back path with the lowest overall metric is selected as the decoded sequence
Advanced Trellis Concepts
Time-Varying Trellis Structures
Time-varying trellises have a structure that changes over time
The number of states, state transitions, or output bits may vary at different stages
Useful for representing convolutional codes with time-varying properties
Examples include punctured codes and codes with periodic structures
Decoding time-varying trellises requires adapting the metric calculations and path management
Trellis Termination Techniques
Trellis termination ensures that the encoder ends in a known state
Achieved by appending a of input bits to the message
The termination sequence drives the encoder to a predetermined final state (usually all-zero state)
Termination reduces the number of possible paths at the end of the trellis
Simplifies the and improves error performance
Common termination techniques include zero-tail termination and tail-biting