(2,1,3) Convolutional Code - Implementation and Analysis

The (2,1,3) convolutional code represents a classical forward error correction scheme particularly effective against channel noise in digital communication systems. This coding structure enhances error correction capability through memory integration, with parameters indicating:

Parameter Significance - Input: 1 bit generates 2 output bits (code rate 1/2) - Constraint length K=3 signifies current output depends on past 2 inputs - The encoder employs 2 shift registers, creating 4 possible states (00,01,10,11)

Implementation Principle The encoder performs modulo-2 addition using two generator polynomials. A typical configuration uses g1=[1,1,1] and g2=[1,0,1], corresponding to specific tap positions. When a new input bit arrives, the system calculates output bit pairs based on current register states while simultaneously updating register contents through shift operations. In MATLAB implementation, this can be modeled using logical XOR operations on tapped register positions.

Visualization Analysis - State Transition Diagram: Directed edges illustrate state transitions with input/output bit annotations (e.g., 0/00) - Trellis Diagram: Expands state transitions along timeline, essential for Viterbi decoding path tracking algorithms - Bit Error Rate Curve: Monte Carlo simulations demonstrate SNR-BER relationships, highlighting coding gain advantages through comparative plotting functions

Application Extensions This code can be converted to higher rates via puncturing techniques, and frequently combines with QAM modulation in practical systems. Understanding its principles provides foundation for modern coding technologies like Turbo codes, where similar convolutional components are interconnected through interleavers.