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Convolutional Codes

Rate 1/2, Constraint Length 3 Encoder (K=3, g1=[1,1,1], g2=[1,0,1])

Input: 1 0 1 1 0
Output: --

Convolutional Encoder (Shift Register + XOR Gates)

Input Sequence (1 bit at a time)

Output Sequence (2 bits per input)

How Convolutional Codes Work

Shift Register: Input bits shift through K-1 memory cells. Current output depends on current and past inputs.

Generator Polynomials: g1=[1,1,1] and g2=[1,0,1] define which taps connect to each output via XOR.

Rate 1/2: For every 1 input bit, we get 2 output bits. More redundancy = better error correction!

Constraint Length K: Determines memory. Longer K = better performance but more complex decoder.

Decoding: Viterbi algorithm finds most likely path through code trellis.

Applications: Deep space (Voyager), GSM, satellite, Wi-Fi (802.11a/g/n).