In telecommunication, a convolutional code is a type of error-
correcting code in which
Each m-bit information symbol (each m-bit string) to be encoded
is transformed into an n-bit symbol, where m/n is the
code rate (n ≥ m) and
The transformation is a function of the last k information symbols,
where k is the constraint length of the code.
Where convolutional codes are used
Convolutional codes are used extensively in numerous
applications in order to achieve reliable data transfer,
including digital video, radio, mobile communication, and satellite
communication. These codes are often implemented
in concatenation with a hard-decision code, particularly Reed
Solomon. Prior to turbo codes, such constructions were the most
efficient, coming closest to the Shannon limit.
Convolutional encoding
To convolutionally encode data, start with k memory registers,
each holding 1 input bit. Unless otherwise specified, all memory
registers start with a value of 0. The encoder has n modulo-
2 adders (a modulo 2 adder can be implemented with a
single Boolean XOR gate, where the logic is: 0+0 = 0, 0+1 = 1,
1+0 = 1, 1+1 = 0), and n generator polynomials — one for each
adder (see figure below). An input bit m1 is fed into the leftmost
register. Using the generator polynomials and the existing values
in the remaining registers, the encoder outputs n bits. Now bit
shift all register values to the right (m1 moves to m0, m0 moves
to m-1) and wait for the next input bit. If there are no remaining
input bits, the encoder continues output until all registers have
returned to the zero state.
The figure below is a rate 1/3 (m/n) encoder with constraint
length (k) of 3. Generator polynomials are G1 = (1,1,1), G2 =
(0,1,1), and G3 = (1,0,1). Therefore, output bits are calculated
(modulo 2) as follows:
n1 = m1 + m0 + m-1
