Convulation Code

Topics: LTI system theory, Convolutional code, Control theory Pages: 3 (647 words) Published: April 28, 2013
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. 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
n2 = m0 + m-1
n3 = m1 + m-1.

One can see that the input being encoded is included in the output sequence too (look at the output 2). Such codes are referred to as systematic; otherwise the code is called non-systematic.

Recursive codes are almost always systematic and, conversely, non-recursive codes are non-systematic. It isn't a strict requirement, but a common practice. A convolutional encoder is called so because it performs a convolution of the input stream with the encoder's impulse responses:

y_i^j=\sum_{k=0}^{\infty} h^j_k...
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