In telecommunications, convolutional codes are a type of error-correcting code that generates parity symbols via the sliding application of a boolean polynomial function to a data stream. The sliding application represents the convolution of the encoder over the data, which gives rise to the term itself. As better explained below, the ability to perform economical maximum likelihood soft decision decoding is one of the major benefits of convolutional codes, besides being especially fit for AWGN channels (unlike, for instance, Reed-Solomon codes suited mainly for burst-error codes). Their main parameters are the base code rate k/n (i.e. the ratio between encoder input and output data rate) and the encoder depth K. (often called constraint length). The output of the encoder at a certain time is a function of the current input as well as the previous K-1 inputs, aspect that characterize them as coding technique with memory (unlike block coding ones that are defined as memoryless). Besides K, the actual code generates depends on on the specific connection made inside the encoder block, and typically summarized in polynomial form in the connection vector. The code rate of a convolutional code is commonly modified via puncturing. For example, a code with base code rate 0.5 can be punctured to a higher rate simply by not transmitting a portion of redundancy symbols [2].

In this project the error-correction capability of convolutional codes is discussed and analyzed. First, a MATLAB script (without employing any of its built-in functions) for the convolutional encoding and Viterbi decoding  is proposed. Then, the same algorithm is adapted for a much more efficient C/C++ implementation.

The developed encoder / decoder pair can handle constrain lengths (K) from 3 up to 8. The optimal connection vectors [1] as a function of K have been chosen as​

• G1 = [1 1 1] and G2 = [1 0 1] for K = 3

• G1 = [1 1 1 1] and G2 = [1 0 1 1] for K = 4

• G1 = [1 0 1 1 1] and G2 = [1 1 0 0 1] for K = 5

• G1 = [1 0 1 1 1 1] and G2 = [1 1 0 1 0 1] for K = 6

• G1 = [1 0 0 1 1 1 1] and G2 = [1 1 0 1 1 0 1] for K = 7

• G1 = [1 0 0 1 1 1 1 1] and G2 = [1 1 1 0 0 1 0 1] for K = 8

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The base code-rate (Rc) is 1/2 (i.e. for each information bit a redundancy bit is added), but can be raised to 2/3, 3/4 or 5/6 by means of puncturing. The puncturing vectors employed for this purpose (in accordance with NASA standards) are

• P = [1 1 0 1] for Rc = 1/2

• P = [1 1 0 1 1 0] for Rc = 2/3

• P = [1 1 0 1 1 0 0 1 1 0] for Rc = 5/6

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Moreover, the decoder is designed to perform both hard and soft decoding and, for the latter case, allow to specify the number of quantization levels. However, keep in mind the implemented soft decoding algorithm only works for binary modulations.

The encoder schemes (applying the information of the aforementioned connection vectors) and the trellis diagrams for K = 3 and 4 (for Rc = 1/2) are depicted below in Fig.1 and Fig.2.