Abstract
It is well known that there is a correspondence between convolutional codes and discrete-time linear systems over finite fields. In this paper, we employ the linear systems representation of a convolutional code to develop a decoding algorithm for convolutional codes over the erasure channel. In this kind of channel, which is important due to its use for data transmission over the Internet, the receiver knows if a received symbol is correct. We study the decoding problem using the state space description of a convolutional code, and this provides in a natural way additional information. With respect to previously known decoding algorithms, our new algorithm has the advantage that it is able to reduce the decoding delay as well as the computational effort in the erasure recovery process. We describe which properties a convolutional code should have in order to obtain a good decoding performance and illustrate it with an example.
Highlights
In modern communication, especially over the Internet, the erasure channel is widely used for data transmission
We study the decoding problem using the state space description of a convolutional code, and this provides in a natural way additional information
We describe which properties a convolutional code should have in order to obtain a good decoding performance and illustrate it with an example
Summary
Especially over the Internet, the erasure channel is widely used for data transmission. The underlying distance measure one uses to identify the closest trajectory (i.e., the closest codeword) depends on the kind of channel that is used for data transmission This decoding process can be interpreted as minimizing a cost function attached to the corresponding linear system, which measures the distance of a received word to a codeword or the distance of a measured trajectory to a possible trajectory, respectively. The big advantage when decoding over an erasure channel is that we know that all received symbols, i.e., all symbols except ∗ in uand v, are correct and we only have to find a way to replace the unknowns ∗ be the original values to bring the cost function to its minimal value, which equals the number of erasures.
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