Talk abstract:

Linear Systems Decoding of Convolutional Codes

Brian M. Allen
Department of Mathematics
University of Notre Dame
ballen1@nd.edu
http://www.nd.edu/~ballen1

The presentation will focus on the decoding of the convolutional codes given by a linear systems representation. In particular, convolutional codes will be defined by an (A,B,C,D) input-state-output linear system where the codewords are the concatenation of the sequences of inputs and outputs. By treating convolutional codes in this way we are able to characterize the codewords using simple algebraic equations. Especially important in this representation are the controllability matrix of the pair (A,B) and the observability matrix of the pair (A,C). These matrices can be seen as a parity check matrix and generator matrix, respectively, for two linear block codes associated with the convolutional code.

We will follow in the line of Rosenthal [1] to develop a sliding window decoding algorithm for convolutional codes using the associated block codes above. By emphasizing the output sequences as much as the input sequences we are able to greatly enhance the previous work. Several variations on the algorithm will be discussed and some examples will be presented. In particular, some theoretical results using the MDS convolutional codes presented by Rosenthal and Smarandache [2] will be given.

Time permitting, we will explore the notion of majority logic decoding utilizing the linear systems representation. The suitability of the MDS codes mentioned above for this type of decoding scheme will be investigated by studying the column distance functions for these codes.

References:

[1] J. Rosenthal. An algebraic decoding algorithm for convolutional codes. In G. Picci and D.S. Gilliam, editors, Dynamical Systems, Control, Coding, Computer Vision: New Trends, Interfaces, and Interplay, pages 343-360. Birkhäuser, Boston-Basel-Berlin, 1999.

[2] J. Rosenthal and R. Smarandache. Maximum distance separable convolutional codes. Technical Report 1998-074, MSRI, Berkeley, California, 1998. To appear in AAECC.

Material used during the talk

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