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A linear algebra approach to minimal convolutional encoders

Johannesson, Rolf LU and Wan, Zhe-Xian LU (1993) In IEEE Transactions on Information Theory 39(4). p.1219-1233
Abstract
The authors review the work of G.D. Forney, Jr., on the algebraic structure of convolutional encoders upon which some new results regarding minimal convolutional encoders rest. An example is given of a basic convolutional encoding matrix whose number of abstract states is minimal over all equivalent encoding matrices. However, this encoding matrix can be realized with a minimal number of memory elements neither in controller canonical form nor in observer canonical form. Thus, this encoding matrix is not minimal according to Forney's definition of a minimal encoder. To resolve this difficulty, the following three minimality criteria are introduced: minimal-basic encoding matrix, minimal encoding matrix, and minimal encoder. It is shown... (More)
The authors review the work of G.D. Forney, Jr., on the algebraic structure of convolutional encoders upon which some new results regarding minimal convolutional encoders rest. An example is given of a basic convolutional encoding matrix whose number of abstract states is minimal over all equivalent encoding matrices. However, this encoding matrix can be realized with a minimal number of memory elements neither in controller canonical form nor in observer canonical form. Thus, this encoding matrix is not minimal according to Forney's definition of a minimal encoder. To resolve this difficulty, the following three minimality criteria are introduced: minimal-basic encoding matrix, minimal encoding matrix, and minimal encoder. It is shown that all minimal-basic encoding matrices are minimal and that there exist minimal encoding matrices that are not minimal-basic. Several equivalent conditions are given for an encoding matrix to be minimal. It is proven that the constraint lengths of two equivalent minimal-basic encoding matrices are equal one by one up to a rearrangement. All results are proven using only elementary linear algebra (Less)
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author
organization
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type
Contribution to journal
publication status
published
subject
in
IEEE Transactions on Information Theory
volume
39
issue
4
pages
1219 - 1233
publisher
IEEE--Institute of Electrical and Electronics Engineers Inc.
external identifiers
  • scopus:0027634632
ISSN
0018-9448
DOI
10.1109/18.243440
language
English
LU publication?
yes
id
160cecfb-634a-4cf1-af71-ed8cee898f26 (old id 1056840)
alternative location
http://ieeexplore.ieee.org/iel1/18/6268/00243440.pdf
date added to LUP
2008-04-11 13:53:48
date last changed
2017-01-01 07:43:19
@article{160cecfb-634a-4cf1-af71-ed8cee898f26,
  abstract     = {The authors review the work of G.D. Forney, Jr., on the algebraic structure of convolutional encoders upon which some new results regarding minimal convolutional encoders rest. An example is given of a basic convolutional encoding matrix whose number of abstract states is minimal over all equivalent encoding matrices. However, this encoding matrix can be realized with a minimal number of memory elements neither in controller canonical form nor in observer canonical form. Thus, this encoding matrix is not minimal according to Forney's definition of a minimal encoder. To resolve this difficulty, the following three minimality criteria are introduced: minimal-basic encoding matrix, minimal encoding matrix, and minimal encoder. It is shown that all minimal-basic encoding matrices are minimal and that there exist minimal encoding matrices that are not minimal-basic. Several equivalent conditions are given for an encoding matrix to be minimal. It is proven that the constraint lengths of two equivalent minimal-basic encoding matrices are equal one by one up to a rearrangement. All results are proven using only elementary linear algebra},
  author       = {Johannesson, Rolf and Wan, Zhe-Xian},
  issn         = {0018-9448},
  language     = {eng},
  number       = {4},
  pages        = {1219--1233},
  publisher    = {IEEE--Institute of Electrical and Electronics Engineers Inc.},
  series       = {IEEE Transactions on Information Theory},
  title        = {A linear algebra approach to minimal convolutional encoders},
  url          = {http://dx.doi.org/10.1109/18.243440},
  volume       = {39},
  year         = {1993},
}