Convolutional coding techniques for data protectio

Massey, J. L.

English

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https://ntrs.nasa.gov/search.jsp?R=19690011548 2020-03-12T07:31:55+00:00Z
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March 4, 1969
QUARTERLY PROGRESS REPORT
CONVOLUTIONAL CODING TECHNIQUES
FOR DATA PROTECTION
NASA GRANT NGL-15-004-026
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and
Flight Data Systems Branch
NASA Goddard Space Flight Center
Greenbelt, Maryland 20771
ATT: Dr. Robert W. Rochelle (Code 710)
National Aeronautics and Space Administration
Washington, D. C. 20546
ATT: Miss Winnie M. Morgan (Code - USI)
Technical Reports Assistant
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Submitted to:
Principal Investigator:
	 Dr. James L. Massey
Professor of Electrical Engineering
University of Notre Dame*
Notre Dame, Ind. 46556
Research Assistants:
	 J. Chang
D. Costello
J. Geist
Research Period Reported: 	 Nov. 16, 1966 to Feb. 15, 1969
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1. Research in Sequential Apcoding
Research has continued into the problem of characterizing the
"typical" behavior of a sequential decoder, i.e. that behavior which
occurs with probability 1-e or more where a is a small number, say
10-3 . The objective is to obtain the probability distribution for
-performance in this typical, region in terms of (a) code parameters
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which can be easily determined, and (b) parameters of the Fan g decoding
algorithm, namely the threshold increment A and the branch metric
values. It is then anticipated that an analytical optimization of all
these parameters can be carried out--a result that could be of con-
siderable practical value either by showing how the typical performance
of existing decoders could be improved or by showing that no additional
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work is needed if existing decoders prove to be already near optimal.
r	 Some tentative results in this direction have been obtained byi
J. Geist in work that is continuing. Realizing that most of the typical
computation is performed in searches along the "correct path" through
the decoding tree (in a sense when the decoder is reassuring itself that
it is indeed on the correct path), Geist has concentrated on this phase
of the decoding effort and as one interesting example of his results
has been able to obtain a geometric construction which might be termed
"superposition of identical triangles" for determining the computation
done at each branch. An example will illustrate the method;
Example: Consider a rate one-half Fano sequential decoder for a
binary symmetric channel with d = 1 and branch weights of +l, -4, and
-9 depending on whether the two branch digits agree exactly, disagree
in one place, or disagree entirely with the received digits. "Then an
isolated single channel error causes 1 extra computation on the third
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tbranch preceding the error, 2 extra on the second, 3 extra on
the first, and 4 extra on the branch with the error. Thl*,-, extra
computation can be represented by the triangular .,,.,rayA
1 2 3 4
The additional computation due to two isolated errors separated
by an error free branch can be determined geometrically as
1 3 4 5 6 3 4
That is, there will be one extra computation fourth branch
preceding the f 4,rst error, 3 extra on the third preceding, etc.
This geometric rule turns out to be completely general for
arbitrary numbers of errors, and is expected to provide needed
insight into the typical "bulges" in computation due to several
clustered errors in a decoding frame.
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2. Research into Convolutional Codes
D. Costello has continued his investigations into the theoretical
structure of convolutional codes. An aspect of particula,n attention
has been the so-called "free distance" dfree of the code which is
, the minimum Hamming distance between two arbitrarily long encoded
sequences with different first information digits (in contrast to
the usual minimum distance dmin in which the minimum is taken only
over the first constraint length.) Simulations as reported in
D. J. Costello, Jr., "A Construction Technique for
Random Error Correcting Convolutional Codes," Memo.
EE-6822, Dept. of Elec. Engr., Univ. of Notre Dame,
Nov. 14, 1968. (Preprint of Correspot,.uance Article to
appear in IEEE Transactions on Information Theory.)
have established a close correlation between dfree and the undetected
error probability of sequential decoding, as had been conjectured
theoretically. Costello has recently obtained an improved Plotkin-
like upper bound on dfree applicable to all. code rates and code types
(systematic or non-systematic) and has obtained an improved Gilbert-
type lower bound which suggests that dfree can be made roughly twice
dmin over an interesting range of rates. This work will be part of
a later report summarizing all our theoretical investigations of con-
volutional codes.
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Convolutional coding techniques for data protection Quarterly progress report, 16 Nov. 1968 - 15 Feb. 1969

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