NDLI: On the Equivalence Between Canonical Forms of Recursive Systematic Convolutional Transducers Based on Single Shift Registers

Content Provider	IEEE Xplore Digital Library
Author	Balta, H. Douillard, C. Isar, A.
Copyright Year	2013
Abstract	Standardized turbo codes (TCs) use recursive systematic convolutional transducers of rate b/(b + d), having a single feedback polynomial (b+dRSCT). In this paper, we investigate the realizability of the b+dRSCT set through two single shift register canonical forms (SSRCFs), called, in the theory of linear systems, constructibility, and controllability. The two investigated SSRCF are the adaptations, for the implementation of b+dRSCT, of the better-known canonical forms controller (constructibility) and observer (controllability). Constructibility is the implementation form actually used for convolutional transducers in TCs. This paper shows that any b+1RSCT can be implemented in a unique SSRCF observer. As a result, we build a function, ξ:H → G, which has as definition domain the set of encoders in SSRCF constructibility, denoted by H, and as codomain a subset of encoders in SSRCF observer, denoted by G. By proving the noninjectivity and nonsurjectivity properties of the function ξ, we prove that H is redundant and incomplete in comparison with G, i.e., the SSRCF observer is more efficient than the SSRCF constructibility for the implementation of b+1RSCT. We show that the redundancy of the set H is dependent on the memory m and on the number of inputs b of the considered b+1RSCT. In addition, the difference between G and ξ(H) contains encoders with very good performance, when used in a TC structure. This difference is consistent for m ≈ b > 1. The results on the realizability of the b+1RSCT allowed us some considerations on b+dRSCT, with b, d > 1, as well, for which we proposed the SSRCF controllability. These results could be useful in the design of TC based on exhaustive search. So, the proposed implementation form permits the design of new TCs, which cannot be conceived based on the actual form. It is possible, even probable, among these new TCs to find better performance than in the current communication standards, such as LTE, DVB, or deep-space communications.
Starting Page	381
Ending Page	394
Page Count	14
File Size	14193194
File Format	PDF
ISSN	21693536
Volume Number	2
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2014-01-01
Publisher Place	U.S.A.
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Convolutional codes Encoding Systematics Decoding Transducers Polynomials Complexity theory turbo codes Canonical forms convolutional encoders equivalence encoding matrix
Content Type	Text
Resource Type	Article
Subject	Computer Science Engineering Materials Science

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