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

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.