Some New Results on Recursive Convolutional Codes and Their Applications

In this paper, a novel decomposition representation is firstly proposed to transform a recursive convolutional code into the serial concatenation of a unit-recursive-convolutional (URC) code with unit nominator polynomial and a non-recursive convolutional (NRC) code. Since the recursive convolutional (RC) code can be regarded as a linear system if all-zero initial state is assumed, the impulse response analysis (IRA) can be applied to reveal the characteristics of the URC code, as well as the RC code. Then, a linear-algebra-like analysis method is developed to enable the computation of the impulse response via the finite field matrix manipulation. It is shown that the equivalent block generation matrix can be used to characterize an RC code completely. Moreover, there is a certain cycle (or period) in the impulse response of any RC code, which corresponds to the state transition cycle of its constituent URC code. It is disclosed that the Hamming weight in every cycle equals half of the cycle length plus one if the denominator polynomial is primitive. Furthermore, the generation matrix of its constituent URC code could be utilized to determine the ending state of an RC code for any input. Finally, it is revealed that the minimum Hamming weight of an RC code can be calculated from the generation matrix of the reverse RC code under some input constraints, thus providing an alternative algorithm for the modified Viterbi algorithm proposed for the weight spectrum calculation of turbo codes