-Convolutional Codes—Part II: Structural Analysis

The structural properties of a noncoherent coded system, which incorporates convolutional codes in conjunction with multiple symbol noncoherent detection, is presented in this second part of a two-part paper, where the performance analysis was provided in Part I. These convolutional codes are referred to as nd-convolutional codes and they provide a general framework for various noncoherent coding systems, including differential systems, for several practical models of the carrier phase. The exponential rate in which the error probability decays to zero, derived in Part I of the paper, is used here to obtain the free equivalent distance of nd-codes, which is the single parameter dominating the error performance at large signal-to-noise ratios. The free equivalent distance is upper-bounded by the free nddistance, which constitutes a more convenient and practical parameter to work with, and it is the basis for a computer search for optimal nd-codes. The resultant codes of the computer search are compared to codes which are optimal for coherent detection, and it is verified that the latter codes are not necessarily optimal for noncoherent detection since they exhibit in many cases a relatively small nd-distance. The ambiguity problem, inherent to noncoherent systems, is also treated in this paper in the general framework of nd-catastrophic codes, and necessary and sufficient conditions for catastrophic error propagation are identified.