Nd-convolutional codes - part I: performance analysis

Yosef Kofman, Ephraim Zehavi, Shlomo Shamai

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15 Scopus citations


A noncoherent coded system, which incorporates convolutional codes in conjunction of multiple symbol noncoherent detection, is presented in this two-part paper, where Part I focuses on the performance analysis of the system and Part II deals with the structural properties of the underlying convolutional codes. These convolutional codes are referred to as nd-convolutional codes. It is shown that nd-convolutional codes provide a general framework for various noncoherent coding systems, including differential systems. Two models of the carrier phase are examined and the relationships between them is established. For the first one, the carrier phase remains constant for L channels signals, whereas for the second one, it unvaries throughout the transmission period. The regular structure of nd-codes facilitates the evaluation of a simple upper bound on the pairwise and bit error probabilities, as well as a simple expression for the generalized cutoff rate. The exponential rate of the error probability, which is the single parameter governing the error performance at large signal-to-noise ratios, is identified via Large Deviations techniques. This parameter leads to the interesting conclusion that increasing L does not necessarily monotonically improve the error performance of the noncoherent system. The same conclusion is reached by examining upper bounds and computer simulation results of several interesting examples. These examples also reveal that optimal codes for coherent detection are not necessarily optimal for noncoherent detection and a search for good codes, some of which are tabulated in Part II of the paper, is required.

Original languageEnglish
Pages (from-to)558-575
Number of pages18
JournalIEEE Transactions on Information Theory
Issue number2
StatePublished - 1997
Externally publishedYes

Bibliographical note

Funding Information:
Manuscript received March 29, 1994; revised January 10, 1996. This work was supported in part by the Chief Scientist Office, Israel Ministry of Communications. The material in this paper was presented in part at the ISSSE, Paris, France, July 1992 and at the 18th Israel Convention of Electrical and Electronic Engineers, Tel-Aviv, Israel, March 1995. Y. Kofman is with Power Spectrum Technology, Ltd., P.O. Box 3819, Kiryat Aryeh, Petach-Tikva 49130, Israel. E. Zehavi is with Qualcomm Israel, P.O. Box 223, Tirat Hacarmel 39026, Israel. S. Shamai is with the Department of Electrical Engineering, Technion–Israel Institute of Technology, Technion City, Haifa 32000, Israel. Publisher Item Identifier S 0018-9448(97)00622-6.


  • BPSK
  • Convolutional codes
  • DPSK
  • Differential decoding
  • Multisymbol detection
  • Nd-codes
  • Noncoherent detection


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