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Unformatted text preview: IEEE COMMUNICATIONS LETTERS, VOL. 3, NO. 8, AUGUST 1999 251 A New Technique for Computing the Weight Spectrum of Turbo-Codes Oscar Y. Takeshita, Member, IEEE , Marc P. C. Fossorier, Member, IEEE , and Daniel J. Costello, Jr., Fellow, IEEE Abstract— We obtain an accurate and simple analytical ap- proximation to the bit error probability of maximum-likelihood decoding of parallel concatenated convolutional codes (Turbo- codes) by extending a result from the paper by Fossorier et al. , which investigates the weight enumerating function of ter- minated convolutional codes, to compute their conditional weight enumerating function. Index Terms— Bit-error rate, cutoff rate, terminated convolu- tional codes, Turbo-codes, weight spectrum. I. INTRODUCTION T HE introduction of the concept of a uniform interleaver in  was an important step in the derivation of an analytical upper bound to the bit-error rate (BER) performance of parallel concatenated convolutional codes (PCCC’s) or Turbo-codes . However, approximations and difficult computational pro- cedures were still needed to derive the bound because, in principle, the complete weight-enumerating function (WEF) of a long terminated convolutional code is needed. In , a more accurate and systematic approximation to the bound is presented. However, due to a numerical precision problem inherent to the methodology, only block lengths up to about 1000 can be considered. It is also mentioned in  that the upper bounds are useful only at signal-to-noise ratios (SNR’s) larger than the channel cutoff rate since, at smaller SNR’s, the usual union bound diverges rapidly, especially for large block lengths. Thus, the traditional union bound is primarily useful for approximating the error-floor region of the BER performance. (We prefer to call these upper bound- ing techniques approximations because they are computed assuming a uniform interleaver and for large block lengths the complete WEF of the terminated component code is difficult to obtain.) Recently, several new bounds that estimate the BER performance more accurately at SNR’s smaller than the cutoff rate have also been proposed –. In this letter we are interested in approximations to the tradi- tional union bound that significantly reduce the computational Manuscript received January 4, 1999. The associate editor coordinating the review of this letter and approving it for publication was Prof. N. C. Beaulieu....
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- Spring '12
- Error detection and correction, Convolutional code, Convolutional Codes, conditional wef