On the Free Distance of TURBO Codes and Related Product Codes

It is the goal of this project to explain the performance of Turbo Codes. An algorithm for determining the free distance of Turbo Codes is proposed and applied to a few examples. The origin of the error oor is explained. The method of random interleaving is corrected and extended to punctured Turbo Codes. The distance spectrum of Turbo Codes is investigated using random interleaving. A theory is developed that explains the performance of Turbo Codes. The decoding complexity of Turbo Codes is reviewed. Constituent encoder optimization and interleaver design are discussed. Chapter 1: Introduction to Turbo Codes introduces the Turbo Code's performance and the Turbo Coding scheme. Chapter 2: The Free Distance of Turbo Codes presents an algorithm to calculate the free distance of a Turbo Code. This algorithm is applied to a few examples. The origin of the error oor is identi ed. The problem of interleaver design for optimal free distance is formalized. Chapter 3: The Distance Spectrum of Turbo Codes explains how the distance spectrum determines the Turbo Code's performance. In a rst analysis, the in uence of the interleaver on the distance spectrum is shown. The method of random interleaving is corrected and extended to punctured Turbo Codes. Theoretical results are derived that show how the distance spectrum changes as the interleaver size increases. A theory is proposed that relates the Turbo Code's performance to its distance spectrum. Chapter 4: The Decoding Complexity of Turbo Codes reviews the decoding complexity. The decoding complexity of a Turbo Code is compared to the decoding complexity of a convolutional code with Viterbi decoding. Chapter 5: Improvements for Turbo Codes discusses constituent encoder selection and interleaver design. The good performance of Turbo Codes with maximum cycle length constituent encoders is explained. Chapter 6: The Relation Between Turbo Codes and Product Codes Chapter 7: Conclusions