In this paper, we study turbo codes from the digital signal processing point of view by defining turbo codes over the complex field. It is known that iterative decoding and interleaving between concatenated parallel codes are two key elements that make turbo codes perform significantly better than the conventional error control codes. This is analytically illustrated in this paper by showing that the decoded noise mean power in the iterative decoding decreases when the number of iterations increases, as long as the interleaving decorrelates the noise after each iterative decoding step. An analytic decreasing rate and the limit of the decoded noise mean power are given. The limit of the decoded noise mean power of the iterative decoding of a turbo code with two parallel codes with their rates less than 1/2 is one third of the noise power before the decoding, which can not be achieved by any non-turbo codes with the same rate. From this study, the role of designing a good interleaver can also be clearly seen.

翻译：本文从数字信号处理的角度出发，通过在复数域上定义Turbo码，对Turbo码进行了研究。众所周知，迭代译码以及级联并行码之间的交织，是使得Turbo码性能显著优于传统纠错码的两个关键要素。本文通过分析表明，只要交织能够在每次迭代译码步骤后对噪声进行去相关，那么随着迭代次数的增加，迭代译码中的译码噪声平均功率就会降低。本文给出了译码噪声平均功率的解析递减速率及其极限值。对于由两个速率均小于1/2的并行码构成的Turbo码，其迭代译码的译码噪声平均功率极限值为译码前噪声功率的三分之一，而任何具有相同速率的非Turbo码均无法达到此极限。通过本研究，还可以清晰地看出设计良好交织器的作用。