  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.
