Analysis of the dynamics of iterative interference cancellation in iterative decoding

Iterative decoding in digital data transmission systems has become very popular with invention of turbo coding [1]. Methods of iterative processing have quickly been realized to be applicable not only to turbo coded signals but also to joint decoding of code-division multiple access (CDMA) systems, intersymbol interference limited channels, multiple-input multiple-output (MIMO) systems, and others. In particular, iterative decoding of CDMA systems is proved to be a very powerful decoding method, which has the potential of realizing a significant portion of the capacity of such a channel [2–8]. Key issues for iterative decoders are convergence conditions. Usually, it is not possible to investigate their convergence by purely analytical methods. For that reason, such methods are typically combined with computer simulations. An example is density evolution analysis, first applied to low-density parity-check (LDPC) codes [9]. It should be noted that in many cases densities involved are either exactly Gaussian, or they are well approximated by Gaussian densities. In these cases, much simpler analysis can be used instead [7, 10–12]. In particular, it has been shown in [11] that simple iterative cancellation receivers (see Section 3) can achieve the capacity of a multiple access channel if powers (or rates) of participating users are chosen in a certain optimal way. In this paper, which is a mathematical supplement to [11], dynamics of the behavior of iterative decoders with iterative cancellation is studied by analytical methods. Some of the results derived are used in [11] to describe the behavior of large-scale CDMA systems under various transmission conditions (without mathematical proofs). Comparison to decoders using linear minimum meansquare error (MMSE) filters is also presented. We consider the model in which the input, d = (d1, . . . , dK) , and the output, y = (y1, . . . , yN )T , channel vectors are related by the formula