Iterative Decoding and Channel Estimation

We investigate iterative decoding and channel estimation for multiple-access channels. Results are obtained concerning the fixed points of such iterations. I. Iterative Receiver Principle In [1] an iterative receiver was proposed for the linear multiple access channel. We now consider an approach for integration of channel estimation into this technique whereby we use the a-posteriori probabilities of the information symbols as uncertain training sequences for the purposes of channel estimation. We investigate the properties of fixed points of such iterations. Let S be a vector space. Unconstrained sequences can take any value u ∈ S as opposed to constrained sequences x ∈ C ⊂ S. We are interested in low-complexity joint detection (or estimation) for sets of constrained sequences observed according to known transition probabilities. These probabilities are defined by some combination of deterministic mappings (e.g. linear combining) and non-deterministic perturbations (e.g. noise). Suppose that the sequences xk, k = 1, . . . , n are each produced by a mapping Ck of an unconstrained sequence uk. The random sequence y is observed according to p(y | x1, . . . , xn). The uk may or may not be independent, but are conditionally dependent given y. This model can be thought of as a multiple-access communications system (the xk are coded information sequences), but is rich enough to describe other systems of interest, such as inter-symbol interference channels (by allowing some of the xk to represent the sequence of channel taps, obeying known spectral constraints) and space-time diversity channels. Optimal detection means the determination of either the posterior density p(u1, u2, . . . , un | y), or its marginals, taking into account the constraints. This is usually an NP-complete problem and we propose a reduced complexity iterative algorithm. The basic principle that we propose for design of such algorithms may be stated concisely as follows. 1. Incorporate dependence, ignore constraints. 2. Incorporate constraints, ignore dependence. We iteratively update the distributions pk(uk). Ideally pk converges over iteration to the k-th marginal of the true posterior distribution p(u1, u2, . . . , un | y). The principle also applies to estimation problems, in which case the distributions are replaced with the current estimates, which we hope converge to some desired estimator e.g. MMSE. Let p = {p1(u1), p2(u2), . . . , pn(un)} be the sequence priors. At the conclusion of any iteration step, the unconstrained joint detector, using as priors the current set of marginal distributions p, produces a new set p, taking into account only the conditional dependencies. All the constraints are relaxed. This results in a p that may place mass on “impossible” events. Relaxation of (especially integer) constraints can result in low-complexity heuristics. An example of this is applying the decorrelator or MMSE filter for detection with a linear model with integer constraints. A bank of constrained detectors ignores the interdependencies between the uk. The detector for uk updates the current prior marginal pk based on the constraint Ck and p(y | uk). For convolutionally coded data, we may use the forward-backward algorithm. For a sequence of channel taps we may use a Kalman filter. II. Convergence Analysis We shall now consider an asynchronous K user CDMA system in the absence of multipath fading. Identical convolutional codes with free distance dfree are used by each transmitter. We are interested in the effective noise variance at the output of each iteration. Considering an input noise variance v to the constrained data estimator (Viterbi decoder), we may bound the output noise variance vd. vd ≥ f(v) = 4dfreeQ (