Optimal Scheduling for Iterative Decoding

We propose an algorithm for finding the optimal decoder activation order in a system with more than two concatenated codes. Optimality is defined as the order that yields convergence using the lowest computational complexity, guided by the extrinsic information transfer (EXIT) charts of the component codes. We also describe how the convergence threshold for the system can be visualized by combining several multi-dimensional EXIT charts into one two-dimensional EXIT chart. I. Optimal Activation Order When two codes are concatenated, iterative decoding is done by alternating between the two component decoders. The average decoding trajectory and the convergence threshold can be visualized in a two-dimensional extrinsic information transfer (EXIT) chart. In multiple parallel concatenated codes [1] and multiple serially concatenated codes [2] it is no longer obvious how to schedule the decoder activations. Furthermore, the EXIT charts in these systems are multidimensional [1, 2] and can become unmanageable without an appropriate approach. Consider a system with N serially concatenated codes, starting with the innermost decoder 1 and ending with the outermost decoder N . An EXIT chart is defined by iterative exchange of mutual information (MI), leading to trajectories where the extrinsic MI is expressed as a function of prior MIs [1], IE(IA(x), IA(y)). Here IA(xn) = I[xn;A(xn)] and IE(xn) = I[xn;E(xn)] denote the MI between the input symbols, xn, and the prior and the extrinsic probabilities, respectively (similarly for the output symbols yn), with 1 ≤ n ≤ N . The following monotonicity assumption will be used. Assumption 1 All extrinsic MIs are monotonically increasing, namely IE(IA(x) + ǫx, IA(y) + ǫy) ≥ IE(IA(x), IA(y)), if ǫx ≥ 0 and ǫy ≥ 0. Under Assumption 1 (reasonable for any useful code), the convergence point is independent of the activation order as long as an unlimited number of decoder activations is allowed. Convergence for N > 2 systems may be visualized using the following projection. Let the MI for code N be fixed, IE(yN ) = I, and activate all the other decoders until their MIs reach fixed values. Do this for all 0 ≤ I ≤ 1 and plot IE(xN−1) F. Brännström and L. K. Rasmussen are supported by the Swedish Research Council under Grant 621-2001-2976. F. Brännström is also supported by Personal Computing and Communication (PCC++) under Grant PCC-0201-09. L. K. Rasmussen and A. Grant are supported by the Australian Government under ARC Grant DP0344856. Part of this work was performed while F. Brännström was visiting the University of South Australia. 1The term activation is used instead of iteration. In a system with two components, one iteration is the same as two activations, one for each of the two decoders. versus I in a two-dimensional EXIT chart. Now do the same with code N−1, letting IE(xN−1) = I and plot I versus IE(yN ) in the same chart. A vertical step in this chart now represents an unspecified number of activations of all decoders except decoder N , until nothing more can be gained. A horizontal step represents activations of all decoders except decoder N − 1. The convergence threshold, but not the number of activations, can now easily be determined. Independently, a similar projection was developed specifically for three serially concatenated codes in [2]. Two of the most commonly used periodic orderings for three serially concatenated codes are 1, 2, 3, 1, 2, 3, . . . and 1, 2, 3, 2, 1, 2, 3, 2, . . .. Another suggestion is to activate the decoder for which the gain in MI is maximal [2]. Let cn be a constant proportional to the computational decoding complexity of decoder n. We can define the optimal order, which reaches the convergence point using the lowest possible decoding complexity. This optimal order depends on the codes involved and the operating signal-to-noise ratio. The collection of all possible activation orders can be described by anN -state trellis (one state for each decoder). Each path through the trellis corresponds to an activation order. To each path we assign a J-dimensional metric, where the first J−1 elements are the MIs and the last element is the total decoding complexity. In a system with N serially concatenated codes there are J − 1 = 2N − 2 different extrinsic MIs, updated according to the activation order and the EXIT charts. According to Assumption 1, the metric is monotonically nondecreasing with the number of activations. Applying an exhaustive search, the total number of paths grows exponentially with the number of activations. The search complexity can be greatly reduced, using an adaptation of the well known Viterbi algorithm modified to use multidimensional metrics. A partial order of the metrics is defined, whereby a path is discarded if there is another path entering the same state with higher MIs in all J − 1 dimensions and lower total complexity. The possibility of exponential growth is not eliminated, but only a small number of retained paths are observed in practice. Numerical examples show that 20– 40% in computational complexity can be saved by choosing the optimal order instead of any of the two periodic orderings mentioned above. The algorithm and the projections of the EXIT charts described above can also be applied in a system with N parallel concatenated codes as long as all codes satisfy Assumption 1.