Receding horizon decoding of convolutional codes

Decoding of convolutional codes poses a significant challenge for coding theory. Classical methods, based on e.g. Viterbi decoding, suffer from being computationally expensive and are restricted therefore to codes of small complexity. Based on analogies with model predictive optimal control, we propose a new iterative method for convolutional decoding that is cheaper to implement than established algorithms, while still offering significant error correction capabilities. The algorithm is particularly well-suited for decoding special types of convolutional codes, such as e.g. cyclic convolutional codes.

💡 Research Summary

The paper addresses the long‑standing problem of high computational complexity in decoding convolutional codes, especially when the code memory (constraint length) is large. Classical optimal decoders such as the Viterbi algorithm require exhaustive traversal of the trellis, leading to exponential growth in both time and memory requirements. To overcome this limitation, the authors introduce a novel “receding horizon decoding” (RHD) scheme that draws inspiration from model predictive control (MPC).

In RHD the received sequence is processed through a sliding window of fixed length N, called the prediction horizon. Within each window the decoder builds a local trellis that only contains the symbols currently under consideration. A dynamic‑programming step analogous to the Viterbi recursion is then applied to this reduced trellis to obtain the minimum‑cost path. The oldest symbol on that path (i.e., the symbol that entered the window earliest) is declared decoded and permanently fixed. The window is then shifted forward by a step size s (typically s = 1), a new received symbol is appended, and the process repeats. By limiting optimality to a finite horizon, the algorithm reduces the per‑iteration complexity from O(L·2^k) (where L is the total block length and k the memory order) to O(N·2^k), which scales linearly with the overall sequence length.

A central theoretical contribution is the proof that, for sufficiently large horizon N, the receding horizon decoder converges to the same error‑correction performance as the full‑trellis Viterbi decoder. The authors also analyze the trade‑off between N and the step size s: larger N yields performance closer to optimal but incurs higher computation, while larger s reduces latency at the cost of potentially higher error propagation. To mitigate the latter, a “weight‑recalibration” mechanism is introduced that dynamically adjusts the branch metrics at the window boundaries, suppressing error accumulation across successive windows.

The paper places particular emphasis on cyclic convolutional codes, a class of codes whose generator matrices repeat periodically. Traditional Viterbi decoding of such codes suffers from severe memory overhead because the trellis must be expanded over many periods. In contrast, RHD only needs to store the local trellis for the current horizon, making it especially attractive for these structures. Simulation results on both additive white Gaussian noise (AWGN) and Rayleigh fading channels demonstrate that RHD achieves bit‑error‑rate (BER) performance within 0.4 dB of the optimal Viterbi decoder while reducing average decoding time by 30–50 %. The gains are even more pronounced for low‑power, resource‑constrained devices such as IoT sensors, where the algorithm’s linear complexity and modest memory footprint enable real‑time operation.

Complexity analysis confirms that the algorithm’s time per window is O(N·2^k) and its memory requirement is O(N·2^k), independent of the total transmitted length. The authors also discuss implementation aspects, noting that the sliding‑window structure maps naturally onto pipelined hardware architectures and can be realized efficiently on ASICs or FPGAs.

Finally, the paper outlines several promising research directions: adaptive selection of horizon length N based on instantaneous channel conditions, extension to multi‑user MIMO scenarios where multiple trellises must be decoded jointly, and the development of hardware prototypes to validate the theoretical gains. In summary, receding horizon decoding offers a compelling compromise between optimal error correction and practical computational demands, opening a new pathway for deploying high‑performance convolutional codes in next‑generation communication systems.