The Trainable BCJR and Its Applications

We propose a trainable version of the Additive Bahl-Cocke-Jelinek-Raviv (A-BCJR) algorithm by interpreting it as a Recurrent Neural Network (RNN). By leveraging the smoothness and differentiability of the max* operator, the core component of the A-BCJR forward and backward recursions, we derive a backpropagation algorithm that enables end-to-end training from the network’s output. The resulting model, referred to as T-BCJR, comprises a linear layer that computes edge metrics from state and input metrics, followed by a nonlinear max* layer that marginalizes these metrics back to the state and output domains. We further derive the corresponding delta backpropagation recursions, which exhibit a structural symmetry with the original forward-backward predictive steps. Unlike prior approaches that only replaced memoryless metric units in BCJR or Viterbi algorithms with Neural Networks (NNs), T-BCJR enables full trainability of the entire recursive structure. We derive a trainable detector, termed T-Detector, by connecting the T-BCJR to the channel output via an additional convolutional linear layer that emulates a channel shortening filter. The TDetector can be trained to operate as a modulation- and channelagnostic detector, capable of performing channel shortening, Maximum Likelihood (ML) sequence detection, and the computation of symbol-level or bit-level Log-Likelihoods (LLs) for softinput decoding. Moreover, it is compatible with iterative receiver schemes involving outer channel decoders. The predictive step of the T-Detector maintains the same Digital Signal Processing (DSP) complexity as a conventional model-based detector, with the added benefit of being trainable from a cost function defined on the generated LLs. Experimental results demonstrate that the T-Detector can be trained to match the performance of the corresponding model-based detector for static and slow-time varying channels with appropriate pilot densities.