In this tutorial paper, we provide an in-depth explanation of a layered architecture to integrate probabilistic shaping and forward error correction (FEC). Probabilistic Amplitude Shaping (PAS) is an exemplary instance of this layered architecture.

The focus of this tutorial is on the development of a practical performance metric that allows to separately assess the practical limitations of shaping (e.g., the rate loss of a finite length distribution matcher) and the FEC code (e.g., the back-off in SNR to achieve a BER of $10^{-15}$). The shaping performance is characterized by the shaping set size, which can be calculated in closed form for many existing distribution matching algorithms. The FEC performance benchmark is characterized by the uncertainty, which incorporates the decoding metric in use (e.g., hard-decision, soft-decision, quantized LLRs, etc) and the channel quality (e.g., via a Gaussian noise decoding model or measurements from transmission experiments). This separation of the performance metric into shaping set size and uncertainty reflects the two layers of the probabilistic shaping architecture and makes it very useful for the design of transceivers that integrate PS and FEC.