On the Equivalence Between Probabilistic Shaping and Geometric Shaping: A Polar Lattice Perspective

This paper aims to build a bridge between the probabilistic shaping and the geometric shaping for lattice codes from the perspective of polar lattices. We prove that when performing the lattice Gaussian shaping on polar lattices, a shaping lattice As which is good for the so-called discrete additive white Gaussian noise (AWGN) channel is constructed indeed, and the shaping process is equivalent to the modulo As operation within a multi-level decoding manner. To achieve the power-constraint AWGN channel capacity or the rate distortion bound of the i.i.d. Gaussian source, one classical approach is to construct two nested lattices where the fine lattice takes care of the Gaussian noise or the target distortion, and the coarse lattice is responsible for the boundary of the lattice codewords. Another approach is to construct a single lattice and then perform the lattice Gaussian shaping. The former approach falls into the category of geometric shaping, while the latter one is regarded as a type of probabilistic shaping. This work proposes a unified perspective of these two approaches, and provides new evidence on why they are both able to achieve the optimal performance of Gaussian channel coding and source coding problems.