Detection by Sampling: Massive MIMO Detector based on Langevin Dynamics

Abstract

We propose a joint channel estimation and data detection algorithm for massive multilple-input multiple-output systems based on diffusion models. Our proposed method solves the blind inverse problem by sampling from the joint posterior distribution of the symbols and channels and computing an approximate maximum a posteriori estimation. To achieve this, we construct a diffusion process that models the joint distribution of the channels and symbols given noisy observations, and then run the reverse process to generate the samples. A unique contribution of the algorithm is to include the discrete prior distribution of the symbols and a learned prior for the channels. Indeed, this is key as it allows a more efficient exploration of the joint search space and, therefore, enhances the sampling process. Through numerical experiments, we demonstrate that our method yields a lower normalized mean squared error than competing approaches and reduces the pilot overhead.

Annealed Langevin Dynamics for Massive MIMO Detection

“…Proposition 1: Given Assumptions 1 and 2 and the Hamiltonian defined in (13), the distribution (4) is the unique invariant distribution of the dynamics (14).…”

“…We consider an annealed version of our dynamic, where random noise of decreasing variance is added to our state x. This procedure helps accelerate convergence and allows us to accommodate discrete states x [1]. First, we define a sequence of noise levels {σ l } L+1 l=1 such that…”

“…In this section, we analyze the proposed methods for the problem of massive MIMO symbol detection. This problem can be cast as (1), where N u are the number of singleantenna transmitters or users and N r the antennas at the base station. In this context, H ∈ C Nr×Nu is the channel matrix, z ∼ CN (0, σ 2 0 I Nr ) is a vector of complex circular Gaussian noise, ȳ ∈ C Nr is the received vector, x ∈ X Nu is the vector of transmitted symbols, which is the unknown variable, and X is a finite set of constellation points.…”

“…Email: {nzilberstein, ashu, segarra}@rice.edu. Preliminary results were published in [1]. is typically intractable.…”

“…In particular, the discretization of the Langevin diffusion, known as unadjusted Langevin algorithm, has been used in image denoising [12], MIMO symbol detection [13], [14], and channel estimation [15], achieving impressive results. The unadjusted Langevin algorithm is an iterative technique that enables sampling from the posterior distribution by leveraging the availability of the gradient of the log-posterior distribution, known as the score function.…”