Covariance-Based Gridless Joint Device Activity Detection and Frequency Offset Estimation for Asynchronous Massive Access with Massive MIMO

Device activity detection has been regarded as a critical task in grant-free random access for massive machine-type communications (mMTC), where only a small portion of the massive potential single-antenna devices are connected to the base station (BS) with a large number of antennas at any given time. However, the inevitable frequency offsets between the Internet of Things (IoT) devices and the BS due to low-cost crystal oscillators and limited mobility make the device activity detection more challenging. To fulfill the device activity detection in the frequency asynchronous scenarios, the existing methods typically discretize the frequency offset interval into a set of equally spaced grid points and search the frequency offsets over the discretized grid set. Such discretization may cause the mismatch between the adopted grid points and the true frequency offsets, and thus degrade the performance of device activity detection. Toward this end, this paper proposes a covariance-based gridless method to perform joint device activity detection and frequency offset estimation directly in the continuous frequency offset range rather than discretizing it. Specifically, the equivalent matrices to be estimated are Hermitian Toeplitz matrices that contain the device activities and the frequency offset parameters. Then, we formulate such Hermitian Toeplitz matrix estimation as the problem of minimizing the covariance fitting (CF) criterion. We further prove that the solution to the formulated problem is a large-sample (in the number of antennas) realization of the maximum likelihood (ML) estimator. By leveraging the block coordinate descent (BCD) framework, we design an algorithm, namely BCD-SDP, to solve each subproblem using a semidefinite programming (SDP) solver. Additionally, we also develop a more computationally efficient algorithm based on the alternating direction method of multipliers (ADMM), i.e., BCD-ADMM, to provide the closed-form iterative expressions for all the variables in each subproblem. Simulation results validate that the proposed covariance-based gridless method significantly improves the performance of device activity detection and frequency offset estimation compared to the baseline schemes, and offers insights for the future use of gridless methods for device activity detection in the frequency asynchronous massive access.