Search Results (5)

In this paper, we study the beamforming design for multiple-input multiple-output (MIMO) ISAC systems, with the weighted mutual information (MI) comprising sensing and communication perspectives adopted as the performance metric. In particular, the weighted sum of the communication mutual information and the sensing mutual information is shown to asymptotically converge to a deterministic limit when the number of transmitting and receiving antennas grow to infinity. This deterministic limit is derived by utilizing the operator-valued free probability theory. Subsequently, an efficient projected gradient ascent (PGA) algorithm is proposed to optimize the transmit beamforming matrix with the aim of maximizing the weighted asymptotic MI. Numerical results validate that the derived closed-form expression matches well with the Monte Carlo simulation results and the proposed optimization algorithm is able to improve the weighted asymptotic MI significantly. We also illustrate the trade-off between asymptotic sensing and asymptotic communication MI. Full article

Physical-layer (PHY) security is widely used as an effective method for ensuring secure wireless communications. However, when the legitimate user (LU) and the eavesdropper (Eve) are in close proximity, the channel coupling can significantly degrade the secure performance of PHY. Frequency diverse array (FDA) technique addresses channel coupling issues by introducing frequency offsets among array elements. However, FDA’s ability to secure communication relies mainly on frequency domain characteristics, lacking the spatial degrees of freedom. The recently proposed movable antenna (MA) technology serves as an effective approach to overcome this limitation. It offers the flexibility to adjust antenna positions dynamically, thereby further decoupling the channels between LU and Eve. In this paper, we propose a novel MA-FDA approach, which offers a comprehensive solution for enhancing PHY security. We aim to maximize the achievable secrecy rate through the joint optimization of all antenna positions at the base station (BS), FDA frequency offsets, and beamformer, subject to the predefined regions for antenna positions, frequency offsets range, and energy constraints. To solve this non-convex optimization problem, which involves highly coupled variables, the alternating optimization (AO) method is employed to cyclically update the parameters, with the projected gradient ascent (PGA) method and block successive upper-bound minimization (BSUM) method being employed to tackle the challenging subproblems. Simulation results demonstrate that the MA-FDA approach can achieve a higher secrecy rate compared to the conventional phased array (PA) or fixed-position antenna (FPA) schemes. Full article

In this paper, based on the novel generalized Hamilton-real (GHR) calculus, we propose for the first time a quaternion Nesterov accelerated projected gradient algorithm for computing the dominant eigenvalue and eigenvector of quaternion Hermitian matrices. By introducing momentum terms and look-ahead updates, the algorithm achieves a faster convergence rate. We theoretically prove the convergence of the quaternion Nesterov accelerated projected gradient algorithm. Numerical experiments show that the proposed method outperforms the quaternion projected gradient ascent method and the traditional algebraic methods in terms of computational accuracy and runtime efficiency. Full article

The current paper proposes and tests algorithms for finding the diameter of a compact convex set and the farthest point in the set to another point. For these two nonconvex problems, I construct Frank–Wolfe and projected gradient ascent algorithms. Although these algorithms are guaranteed to go uphill, they can become trapped by local maxima. To avoid this defect, I investigate a homotopy method that gradually deforms a ball into the target set. Motivated by the Frank–Wolfe algorithm, I also find the support function of the intersection of a convex cone and a ball centered at the origin and elaborate a known bisection algorithm for calculating the support function of a convex sublevel set. The Frank–Wolfe and projected gradient algorithms are tested on five compact convex sets: (a) the box whose coordinates range between −1 and 1, (b) the intersection of the unit ball and the non-negative orthant, (c) the probability simplex, (d) the Manhattan-norm unit ball, and (e) a sublevel set of the elastic net penalty. Frank–Wolfe and projected gradient ascent are about equally fast on these test problems. Ignoring homotopy, the Frank–Wolfe algorithm is more reliable. However, homotopy allows projected gradient ascent to recover from its failures. Full article

The Hausdorff distance between two closed sets has important theoretical and practical applications. Yet apart from finite point clouds, there appear to be no generic algorithms for computing this quantity. Because many infinite sets are defined by algebraic equalities and inequalities, this a huge gap. The current paper constructs Frank–Wolfe and projected gradient ascent algorithms for computing the Hausdorff distance between two compact convex sets. Although these algorithms are guaranteed to go uphill, they can become trapped by local maxima. To avoid this defect, we investigate a homotopy method that gradually deforms two balls into the two target sets. The Frank–Wolfe and projected gradient algorithms are tested on two pairs $(A,B)$ of compact convex sets, where: (1) A is the box $[-\mathbf{1},\mathbf{1}]$ translated by $\mathbf{1}$ and B is the intersection of the unit ball and the non-negative orthant; and (2) A is the probability simplex and B is the ${\ell }_1$ unit ball translated by $\mathbf{1}$. For problem (2), we find the Hausdorff distance analytically. Projected gradient ascent is more reliable than the Frank–Wolfe algorithm and finds the exact solution of problem (2). Homotopy improves the performance of both algorithms when the exact solution is unknown or unattained. Full article