Optimal Placement of Movable Antennas for Angle-of-Departure Estimation Under User Location Uncertainty

Movable antennas (MA) have gained significant attention in recent years to overcome the limitations of extremely large antenna arrays in terms of cost and power consumption. In this paper, we investigate the use of MA arrays at the base station (BS) for angle-of-departure (AoD) estimation under uncertainty in the user equipment (UE) location. Specifically, we (i) derive the theoretical performance limits through the Cramér-Rao bound (CRB) and (ii) optimize the antenna positions to ensure robust performance within the UE’s uncertainty region. Numerical results show that dynamically optimizing antenna placement by explicitly considering the uncertainty region yields superior performance compared to fixed arrays, demonstrating the ability of MA systems to adapt and outperform conventional arrays.

💡 Research Summary

The paper addresses the challenge of accurately estimating the angle‑of‑departure (AoD) in downlink communications when the user equipment (UE) location is uncertain. It proposes to exploit movable antennas (MAs) at the base station (BS) and to jointly design the antenna positions and the transmit precoding to minimize the worst‑case Cramér‑Rao bound (CRB) over the UE’s uncertainty region.

First, a signal model is introduced in which a BS equipped with an L‑element linear MA array can reposition its antennas along a one‑dimensional segment of length D, respecting a minimum inter‑element spacing d. The BS transmits K pilot symbols over G transmissions, each using a precoding vector f_g. The received baseband signal at the UE follows a standard narrow‑band model with a steering vector a(θ,r) that depends on the AoD θ and the antenna position vector r.

Using the Slepian‑Bangs formula, the authors derive a closed‑form expression for the AoD CRB that depends on the signal‑to‑noise ratio (SNR), the precoding matrix F, and the antenna positions r. They show that the optimal precoder lies in the two‑dimensional subspace spanned by the steering vector and its derivative with respect to θ. By allocating a fraction γ of the transmit power to the directional beam and (1‑γ) to the derivative beam, the CRB simplifies to a term proportional to 1/(2 SNR (1‑γ) (2πλ cosθ)² ‖r‖²). Consequently, minimizing the CRB is equivalent to maximizing the squared norm of r, i.e., the variance of the antenna position vector.

When the UE location is perfectly known, the optimal placement reduces to the “maximum‑variance” (Max‑Var) solution previously reported for AoA estimation: the antennas are split into two groups placed at opposite ends of the available aperture while respecting the spacing constraint. However, in realistic scenarios the UE is only known to lie within an angular uncertainty region P. To handle this, the authors fix the precoder to steer toward the centre angle θ_c of P and introduce an additional spatial‑correlation‑coefficient (SCC) constraint that limits the correlation between the steering vector at θ_c and those at any other angle inside P. This constraint prevents high‑gain side lobes that could cause ambiguities in AoD estimation.

The resulting optimization problem is non‑convex because the objective involves a ratio of quadratic forms in r and the SCC constraint is also nonlinear. The authors therefore perform an exhaustive grid search over two design parameters a and b that define the inter‑antenna distances for a six‑element array (L = 6, D = 10λ, d = λ/2). Two uncertainty regions are considered: P₁ =