Hybrid Beamforming Optimization for MIMO ISAC based on Prior Distribution Information

This paper considers a multiple-input multiple-output (MIMO) integrated sensing and communication (ISAC) system, where a multi-antenna base station (BS) with transceiver hybrid analog-digital arrays transmits dual-functional signals to communicate with a multi-antenna user and simultaneously sense the unknown and random location information of a target based on the reflected echo signals and the prior distribution information on the target’s location. Under transceiver hybrid arrays, we characterize the sensing performance by deriving the posterior Cramér-Rao bound (PCRB) of the mean-squared error which is a function of the transmit hybrid beamforming and receive analog beamforming. We study joint transmit hybrid beamforming and receive analog beamforming optimization to minimize the PCRB subject to a communication rate requirement. We first consider a sensing-only system and derive the optimal solution to each element in the transmit/receive analog beamforming matrices that minimizes the PCRB in closed form. Then, we develop an alternating optimization (AO) based algorithm. Next, we study a narrowband MIMO ISAC system and devise an efficient AO-based hybrid beamforming algorithm by leveraging weighted minimum mean-squared error and feasible point pursuit successive convex approximation methods. Furthermore, we extend the results for narrowband systems to a MIMO orthogonal frequency-division multiplexing (OFDM) ISAC system. Numerical results validate the effectiveness of our proposed hybrid beamforming designs. It is revealed that the number of receive RF chains has more significant impact on the sensing performance than its transmit counterpart. Under a given budget on the total number of transmit/receive RF chains at the BS, the optimal number of transmit RF chains increases as the communication rate target increases due to the non-trivial PCRB-rate trade-off.

💡 Research Summary

This paper addresses the design of hybrid analog‑digital beamforming for a multiple‑input multiple‑output (MIMO) integrated sensing and communication (ISAC) system in which both the base‑station (BS) transmitter and receiver are equipped with limited radio‑frequency (RF) chains. The key novelty lies in exploiting prior statistical knowledge of the target’s angular location, represented by a probability density function (PDF), to formulate the posterior Cramér‑Rao bound (PCR‑B) as the sensing performance metric. Unlike the conventional CRB, which requires the exact unknown parameters, the PCR‑B depends only on the prior PDF and thus provides a realistic lower bound on the mean‑squared error (MSE) of any unbiased estimator.

The authors first derive an explicit expression for the PCR‑B under hybrid transceiver architectures. Because the receiver performs analog combining before any digital processing, the PCR‑B is shown to be a function of the receive analog beamforming matrix, a departure from the fully‑digital case where the bound is independent of the receiver processing. This insight highlights the critical role of analog beamforming in determining sensing accuracy.

A sensing‑only scenario (no communication constraint) is examined next. Despite the non‑convex unit‑modulus constraints on the analog beamformers and the complicated PCR‑B expression, the authors obtain closed‑form optimal values for each element of the transmit and receive analog beamforming matrices. These per‑element solutions reveal a probability‑dependent power‑focusing effect: the beamformer steers more energy toward angular regions with higher prior probability. Building on these results, an alternating optimization (AO) algorithm is proposed that iteratively updates the analog and digital beamformers; convergence to a Karush‑Kuhn‑Tucker (KKT) point is proved.

The paper then tackles the full ISAC problem where a minimum MIMO communication rate must be satisfied. The rate constraint is transformed using the weighted minimum mean‑squared error (WMMSE) method, converting the original non‑convex problem into a sequence of tractable sub‑problems. The analog beamforming design, still subject to unit‑modulus constraints, is handled via feasible‑point pursuit successive convex approximation (FPP‑SCA). The resulting AO‑WMMSE‑FPP‑SCA framework yields high‑quality solutions in polynomial time and is shown to outperform several benchmark schemes.

Extending the methodology to wideband orthogonal frequency‑division multiplexing (OFDM) ISAC, the authors recognize that the analog beamformer must be common across all sub‑carriers, while digital beamformers can be optimized per sub‑carrier. They derive the PCR‑B for the OFDM case, which aggregates contributions from each sub‑carrier, and adapt the AO‑WMMSE‑FPP‑SCA algorithm accordingly. This extension demonstrates that the proposed design remains effective even when frequency selectivity is present.

A further contribution is the treatment of a fully‑connected receiver architecture. By employing a discrete Fourier transform (DFT) codebook, the analog beamformer can be selected from a finite set, dramatically reducing computational complexity while preserving performance.

Numerical simulations explore various allocations of RF chains between transmitter and receiver under a fixed total RF budget. The results reveal that the number of receive RF chains has a more pronounced impact on PCR‑B than the number of transmit RF chains, confirming the importance of analog combining for sensing. For low to moderate communication‑rate requirements, allocating the smallest feasible number of transmit RF chains (often one or two) and the remainder to the receiver yields the lowest PCR‑B. As the required communication rate increases, more transmit RF chains become necessary to support spatial multiplexing, and the optimal allocation shifts accordingly. Moreover, with moderate numbers of RF chains (e.g., 4–8), the hybrid designs achieve performance close to that of an optimal fully‑digital solution, validating their practicality.

In summary, this work provides a comprehensive, analytically grounded framework for hybrid beamforming in MIMO ISAC systems that leverages prior distribution information. By jointly optimizing transmit and receive analog/digital beamformers, the proposed algorithms achieve near‑optimal sensing accuracy while satisfying communication constraints, and they elucidate the trade‑off between RF‑chain allocation and overall system performance—insights that are highly relevant for the design of cost‑effective, high‑performance 6G ISAC networks.