Diagonal Beam Selection for MIMO ISAC

• Diagonal Beam Selection (DBS) is a low-complexity beam selection method in beamspace MIMO ISAC that maximizes a unified mutual-information metric using a diagonal approximation.
• The method computes per-beam contributions from diagonalized Gram matrices, enabling rapid selection of K beams without iterative subset updates.
• Simulations show that DBS attains near-optimal performance with linear complexity, offering significant gains in energy efficiency over conventional exhaustive and greedy methods.

Searching arXiv for the specified paper and related DBS context. arXiv search query: (Shin et al., 14 Jul 2025) Diagonal Beam Selection beamspace MIMO ISAC Diagonal Beam Selection (DBS) is a beam-selection method introduced for beamspace multiple-input multiple-output integrated sensing and communication (MIMO ISAC) systems as a simplified extension of a greedy RF-chain selection framework that maximizes a unified mutual-information (MI)-based performance metric applicable to both communication and sensing. In the reported formulation, DBS operates under a diagonal approximation in which beam-domain Gram matrices are approximately diagonal, so the contribution of each beam can be computed once and the system may simply pick the $K$ beams with the largest contribution values. The method is presented as a low-complexity alternative to exhaustive search and to greedy eigen-based selection (GES) and greedy cofactor-based selection (GCS), with simulation results showing near-optimal performance at substantially lower complexity in beamspace MIMO ISAC (Shin et al., 14 Jul 2025).

1. Beamspace MIMO ISAC setting

DBS is defined in a hybrid analog-digital MIMO ISAC architecture in which a base-station (BS) with $N_t$ antennas is equipped with an analog beamforming network that forms $M$ predefined beams, for example columns of a size-$N_t$ DFT matrix. Each of the $M$ RF chains excites one beam. The $m$th beamforming vector is

$$\mathbf{u}_m = \sqrt{\tfrac1{N_t}} \bigl[ 1,\;e^{-j2\pi\frac{m-1}{M}},\;\dots,\; e^{-j2\pi\frac{(N_t-1)(m-1)}M} \bigr]^{T},$$

and the analog beamforming matrix is

$$\mathbf{U}=[\mathbf{u}_1,\dots,\mathbf{u}_M]\in\mathbb{C}^{N_t\times M}.$$

On the digital side, the BS selects $K$ out of these $M$ beam-ports (RF chains) to transmit information/sensing waveforms. The UE has $N_t$0 antennas, and the ISAC sensing receiver has $N_t$1 separate antennas. The model assumes narrowband operation, known channel statistics, and standard AWGN at all receivers (Shin et al., 14 Jul 2025).

Within this setting, DBS is specifically a beamspace selection rule. A plausible implication is that its role is not to redesign the analog beam codebook, but to exploit a predefined beam basis and then perform low-complexity digital activation of a subset of beam-ports.

2. Unified MI criterion for communication and sensing

The beamspace formulation uses the transformed channel and sensing quantities

$N_t$2

For any beam-subset $N_t$3 of size $N_t$4, the communication MI is

$N_t$5

and the sensing MI is

$N_t$6

where $N_t$7 is the transmit-SNR, $N_t$8 the number of snapshots, and

$N_t$9

is the transmit-covariance of the $M$0th sensing path in beam-domain. Each MI is normalized, and the weighted sum is

$M$1

This criterion is the basis on which DBS is constructed (Shin et al., 14 Jul 2025).

The significance of this formulation is explicit in the source: the framework maximizes a unified MI-based performance metric applicable to both functions. This suggests that DBS inherits a single objective spanning both communication and sensing, rather than alternating between disparate criteria such as MI for communication and beam-pattern mean-squared error or the Cramér–Rao lower bound for sensing.

3. Diagonal approximation and per-beam contribution

The decisive step behind DBS is the diagonal approximation. When $M$2, $M$3 and $M$4 are large and the beam-vectors become nearly orthogonal to each other, as in uniform linear arrays with DFT beams, the Gram matrices $M$5 and each $M$6 are approximately diagonal. The diagonal entries are defined as

$M$7

Under this approximation, the MI expressions simplify to

$M$8

Accordingly, the per-beam contribution to the normalized weighted MI is

$M$9

The article’s key simplification follows directly: under the diagonal approximation, these per-beam contributions $N_t$0 do not change when beams are removed, so one may simply pick the $N_t$1 beams with largest $N_t$2. Equivalently, one can think of iteratively removing the beam with smallest $N_t$3 (Shin et al., 14 Jul 2025).

A common misunderstanding would be to treat DBS as merely a heuristic sorting rule unrelated to the original MI objective. The formulation in fact derives the score $N_t$4 from the diagonalized communication and sensing MI terms. At the same time, the method depends on the approximation that the relevant beam-domain matrices are approximately diagonal; this dependence is part of the construction, not an incidental implementation detail.

4. Selection procedure

The paper gives DBS in a single-shot selection version. Its inputs are the number of beams $N_t$5, desired active beams $N_t$6, and the weights $N_t$7. The required precomputations are

$N_t$8

and

$N_t$9

The beam score is then evaluated as

$M$0

Finally, the procedure sorts $M$1 in descending order and selects the top-$M$2 indices, returning $M$3 as the indices of the selected beams (Shin et al., 14 Jul 2025).

The algorithmic interpretation is straightforward: DBS replaces repeated subset-dependent updates with a one-time computation of beam scores. A plausible implication is that the method is especially attractive in regimes where the beam codebook is fixed and selection must be performed rapidly across different operating points or weighting choices.

5. Complexity profile relative to other selectors

The reported complexity comparison places DBS alongside exhaustive search and the full greedy methods GES and GCS.

Method	Complexity	Characterization
Exhaustive Search	$M$4	approximately exponential in $M$5
GES / GCS	$M$6	$M$7 iterations with rank-one updates and small-matrix inversions
DBS	$M$8	linear in $M$9

For exhaustive search, the source specifies search over $m$0 subsets, each evaluating a full MI with $m$1 for determinants. For GES or GCS, the source specifies $m$2 iterations, each performing rank-one updates and small-matrix inversions, with $m$3 the small rank of each sensing covariance. For DBS, the total consists of computing $m$4 values of $m$5 in $m$6, computing $m$7 values $m$8 in $m$9, and sorting $$\mathbf{u}_m = \sqrt{\tfrac1{N_t}} \bigl[ 1,\;e^{-j2\pi\frac{m-1}{M}},\;\dots,\; e^{-j2\pi\frac{(N_t-1)(m-1)}M} \bigr]^{T},$$0 numbers in $$\mathbf{u}_m = \sqrt{\tfrac1{N_t}} \bigl[ 1,\;e^{-j2\pi\frac{m-1}{M}},\;\dots,\; e^{-j2\pi\frac{(N_t-1)(m-1)}M} \bigr]^{T},$$1 (Shin et al., 14 Jul 2025).

The complexity reduction is therefore not described only qualitatively. The source explicitly states that DBS has total complexity linear in $$\mathbf{u}_m = \sqrt{\tfrac1{N_t}} \bigl[ 1,\;e^{-j2\pi\frac{m-1}{M}},\;\dots,\; e^{-j2\pi\frac{(N_t-1)(m-1)}M} \bigr]^{T},$$2, whereas exhaustive search is approximately exponential in $$\mathbf{u}_m = \sqrt{\tfrac1{N_t}} \bigl[ 1,\;e^{-j2\pi\frac{m-1}{M}},\;\dots,\; e^{-j2\pi\frac{(N_t-1)(m-1)}M} \bigr]^{T},$$3. This suggests why DBS is positioned as a simplified solution for beamspace MIMO ISAC rather than merely a lower-constant-factor implementation of the full greedy framework.

6. Reported numerical behavior, trade-offs, and limitations

The numerical section summarized for DBS uses a beamspace system with $$\mathbf{u}_m = \sqrt{\tfrac1{N_t}} \bigl[ 1,\;e^{-j2\pi\frac{m-1}{M}},\;\dots,\; e^{-j2\pi\frac{(N_t-1)(m-1)}M} \bigr]^{T},$$4, $$\mathbf{u}_m = \sqrt{\tfrac1{N_t}} \bigl[ 1,\;e^{-j2\pi\frac{m-1}{M}},\;\dots,\; e^{-j2\pi\frac{(N_t-1)(m-1)}M} \bigr]^{T},$$5, and $$\mathbf{u}_m = \sqrt{\tfrac1{N_t}} \bigl[ 1,\;e^{-j2\pi\frac{m-1}{M}},\;\dots,\; e^{-j2\pi\frac{(N_t-1)(m-1)}M} \bigr]^{T},$$6. In the weighted sum of normalized MI versus SNR, at SNR $$\mathbf{u}_m = \sqrt{\tfrac1{N_t}} \bigl[ 1,\;e^{-j2\pi\frac{m-1}{M}},\;\dots,\; e^{-j2\pi\frac{(N_t-1)(m-1)}M} \bigr]^{T},$$7 dB, GES/GCS achieve within $$\mathbf{u}_m = \sqrt{\tfrac1{N_t}} \bigl[ 1,\;e^{-j2\pi\frac{m-1}{M}},\;\dots,\; e^{-j2\pi\frac{(N_t-1)(m-1)}M} \bigr]^{T},$$8 dB of Exhaustive Search, while DBS remains within $$\mathbf{u}_m = \sqrt{\tfrac1{N_t}} \bigl[ 1,\;e^{-j2\pi\frac{m-1}{M}},\;\dots,\; e^{-j2\pi\frac{(N_t-1)(m-1)}M} \bigr]^{T},$$9 of optimal and outperforms Random and Fixed by $$\mathbf{U}=[\mathbf{u}_1,\dots,\mathbf{u}_M]\in\mathbb{C}^{N_t\times M}.$$0. In the weighted sum of normalized MI versus number of active beams, for $$\mathbf{U}=[\mathbf{u}_1,\dots,\mathbf{u}_M]\in\mathbb{C}^{N_t\times M}.$$1 beams, DBS lags GES/GCS by only $$\mathbf{U}=[\mathbf{u}_1,\dots,\mathbf{u}_M]\in\mathbb{C}^{N_t\times M}.$$2, while Random selection suffers up to $$\mathbf{U}=[\mathbf{u}_1,\dots,\mathbf{u}_M]\in\mathbb{C}^{N_t\times M}.$$3 loss relative to GES/GCS when $$\mathbf{U}=[\mathbf{u}_1,\dots,\mathbf{u}_M]\in\mathbb{C}^{N_t\times M}.$$4. In energy efficiency, DBS achieves higher EE than full-activation by up to $$\mathbf{U}=[\mathbf{u}_1,\dots,\mathbf{u}_M]\in\mathbb{C}^{N_t\times M}.$$5, since it uses fewer RF chains for near-optimal MI, whereas Random selection yields poor EE across all $$\mathbf{U}=[\mathbf{u}_1,\dots,\mathbf{u}_M]\in\mathbb{C}^{N_t\times M}.$$6. For the Pareto trade-off between communication MI and sensing MI, DBS produces a narrower Pareto front than GES/GCS; it cannot adjust inter-beam correlations but still outperforms non-optimized baselines (Shin et al., 14 Jul 2025).

These results locate DBS in a specific complexity-performance regime. The paper’s own summary is that DBS offers a compelling complexity-performance trade-off: a few percent MI loss for orders-of-magnitude reduction in selection complexity. The reported limitation is equally specific: DBS produces a narrower Pareto front than GES/GCS because it cannot adjust inter-beam correlations. A plausible implication is that DBS is best matched to settings where selection latency and implementation simplicity dominate, while GES/GCS remain preferable when fine control of the communication-sensing trade-off is required.

7. Position within the broader selection framework

DBS is not presented as an isolated method. It is an extension of a low-complexity greedy RF-chain selection framework that decomposes the total MI into individual contributions of each RF chain, yielding GES and GCS for general MIMO ISAC, and then extends the framework to beam selection for beamspace MIMO ISAC systems by introducing DBS as a simplified solution (Shin et al., 14 Jul 2025).

In that framework, DBS occupies the most aggressively simplified end of the design space. Exhaustive search is the combinatorial benchmark; GES and GCS preserve subset-adaptive greedy updates; DBS instead relies on the fact that, under the diagonal approximation, beam contributions do not change when beams are removed. This suggests a hierarchy of selectors: exhaustive enumeration for optimality, greedy contribution-based removal for near-optimality with moderate complexity, and diagonal beam selection for linear-in-$$\mathbf{U}=[\mathbf{u}_1,\dots,\mathbf{u}_M]\in\mathbb{C}^{N_t\times M}.$$7 scaling. The principal technical distinction is therefore not only computational cost, but also whether the selector models and exploits inter-beam coupling during the selection process.