DAOR in ISAC Beamforming

• DAOR is a privacy metric that quantifies the ratio of received angular power between a fake direction and the true LOS, ensuring location privacy without nulling the LOS path.
• The framework embeds DAOR as a constraint in a rate-maximization problem solved via semidefinite relaxation, eigenmode selection, and optimal power allocation.
• Numerical studies reveal a privacy-throughput trade-off where increasing DAOR reduces achievable rate while preserving ISAC sensing and communication performance.

Direction of Arrival Obfuscation Ratio (DAOR) is a privacy metric introduced for location privacy-enabled beamforming in integrated sensing and communication (ISAC) scenarios. It quantifies the ratio between the received angular power at an intentionally obfuscated direction and the received angular power at the true line-of-sight (LOS) direction, thereby enabling beamforming designs that protect transmitter location privacy by making a false direction appear dominant at the receiver without suppressing the LOS component (Khan et al., 13 Aug 2025). In the formulation proposed in "Location Privacy-Enabled Beamforming in ISAC Scenarios" (Khan et al., 13 Aug 2025), DAOR is embedded as a constraint in an achievable rate-maximization problem, with feasible values characterized by generalized eigenvalue bounds and with solutions obtained through semidefinite relaxation, eigenmode selection, and optimal power allocation.

1. Definition and operational meaning

In the underlying ISAC model, the true direction is denoted by $\theta_{\mathrm{true}} \equiv \phi$, which is the LOS DOA/DOD of the direct Tx-Rx path, and the obfuscated direction is denoted by $\theta_{\mathrm{fake}} \equiv \hat{\phi}$. The received angular power, or beampattern, in direction $\theta$ is defined as

$$P(\theta) \triangleq \mathbf{a}_R^H(\theta)\,\mathbf{R}\,\mathbf{a}_R(\theta),$$

where

$$\mathbf{R} \triangleq \mathbb{E}[\mathbf{y}\mathbf{y}^H] = \mathbf{H}\mathbf{W}\mathbf{W}^H\mathbf{H}^H + N_0\,\mathbf{I}_{N_R}.$$

DAOR is then defined as

$$\gamma \triangleq \frac{\mathbb{E}\!\left[(\mathbf{a}_R^H(\hat{\phi})\mathbf{y})(\mathbf{a}_R^H(\hat{\phi})\mathbf{y})^H\right]} {\mathbb{E}\!\left[(\mathbf{a}_R^H(\phi)\mathbf{y})(\mathbf{a}_R^H(\phi)\mathbf{y})^H\right]} = \frac{\mathbf{a}_R^H(\hat{\phi})\mathbf{R}\mathbf{a}_R(\hat{\phi})} {\mathbf{a}_R^H(\phi)\mathbf{R}\mathbf{a}_R(\phi)} = \frac{P(\hat{\phi})}{P(\phi)}.$$

The privacy interpretation is explicit. DAOR $>1$ implies that the receiver observes more power arriving from $\hat{\phi}$ than from $\phi$. Under common DOA estimation strategies, including beam scanning, Capon/MVDR peak search, Bartlett, or energy detectors, the peak at $\hat{\phi}$ tends to dominate, increasing the likelihood of misestimating the transmitter direction. The paper further states that, while it does not link DAOR to a closed-form CRB, a larger DAOR generally correlates with increased bias or ambiguity in peak-based DOA estimators and reduces the chance that the true LOS direction is identified (Khan et al., 13 Aug 2025).

A frequent misconception in this setting is that location privacy necessarily requires nullifying the LOS path. DAOR is introduced precisely to avoid that design choice. The metric is operational because it is defined directly from the receiver-side angular power distribution and is tunable because the privacy target is specified through a threshold $\theta_{\mathrm{fake}} \equiv \hat{\phi}$0.

2. System and signal model

The proposed framework considers a transmitter with $\theta_{\mathrm{fake}} \equiv \hat{\phi}$1 antennas and a receiver with $\theta_{\mathrm{fake}} \equiv \hat{\phi}$2 antennas. Both use ULAs with inter-element spacing $\theta_{\mathrm{fake}} \equiv \hat{\phi}$3, typically $\theta_{\mathrm{fake}} \equiv \hat{\phi}$4. For a direction $\theta_{\mathrm{fake}} \equiv \hat{\phi}$5, the transmit steering vector is written as

$\theta_{\mathrm{fake}} \equiv \hat{\phi}$6

and analogously for $\theta_{\mathrm{fake}} \equiv \hat{\phi}$7.

The transmitter maps $\theta_{\mathrm{fake}} \equiv \hat{\phi}$8 data streams $\theta_{\mathrm{fake}} \equiv \hat{\phi}$9, with $\theta$0, to

$\theta$1

where $\theta$2 is the precoder and

$\theta$3

The received signal is

$\theta$4

with $\theta$5.

The channel is a narrowband Rician channel with LOS plus multipath NLOS and $\theta$6 NLOS paths: $\theta$7 The LOS component is

$\theta$8

where $\theta$9 is both the DOD and DOA of the LOS path. The NLOS component is

$$P(\theta) \triangleq \mathbf{a}_R^H(\theta)\,\mathbf{R}\,\mathbf{a}_R(\theta),$$0

with $$P(\theta) \triangleq \mathbf{a}_R^H(\theta)\,\mathbf{R}\,\mathbf{a}_R(\theta),$$1 and random angles $$P(\theta) \triangleq \mathbf{a}_R^H(\theta)\,\mathbf{R}\,\mathbf{a}_R(\theta),$$2.

The achievable rate is

$$P(\theta) \triangleq \mathbf{a}_R^H(\theta)\,\mathbf{R}\,\mathbf{a}_R(\theta),$$3

and the signal-to-noise ratio is defined as $$P(\theta) \triangleq \mathbf{a}_R^H(\theta)\,\mathbf{R}\,\mathbf{a}_R(\theta),$$4. Within this model, the sensing capability is implicitly preserved by avoiding LOS nulling and retaining a strong LOS response in the beampattern (Khan et al., 13 Aug 2025).

3. Generalized Rayleigh formulation and feasible DAOR range

A central analytical step is the reformulation of DAOR as a generalized Rayleigh quotient. The paper derives

$$P(\theta) \triangleq \mathbf{a}_R^H(\theta)\,\mathbf{R}\,\mathbf{a}_R(\theta),$$5

with

$$P(\theta) \triangleq \mathbf{a}_R^H(\theta)\,\mathbf{R}\,\mathbf{a}_R(\theta),$$6

and

$$P(\theta) \triangleq \mathbf{a}_R^H(\theta)\,\mathbf{R}\,\mathbf{a}_R(\theta),$$7

The additive identity term embeds the receiver-noise contribution into the trace representation. The paper states that including $$P(\theta) \triangleq \mathbf{a}_R^H(\theta)\,\mathbf{R}\,\mathbf{a}_R(\theta),$$8 ensures that $$P(\theta) \triangleq \mathbf{a}_R^H(\theta)\,\mathbf{R}\,\mathbf{a}_R(\theta),$$9 and $$\mathbf{R} \triangleq \mathbb{E}[\mathbf{y}\mathbf{y}^H] = \mathbf{H}\mathbf{W}\mathbf{W}^H\mathbf{H}^H + N_0\,\mathbf{I}_{N_R}.$$0 are positive definite Hermitian, so the denominator cannot be zero and the LOS component is never nulled by construction.

Because $$\mathbf{R} \triangleq \mathbb{E}[\mathbf{y}\mathbf{y}^H] = \mathbf{H}\mathbf{W}\mathbf{W}^H\mathbf{H}^H + N_0\,\mathbf{I}_{N_R}.$$1 is a generalized Rayleigh quotient over positive definite Hermitian matrices, it is bounded by the extremal generalized eigenvalues of the matrix pair $$\mathbf{R} \triangleq \mathbb{E}[\mathbf{y}\mathbf{y}^H] = \mathbf{H}\mathbf{W}\mathbf{W}^H\mathbf{H}^H + N_0\,\mathbf{I}_{N_R}.$$2: $$\mathbf{R} \triangleq \mathbb{E}[\mathbf{y}\mathbf{y}^H] = \mathbf{H}\mathbf{W}\mathbf{W}^H\mathbf{H}^H + N_0\,\mathbf{I}_{N_R}.$$3 These bounds are attained by generalized eigenvectors solving

$$\mathbf{R} \triangleq \mathbb{E}[\mathbf{y}\mathbf{y}^H] = \mathbf{H}\mathbf{W}\mathbf{W}^H\mathbf{H}^H + N_0\,\mathbf{I}_{N_R}.$$4

The extremal beamformers are rank-1: $$\mathbf{R} \triangleq \mathbb{E}[\mathbf{y}\mathbf{y}^H] = \mathbf{H}\mathbf{W}\mathbf{W}^H\mathbf{H}^H + N_0\,\mathbf{I}_{N_R}.$$5 and

$$\mathbf{R} \triangleq \mathbb{E}[\mathbf{y}\mathbf{y}^H] = \mathbf{H}\mathbf{W}\mathbf{W}^H\mathbf{H}^H + N_0\,\mathbf{I}_{N_R}.$$6

This characterization gives DAOR a precise feasible interval for a given channel realization, array configuration, and power budget. A plausible implication is that privacy targets outside this interval are not design issues but feasibility violations, since no precoder can satisfy them under the stated model.

4. Rate maximization under a DAOR constraint

The main design problem is the achievable rate-maximization problem under a DAOR threshold: $$\mathbf{R} \triangleq \mathbb{E}[\mathbf{y}\mathbf{y}^H] = \mathbf{H}\mathbf{W}\mathbf{W}^H\mathbf{H}^H + N_0\,\mathbf{I}_{N_R}.$$7

The paper identifies four feasibility cases in terms of $$\mathbf{R} \triangleq \mathbb{E}[\mathbf{y}\mathbf{y}^H] = \mathbf{H}\mathbf{W}\mathbf{W}^H\mathbf{H}^H + N_0\,\mathbf{I}_{N_R}.$$8 and the generalized eigenvalue bounds. If $$\mathbf{R} \triangleq \mathbb{E}[\mathbf{y}\mathbf{y}^H] = \mathbf{H}\mathbf{W}\mathbf{W}^H\mathbf{H}^H + N_0\,\mathbf{I}_{N_R}.$$9, the problem is infeasible. If $$\gamma \triangleq \frac{\mathbb{E}\!\left[(\mathbf{a}_R^H(\hat{\phi})\mathbf{y})(\mathbf{a}_R^H(\hat{\phi})\mathbf{y})^H\right]} {\mathbb{E}\!\left[(\mathbf{a}_R^H(\phi)\mathbf{y})(\mathbf{a}_R^H(\phi)\mathbf{y})^H\right]} = \frac{\mathbf{a}_R^H(\hat{\phi})\mathbf{R}\mathbf{a}_R(\hat{\phi})} {\mathbf{a}_R^H(\phi)\mathbf{R}\mathbf{a}_R(\phi)} = \frac{P(\hat{\phi})}{P(\phi)}.$$0, a closed-form rank-1 solution is obtained via $$\gamma \triangleq \frac{\mathbb{E}\!\left[(\mathbf{a}_R^H(\hat{\phi})\mathbf{y})(\mathbf{a}_R^H(\hat{\phi})\mathbf{y})^H\right]} {\mathbb{E}\!\left[(\mathbf{a}_R^H(\phi)\mathbf{y})(\mathbf{a}_R^H(\phi)\mathbf{y})^H\right]} = \frac{\mathbf{a}_R^H(\hat{\phi})\mathbf{R}\mathbf{a}_R(\hat{\phi})} {\mathbf{a}_R^H(\phi)\mathbf{R}\mathbf{a}_R(\phi)} = \frac{P(\hat{\phi})}{P(\phi)}.$$1 or $$\gamma \triangleq \frac{\mathbb{E}\!\left[(\mathbf{a}_R^H(\hat{\phi})\mathbf{y})(\mathbf{a}_R^H(\hat{\phi})\mathbf{y})^H\right]} {\mathbb{E}\!\left[(\mathbf{a}_R^H(\phi)\mathbf{y})(\mathbf{a}_R^H(\phi)\mathbf{y})^H\right]} = \frac{\mathbf{a}_R^H(\hat{\phi})\mathbf{R}\mathbf{a}_R(\hat{\phi})} {\mathbf{a}_R^H(\phi)\mathbf{R}\mathbf{a}_R(\phi)} = \frac{P(\hat{\phi})}{P(\phi)}.$$2. If $$\gamma \triangleq \frac{\mathbb{E}\!\left[(\mathbf{a}_R^H(\hat{\phi})\mathbf{y})(\mathbf{a}_R^H(\hat{\phi})\mathbf{y})^H\right]} {\mathbb{E}\!\left[(\mathbf{a}_R^H(\phi)\mathbf{y})(\mathbf{a}_R^H(\phi)\mathbf{y})^H\right]} = \frac{\mathbf{a}_R^H(\hat{\phi})\mathbf{R}\mathbf{a}_R(\hat{\phi})} {\mathbf{a}_R^H(\phi)\mathbf{R}\mathbf{a}_R(\phi)} = \frac{P(\hat{\phi})}{P(\phi)}.$$3, the privacy constraint is inactive and conventional SVD water-filling over $$\gamma \triangleq \frac{\mathbb{E}\!\left[(\mathbf{a}_R^H(\hat{\phi})\mathbf{y})(\mathbf{a}_R^H(\hat{\phi})\mathbf{y})^H\right]} {\mathbb{E}\!\left[(\mathbf{a}_R^H(\phi)\mathbf{y})(\mathbf{a}_R^H(\phi)\mathbf{y})^H\right]} = \frac{\mathbf{a}_R^H(\hat{\phi})\mathbf{R}\mathbf{a}_R(\hat{\phi})} {\mathbf{a}_R^H(\phi)\mathbf{R}\mathbf{a}_R(\phi)} = \frac{P(\hat{\phi})}{P(\phi)}.$$4 is used. If $$\gamma \triangleq \frac{\mathbb{E}\!\left[(\mathbf{a}_R^H(\hat{\phi})\mathbf{y})(\mathbf{a}_R^H(\hat{\phi})\mathbf{y})^H\right]} {\mathbb{E}\!\left[(\mathbf{a}_R^H(\phi)\mathbf{y})(\mathbf{a}_R^H(\phi)\mathbf{y})^H\right]} = \frac{\mathbf{a}_R^H(\hat{\phi})\mathbf{R}\mathbf{a}_R(\hat{\phi})} {\mathbf{a}_R^H(\phi)\mathbf{R}\mathbf{a}_R(\phi)} = \frac{P(\hat{\phi})}{P(\phi)}.$$5, the design is governed by a non-convex trace-ratio constraint (Khan et al., 13 Aug 2025).

The semidefinite relaxation reformulation introduces

$$\gamma \triangleq \frac{\mathbb{E}\!\left[(\mathbf{a}_R^H(\hat{\phi})\mathbf{y})(\mathbf{a}_R^H(\hat{\phi})\mathbf{y})^H\right]} {\mathbb{E}\!\left[(\mathbf{a}_R^H(\phi)\mathbf{y})(\mathbf{a}_R^H(\phi)\mathbf{y})^H\right]} = \frac{\mathbf{a}_R^H(\hat{\phi})\mathbf{R}\mathbf{a}_R(\hat{\phi})} {\mathbf{a}_R^H(\phi)\mathbf{R}\mathbf{a}_R(\phi)} = \frac{P(\hat{\phi})}{P(\phi)}.$$6

and yields

$$\gamma \triangleq \frac{\mathbb{E}\!\left[(\mathbf{a}_R^H(\hat{\phi})\mathbf{y})(\mathbf{a}_R^H(\hat{\phi})\mathbf{y})^H\right]} {\mathbb{E}\!\left[(\mathbf{a}_R^H(\phi)\mathbf{y})(\mathbf{a}_R^H(\phi)\mathbf{y})^H\right]} = \frac{\mathbf{a}_R^H(\hat{\phi})\mathbf{R}\mathbf{a}_R(\hat{\phi})} {\mathbf{a}_R^H(\phi)\mathbf{R}\mathbf{a}_R(\phi)} = \frac{P(\hat{\phi})}{P(\phi)}.$$7

The linearized form of the DAOR constraint,

$$\gamma \triangleq \frac{\mathbb{E}\!\left[(\mathbf{a}_R^H(\hat{\phi})\mathbf{y})(\mathbf{a}_R^H(\hat{\phi})\mathbf{y})^H\right]} {\mathbb{E}\!\left[(\mathbf{a}_R^H(\phi)\mathbf{y})(\mathbf{a}_R^H(\phi)\mathbf{y})^H\right]} = \frac{\mathbf{a}_R^H(\hat{\phi})\mathbf{R}\mathbf{a}_R(\hat{\phi})} {\mathbf{a}_R^H(\phi)\mathbf{R}\mathbf{a}_R(\phi)} = \frac{P(\hat{\phi})}{P(\phi)}.$$8

is the basis for the subsequent algorithmic decomposition. This suggests that the privacy target can be enforced at the covariance level rather than directly on the original precoder.

5. Optimal and reduced-complexity solution strategies

For the non-convex regime, the paper performs the eigendecomposition

$$\gamma \triangleq \frac{\mathbb{E}\!\left[(\mathbf{a}_R^H(\hat{\phi})\mathbf{y})(\mathbf{a}_R^H(\hat{\phi})\mathbf{y})^H\right]} {\mathbb{E}\!\left[(\mathbf{a}_R^H(\phi)\mathbf{y})(\mathbf{a}_R^H(\phi)\mathbf{y})^H\right]} = \frac{\mathbf{a}_R^H(\hat{\phi})\mathbf{R}\mathbf{a}_R(\hat{\phi})} {\mathbf{a}_R^H(\phi)\mathbf{R}\mathbf{a}_R(\phi)} = \frac{P(\hat{\phi})}{P(\phi)}.$$9

where $>1$0 is unitary and $>1$1 is real, and restricts the covariance to the form

$>1$2

with $>1$3, $>1$4, and $>1$5.

The design is decomposed into two subproblems. In P2-A, the method selects an index set $>1$6 with $>1$7 and forms $>1$8 from the corresponding columns of $>1$9. In P2-B, for a fixed $\hat{\phi}$0, it allocates power over the selected modes by solving

$\hat{\phi}$1

The paper states that this problem is concave and solved efficiently with off-the-shelf convex solvers such as CVX. The final precoder is

$\hat{\phi}$2

The optimal solution strategy enumerates all $\hat{\phi}$3 index sets. Its algorithmic steps are: compute $\hat{\phi}$4, $\hat{\phi}$5, and $\hat{\phi}$6; construct $\hat{\phi}$7 and $\hat{\phi}$8; compute $\hat{\phi}$9 and $\phi$0 via GEVD to test feasibility and closed-form cases; if $\phi$1, perform the EVD of $\phi$2; enumerate all index sets, solve P2-B for each, evaluate the rate, and keep the best (Khan et al., 13 Aug 2025).

The reduced-complexity suboptimal strategy is motivated by the combinatorial cost of exhaustive eigenmode selection. It uses two phases. In Phase I, each index set is scored by setting equal power $\phi$3 and evaluating the achievable rate; the top-$\phi$4 index sets are retained, with $\phi$5. In Phase II, P2-B is solved only for those $\phi$6 shortlisted sets, and the best resulting rate determines the precoder. The paper states that the suboptimal strategy coincides with the optimal strategy when $\phi$7 equals the total number of combinations. For fixed $\phi$8, P2-B is solved to global optimality, while exhaustive search over $\phi$9 guarantees global optimality of P2-A. The dominant operations are EVD or GEVD on $\hat{\phi}$0 matrices, log-det evaluations on $\hat{\phi}$1 matrices, and repeated solutions of P2-B, so the overall complexity scales like $\hat{\phi}$2 times the cost of P2-B per channel realization.

6. Numerical behavior, comparison to prior approaches, and limitations

The reported numerical study uses the baseline parameters $\hat{\phi}$3, $\hat{\phi}$4, $\hat{\phi}$5, $\hat{\phi}$6, $\hat{\phi}$7, $\hat{\phi}$8, $\hat{\phi}$9, $\theta_{\mathrm{fake}} \equiv \hat{\phi}$00, and $\theta_{\mathrm{fake}} \equiv \hat{\phi}$01, with results averaged over independent channel realizations. Unless otherwise specified, the optimal solution is used (Khan et al., 13 Aug 2025).

For the DAOR-rate trade-off, the paper reports results for $\theta_{\mathrm{fake}} \equiv \hat{\phi}$02. When $\theta_{\mathrm{fake}} \equiv \hat{\phi}$03, the design reduces to conventional rate-maximizing beamforming and the beampattern peak is at $\theta_{\mathrm{fake}} \equiv \hat{\phi}$04. When $\theta_{\mathrm{fake}} \equiv \hat{\phi}$05, equal power appears in the true and obfuscated directions. When $\theta_{\mathrm{fake}} \equiv \hat{\phi}$06, the fake direction dominates and DAOR-based privacy is achieved. When $\theta_{\mathrm{fake}} \equiv \hat{\phi}$07, maximal obfuscation is obtained under the given channel and arrays. Across these cases, the achievable rate decreases as $\theta_{\mathrm{fake}} \equiv \hat{\phi}$08 increases, which quantifies the privacy-throughput trade-off.

Array configuration and stream count materially affect the feasible range. Both $\theta_{\mathrm{fake}} \equiv \hat{\phi}$09 and $\theta_{\mathrm{fake}} \equiv \hat{\phi}$10 increase with SNR. Increasing $\theta_{\mathrm{fake}} \equiv \hat{\phi}$11 yields higher $\theta_{\mathrm{fake}} \equiv \hat{\phi}$12, whereas increasing $\theta_{\mathrm{fake}} \equiv \hat{\phi}$13 tends to lower $\theta_{\mathrm{fake}} \equiv \hat{\phi}$14 with other parameters fixed. Larger arrays improve achievable rates. For the number of data streams, fewer streams concentrate more power in directions that enforce privacy, improving obfuscation but reducing rate; increasing $\theta_{\mathrm{fake}} \equiv \hat{\phi}$15 enhances rate, with diminishing returns at high $\theta_{\mathrm{fake}} \equiv \hat{\phi}$16, and weakens obfuscation for a given $\theta_{\mathrm{fake}} \equiv \hat{\phi}$17.

The comparison between the optimal and suboptimal strategies is likewise explicit. At $\theta_{\mathrm{fake}} \equiv \hat{\phi}$18, the suboptimal strategy with $\theta_{\mathrm{fake}} \equiv \hat{\phi}$19 achieves nearly an $\theta_{\mathrm{fake}} \equiv \hat{\phi}$20 reduction in computation time relative to the optimal strategy, with about $\theta_{\mathrm{fake}} \equiv \hat{\phi}$21 loss in achievable rate. The paper also notes that $\theta_{\mathrm{fake}} \equiv \hat{\phi}$22 further reduces cost but with larger rate degradation.

Relative to prior LOS-suppressing methods, the distinguishing feature of the DAOR framework is that it preserves LOS rather than nulling it. The paper states that LOS-suppressing methods can conceal the true direction but substantially degrade rate and/or require path separation in time or specialized pilots. By contrast, DAOR-based beamforming preserves LOS, maintains ISAC sensing viability and communication robustness, reshapes the receiver angular power distribution so that a false direction appears dominant, and requires only channel knowledge at the transmitter and standard array processing, with no extra infrastructure such as IRS or fake-delay injection (Khan et al., 13 Aug 2025).

Several practical considerations and limitations are identified. The obfuscated direction $\theta_{\mathrm{fake}} \equiv \hat{\phi}$23 should be sufficiently separated from $\theta_{\mathrm{fake}} \equiv \hat{\phi}$24 to avoid beampattern overlap, and the achievable $\theta_{\mathrm{fake}} \equiv \hat{\phi}$25 depends on array aperture, SNR, and multipath richness. The approach assumes that $\theta_{\mathrm{fake}} \equiv \hat{\phi}$26 is known at the transmitter; robust extensions could incorporate uncertainty sets for $\theta_{\mathrm{fake}} \equiv \hat{\phi}$27 and the angles $\theta_{\mathrm{fake}} \equiv \hat{\phi}$28 through worst-case or chance-constrained variants of P2-B. The paper also states that multi-user extensions would require managing several DAOR constraints or a weighted privacy objective across users or angles. Not nulling LOS helps preserve sensing, but very large $\theta_{\mathrm{fake}} \equiv \hat{\phi}$29 may de-emphasize power in the true LOS direction and degrade radar performance. The stated assumptions are a narrowband model, ULAs, a Rician channel, a single intended receiver, and transmitter-side CSI. Extensions to wideband OFDM, hybrid precoding, multi-antenna sensing receivers, and multiple adversaries are described as possible but not covered.