Recursive Rotation Root-MUSIC

• Recursive Rotation Root-MUSIC is a DOA estimation technique that iteratively rotates an antenna array to align its boresight with the true emitter direction, reducing gain loss.
• The method integrates classical Root-MUSIC subspace processing with mechanical rotation to achieve mean-square-error performance approaching the Cramér–Rao lower bound.
• Empirical results show orders-of-magnitude MSE improvement over fixed Root-MUSIC, especially for off-boresight targets and low-elevation angles.

Recursive Rotation Root-MUSIC (RR-Root-MUSIC) is a direction-of-arrival (DOA) estimation method for antenna arrays that combines mechanical rotation of a directive array with classical Root-MUSIC subspace processing. The central innovation is an iterative algorithm that physically reorients a uniform planar array (UPA) to align its boresight with the true emitter direction, thereby mitigating gain loss and performance degradation associated with large initial boresight deflections. When implemented with moderate actuator precision and accurate calibration, RR-Root-MUSIC achieves mean-square-error (MSE) performance approaching the Cramér–Rao lower bound (CRLB), substantially outperforming conventional fixed Root-MUSIC, especially for off-boresight targets and low-elevation angles (Jiang et al., 29 Nov 2025).

1. Signal and Array Model

The system model centers on a UPA with $M\times N$ identical, directive antenna elements arrayed in the $x$–$z$ plane. The $(n,m)$-th element is positioned at: $$\mathbf{p}_{n,m} = [\,x_n,\;0,\;z_m\,]^T,\quad x_n = \left(n-\frac{N-1}{2}\right)d_x,\quad z_m = \left(m-\frac{M-1}{2}\right)d_z$$ where $d_x$ and $d_z$ are the inter-element spacings along $x$ and $z$. The emitter direction is parameterized by elevation $\theta$ and azimuth $\phi$: $$\overrightarrow{u}(\theta,\phi) = [\sin\theta\cos\phi,\; \sin\theta\sin\phi, \;\cos\theta]^T$$ Each element exhibits directive gain

$$g(\varphi) = g_0 \cos^p(\varphi), \qquad g_0 = \sqrt{\frac{A}{4\pi r^2}G_0}, \quad G_0 = 2(2p+1)$$

where $\varphi$ is the angle between emitter and element boresight, $r$ the emitter range, $A$ the array's physical aperture, and $p$ the directivity exponent. The array normal $\overrightarrow{n}$ defines the boresight, with

$$\varphi = \arccos\!\left(\overrightarrow{u}(\theta,\phi)^{T} \,\overrightarrow{n}\right)$$

After mechanical rotation ($\Delta_\theta$, $\Delta_\phi$ about the $x$ and $z$ axes, respectively), the received signal at $(n,m)$ is modeled as: $$\tilde y_{n,m} = s\,g\bigl(\varphi(\Delta_\theta,\Delta_\phi)\bigr)\, e^{\,j\,\psi_{n,m}(\Delta_\theta,\Delta_\phi;\theta,\phi)} + n_{n,m}$$ with $$\tilde{\mathbf y}=s\,g(\varphi)\,\tilde{\mathbf a}(\theta,\phi)+\mathbf n$$, where $\tilde{\mathbf a}$ is the rotated array manifold.

2. Cramér–Rao Lower Bound Analysis

The performance lower bound for unbiased DOA estimation is given by the CRLB, derived via the Fisher information matrix (FIM). For $K$ snapshots, the FIM is: $$\mathbf{F}(\theta,\phi)=K\,\mathbf{F}_{\rm (1\,snap)}(\theta,\phi) = \frac{2K|s|^2}{\sigma^2}\begin{bmatrix}P & Q \ Q & R\end{bmatrix}$$ The bound for the DOA estimates is thus: $$\mathrm{CRLB}(\theta,\phi) = \mathbf{F}^{-1}(\theta,\phi) = \frac{\sigma^2}{2K\,|s|^2} \frac{1}{PR-Q^2} \begin{bmatrix}R & -Q \ -Q & P\end{bmatrix}$$ This yields marginal bounds: $$\mathrm{Var}\{\hat\theta\} \ge \frac{\sigma^2}{2K|s|^2}\,\frac{R}{PR - Q^2},\qquad \mathrm{Var}\{\hat\phi\} \ge \frac{\sigma^2}{2K|s|^2}\,\frac{P}{PR - Q^2}$$ The parameters $P$, $Q$, $R$ incorporate array geometry, antenna pattern derivative, and steering vector phase response. When array boresight aligns with the emitter ($\varphi\to 0$), $g(\varphi)$ increases and the bound tightens, reflecting improved estimation precision (Jiang et al., 29 Nov 2025).

3. RR-Root-MUSIC Algorithmic Procedure

Initialization and Subspace Steps

• The array is initially placed with boresight along the $y$-axis ($\Delta_\theta=\Delta_\phi=0$).
• $K$ signal snapshots are acquired; the sample covariance matrix $\hat{\mathbf R}$ is formed and eigendecomposed.
• The noise subspace $\mathbf U_n$ is extracted.
• 2D Root-MUSIC proceeds by reparameterizing steering vectors as

$$z_x = e^{j\frac{2\pi}{\lambda}d_x\sin\theta\cos\phi}, \quad z_z = e^{j\frac{2\pi}{\lambda}d_z\cos\theta}$$

• Minimization of $$f(z_x,z_z) = \mathbf a^H(z_x,z_z) \mathbf U_n\mathbf U_n^H \mathbf a(z_x,z_z)$$ on the unit circle yields $(\hat\theta, \hat\phi)$.

Recursive Mechanical Rotation

For each iteration $i$:

1. Set new rotation targets: $\Delta_{\theta_i}=90^\circ-\hat\theta_i$, $\Delta_{\phi_i}=90^\circ-\hat\phi_i$.
2. Mechanically rotate array.
3. Acquire fresh snapshots and repeat Root-MUSIC estimation.
4. Transform estimated angles back to original coordinate frame.
5. Halt if $$|\bar\theta_{\,i+1}-\bar\theta_{\,i}| \le\epsilon_e$$ and $|\bar\phi_{\,i+1}-\bar\phi_{\,i}| \le\epsilon_e$, where $\epsilon_e=0.01^\circ$.
6. Otherwise, repeat with updated estimates.

This iterative procedure ensures the array's boresight converges towards the emitter, incrementally increasing the array's effective gain and SNR for the signal of interest.

4. Estimation Accuracy and MSE Performance

Let $\hat\theta_{\rm RR}$ denote the final elevation estimate. The RR-Root-MUSIC MSE,

$$\mathrm{MSE}_{\rm RR} = \mathbb{E}[(\hat\theta_{\rm RR}-\theta)^2],$$

is bounded below by the CRLB. In contrast, conventional (fixed) Root-MUSIC, in the presence of large off-boresight deflection, demonstrates MSEs orders of magnitude above the CRLB. Upon convergence, RR-Root-MUSIC MSE closely tracks the bound, maintaining high accuracy across substantial angular deviations (Jiang et al., 29 Nov 2025).

5. Empirical Results and Illustration

All simulation outcomes are averaged over 2000 Monte-Carlo trials:

• UPA with $M=N=6$, $\lambda=0.125$ m, $r=250$ m, noise $\sigma^2=-100$ dBm, snapshot size $K=1000$.
• At low SNR ($-10$ dB), for $\theta=15^\circ$, fixed Root-MUSIC yields MSE $\sim 10^{-2}$ rad$^2$, whereas RR-Root-MUSIC achieves $\sim 10^{-8}$ rad$^2$ within $\approx13$ iterations.
• At moderate SNR ($10$ dB) and high SNR ($30$ dB), convergence accelerates to $\approx5$ and $\approx1$ iterations, respectively.
• Across $\theta\in[0,90^\circ]$, RR-Root-MUSIC outperforms fixed Root-MUSIC by up to 6–8 orders of magnitude in MSE, particularly in the low-elevation regime ($\theta\le15^\circ$).
• For a semi-circular UAV flight path, RR-Root-MUSIC maintains near-CRLB performance even at $\theta\to 0^\circ$, while fixed MUSIC fails due to array “blind spots.”
• Increasing directivity exponent $p$ enhances performance at $\varphi=0$ but intensifies loss at larger deflections, further motivating the RR-Root-MUSIC approach.

Scenario	Fixed Root-MUSIC MSE	RR-Root-MUSIC MSE	# Iterations (RR)
$\theta=15^\circ$, SNR $-10$ dB	$\sim 10^{-2}$ rad$^2$	$\sim 10^{-8}$ rad$^2$	$\approx 13$
$\theta=45^\circ$, SNR $10$ dB	Above CRLB	Near CRLB	$\approx 5$
UAV overhead, SNR any	Failure	Near CRLB	$\leq 10$

6. Practical Considerations and Deployment Implications

RR-Root-MUSIC enables robust DOA estimation for low-altitude and steep-angle tracking scenarios—regions where fixed arrays are subject to severe performance degradation. Benefits include:

• Full directional coverage including “above-BS blind spots.”
• Orders-of-magnitude MSE improvement in off-boresight or high directivity settings.
• Rapid convergence (≤10 mechanical rotations), compatible with moderate-rate actuator systems.

Trade-offs include:

• Added mechanical complexity, cost, and maintenance associated with rotary stages.
• Increased measurement latency per iteration due to physical movement and data acquisition.
• Requirement for accurate calibration of element patterns and rotation angles.

A plausible implication is the suitability of RR-Root-MUSIC for dynamic 5G/6G base-station deployments, UAV command-and-control, and emerging low-altitude communication networks where sensing reliability and angular coverage are critical (Jiang et al., 29 Nov 2025).