Uniform Circular Arrays: Theory & Applications

• Uniform Circular Arrays (UCAs) are antenna configurations with elements evenly distributed along a circle, offering rotational symmetry and uniform spatial coverage.
• Their circulant structure enables DFT-diagonalization, achieving full spatial multiplexing in LoS MIMO and efficient OAM multiplexing through helical phasefronts.
• Advanced techniques like delay-phase precoding and precise calibration allow UCAs to maintain near-optimal performance in wideband and dense network scenarios.

A uniform circular array (UCA) consists of $N$ antenna elements distributed equidistantly along the circumference of a circle with radius $R$ in a plane, most commonly the $xy$-plane. Each element’s position can be described by a polar angle (azimuthal position) $\psi_n = 2\pi n/N$ for $n=0,1,\dots,N-1$, yielding a 2D spatial distribution with complete rotational symmetry. This geometry endows UCAs with key properties: identical array response for all azimuth angles, structure amenable to DFT-based diagonalization, and unique channel characteristics in LoS, near-field, wideband, and OAM-multiplexed communications.

1. Geometric Model, Array Manifold, and Channel Structure

Each element of a UCA lies at $\mathbf{p}_n = [R\cos\psi_n,\, R\sin\psi_n,\,0]^T$. For plane wave incidence from azimuth angle $\theta$ (with elevation $\phi$), the array manifold (steering vector) is

$$\mathbf{a}(\theta, \phi) = \left[e^{j k R \cos(\theta-\psi_0)}, \dots, e^{j k R \cos(\theta-\psi_{N-1})}\right]^T$$

where $k=2\pi/\lambda$. For LoS MIMO systems utilizing transmit and receive UCAs separated by $D$, the element-to-element distance under far-field $D\gg R$ is

$$d_{nm} \approx D - \frac{R_t R_r}{D} \cos(\psi_n - \psi_m + \theta_o)$$

with misalignment terms for rotation $\theta_o$, tilting $(\phi_x, \phi_y)$, and center-shift vector $\mathbf{c}$ included as per the generic model (Jeon et al., 2020).

The normalized LOS channel coefficient is

$h(n,m) = e^{-j \frac{2\pi}{\lambda} d(n,m)}$

with the channel matrix $H$ factoring into circulant (DFT-diagonalizable) forms. Channel singular values depend only on the radii-product-to-distance ratio (RPDR) $\beta = 2\pi R_t R_r / (\lambda D)$ and relative array rotation, remaining independent of tilting and center-shift (Jeon et al., 2020).

2. DFT-Diagonalization, Multiplexing, and Optimal Design

A core property, the circulant structure, allows $H$ to be diagonalized by the DFT matrix $Q$: $H = Q \Delta_A Q^H$ which immediately yields singular values $\sigma_k$ as explicit functions of $\beta$ and $\theta_o$: $$\sigma_k(\beta, \theta_o) = \left| \sum_{i=0}^{N-1} e^{-j(2\pi i(k-1)/N - \beta \cos(2\pi i/N + \theta_o))} \right|$$ Thus, UCA-based LoS MIMO achieves full spatial multiplexing with $N$ streams per symbol, when $\beta$ is set optimally. Practically, $\beta^\circ$ is found by 1D search to maximize the sum capacity: $$C(\beta) = \sum_{k=1}^N \log_2(1 + p_k \sigma_k^2(\beta)/N_0)$$ with power allocation $\sum_k p_k = P_T$. Selecting $\beta^\circ$ (offline for $(N, \mathrm{SNR})$), then choosing $R_t, R_r$ so that $R_t R_r = \beta^\circ \lambda D/(2\pi)$, nearly achieves orthogonal channel conditions; ZF and water-filling receivers then deliver maximal throughput without CSI feedback (Jeon et al., 2020).

$N$	Optimal $\beta^\circ$ (SNR 15dB)	$R_t = R_r$ at $D=100$m
4	1.54	0.31 m
8	3.09	0.44 m
12	4.57	0.54 m
16	5.98	0.62 m

With optimal $\beta$, the channel matrix approaches unitarity, enabling robust spatial multiplexing with ZF (or ZF+SIC) processing (Jeon et al., 2020).

3. Channel-Independent Beamforming for UCA LoS MIMO

Channel-independent beamforming in UCA systems exploits the circularly symmetric geometry to enable fixed, DFT/IDFT-based precoding and combining that "decouples" the MIMO channel into parallel links. With parallel or aligned UCAs (with or without coaxiality), the fixed transmit matrix $W_t = T^* W$ and receive matrix $W_r = W^* A^*$ (where $W$ is the DFT/IDFT, $A/T$ are deterministic phase precompensation) reduce the effective channel $H_{\text{eff}}$ to a diagonal form, enabling symbol-wise ML detection at extremely reduced complexity: $$y'_n = H_n s_n + z'_n;\qquad \hat{s}_n = \arg\min_{c \in \mathcal{C}} |y'_n - H_n c|^2$$ Bit-error-rate performance matches that of full CSI-based MIMO processing, while computational cost drops by several orders of magnitude for moderate $N$ (Jing et al., 2018, Jing et al., 2024). This approach extends to both coaxial and laterally shifted UCA pairs, provided far-field ($D \gg R$) holds.

4. Orbital Angular Momentum (OAM) Multiplexing with UCAs

UCAs are the canonical structure for generating and detecting radio OAM modes, where feeding element $n$ with phase $e^{i \ell \psi_n}$ realizes helical phasefronts indexed by integer $\ell$. The array factor for OAM mode $\ell$ is

$$AF_\ell(\theta, \phi) = N e^{i \ell \phi} J_\ell(k R \sin \theta)$$

where $J_\ell$ is the Bessel function of order $\ell$, yielding a doughnut-shaped beam with central null for $\ell \ne 0$ (Gaffoglio et al., 2015, Chen et al., 2020). The link budget for OAM transmission acquires an extra decay $\propto d^{-2|\ell|-2}$ with distance. Mode isolation is high—mode sorters and precise alignment (mechanical tolerance $\le 1^\circ$) achieve $>15$ dB inter-mode isolation in field experiments. For high-order OAM modes, divergence and attenuation grow rapidly; concentric UCAs (multiple rings) enable capacity-optimized multiplexing using several parallel low-order modes, with water-filling power allocation across rings and modes (Jing et al., 2024, Jing et al., 2024).

5. Wideband Beamforming, Spatial Effects, and Delay-Phase Precoding

UCA hybrid precoding architectures for mmWave/THz operate under spatial-wideband impairments. Unlike ULAs (which suffer beam split), UCAs manifest a "beam defocus" effect: analog phase shifters cannot maintain constructive interference across ultra-large bandwidths, so the main-lobe gain drops at frequencies away from the carrier. The frequency-domain beam pattern is analytically

$G(f) = | J_0(2\pi R \Delta f/c) |$

where $\Delta f = f - f_c$. Delay-phase-precoding (DPP) schemes remedy defocus by integrating true-time-delay (TTD) devices per element or subarray, producing frequency-dependent phase shifts and restoring constructive summation over wideband. Analytical and simulation results show DPP with $K$ TTD taps recovers $>90\%$ of the optimum gain and achieves near-ideal spectral efficiency across multi-GHz bandwidths; narrowband PS-only architectures suffer bandwidth-dependent loss (Wu et al., 2023).

6. Near-Field, XL-MIMO, and Localization

UCAs, due to their rotational symmetry, support angle-independent and omnidirectional near-field beamforming and localization. Key metrics such as effective Rayleigh distance (ERD) quantify the spatial region for beamfocusing; for UCAs,

$$r_\text{ERD}^{(C)} = \frac{\pi R^2}{2\lambda J_0^{-1}(1-\delta)}$$

is angle-invariant, in contrast to ULAs, whose ERD shrinks at off-broadside angles (Wu et al., 2022). In radiative near-field, closed-form expressions for beamdepth and EBRD enable analytic trade-off of coverage versus capacity under element-count or fixed-aperture constraints (Hussain et al., 16 Nov 2025). FFT-accelerated backprojection on sectored UCAs achieves ML-consistent localization with nearly linear complexity, and exact angle quantization with massive UCAs yields real-time 2D-DOA estimation robust to nonuniform noise (Liu et al., 2024, Gong, 17 Jul 2025).

7. Design Guidelines, Implementation, and Calibration

Optimal UCA LoS MIMO mandates setting the RPDR near $\beta^\circ$ for orthogonalizable channels; this tunes $R_t, R_r$ for a given $D$ (Jeon et al., 2020). When mechanical constraints limit $R$, codebook-based phase precoding with a small feedback overhead (6–10 bits) can recover most capacity loss incurred by sub-optimal $\beta$.

Odd-element UCAs, particularly $N=3$, facilitate wideband decoupling and matching with compact DMNs; advanced microstrip designs (two-stage or star-triangle) extend matching/decoupling bandwidth to several percent RF BW, outperforming simple neutralization-line DMNs (Kornprobst et al., 2021). Calibration—including mutual coupling—can be performed via sparse recovery with an integrated wideband dictionary, combining subspace SVD projection, iterative LASSO, and non-numerical atomic construction (Bozorgasl et al., 2024).

In massive MIMO, UCAs guarantee "favorable propagation": inter-user interference decays as $O(N^{-1/3})$ for fixed spacing $d$ in pure-LoS, an asymptotic property derived via Bessel expansion. Stacking UCAs vertically (cylindrical arrays) achieves double-sided FP for distinct elevation and azimuth (Anarakifirooz et al., 2021).

8. Beamforming, User-Dense Networks, and Concentric UCAs

Concentric UCAs (UCCAs) with multiple rings enable sharper beams, higher beam-packing gains, and enhanced spatial multiplexing. Large-aperture arrays with spacing $d > \lambda/2$ yield narrower HPBW and up to $30\times$ higher angular packing capacity than conventional planar arrays. SINR and spectral efficiency for UCCAs in dense 5G scenarios exceed planar arrays by up to $5\times$; moderate sidelobe levels and high efficiency are retained provided amplitude tapering and calibration are implemented (Hasan et al., 2022).

Conclusion

Uniform circular arrays represent a highly symmetric, analytically tractable transceiver architecture offering unique advantages in LoS MIMO, OAM multiplexing, wideband beamforming, and near-field spatial sensing. Their channel structure, when properly exploited (by RPDR tuning or DFT-based beamforming), achieves maximal multiplexing rate and computational efficiency. UCA’s omnidirectional symmetry underpins angle-invariant coverage, favorable propagation, and simplified calibration. Next-generation enhancements—multi-ring concentric architectures, delay-phase precoding, and sparse calibration—continue expanding UCA's practical relevance in ultra-dense, bandwidth-rich communication paradigms.