Spatially Isotropic Constellations in mmWave OAM

• Spatially isotropic constellations are signaling schemes that maintain uniform minimum Euclidean distance across spatial regions by harnessing proportional sub-channel gain matrices.
• They employ constant-beta contours and map-assisted partitioning to design robust constellations for mmWave WDM with orbital angular momentum in short-range LOS links.
• Simulation results and fixed-power allocation strategies demonstrate stable error-rate performance, enabling efficient and scalable communication across the defined spatial domain.

Spatially isotropic constellations in the context of mmWave WDM with OAM for short-range LOS links are multi-dimensional signaling schemes whose minimum Euclidean distance (MED), and thus error-rate performance, is uniform across defined regions of space due to the proportionality or near-proportionality of the underlying sub-channel gain matrices. This spatial isotropy is operationalized by partitioning the receiver’s $(r,z)$ domain—using constant-$\beta$ contours—into bands wherein the constellation design remains unchanged or whose MED degradation remains within rigorously bounded thresholds. Collectively, such constellations exploit the rotational symmetry and channel structure inherent in OAM-mmWave links to provide a small library of signaling patterns indexed by $\beta$-bands, obviating the need for real-time optimization and ensuring stable communication rates throughout the operational area (Wang et al., 2021).

1. System Model and Notation

The transmitter and receiver are aligned along the $z$-axis, with the receiver possibly displaced off-axis by a transverse distance $r$ in the $(x,y)$ plane. The system uses $I$ carrier frequencies $\{f_i=i\triangle f+f_0\}$ and $L$ orbital angular momentum modes $\mathcal L=\{l_1,\dots,l_L\}$, forming $U=I\cdot L$ parallel sub-channels. Denote channel input and output vectors as $\mathbf x,\mathbf y\in\mathbb C^U$, and additive noise as $\mathbf n$. The channel transfer matrix is diagonal: $$\mathbf y = \mathbf H\,\mathbf x + \mathbf n,\quad \mathbf H = \operatorname{diag}(h_{1}^{l_1}, \dots, h_{I}^{l_L}),$$ where

$h_i^l(r,\phi,z) = h_i^l(r,z)e^{-jl\phi}$

and

$$h_i^l(r,z) = \frac{\sqrt{\zeta_i^l}\,\lambda_i}{4\pi\,d_{i,m}^l(z)}\left(\frac{r}{r_{i,\max}^l(z)}\right)^{|l|} \exp\left(\frac{r_{i,\max}^l(z)^2 - r^2}{\omega_i^2(z)}\right) \exp\left(-j \frac{2\pi d_{i,m}^l(z)}{\lambda_i}\right).$$

This construction adheres to Eq. (1) and related notation in (Wang et al., 2021).

The power (link) gain is

$$g_i^l(r,z) = |h_i^l(r,z)|^2 = \frac{\zeta_i^l \lambda_i^2}{(4\pi d_{i,m}^l(z))^2}\left(\frac{r}{r_{i,\max}^l(z)}\right)^{2|l|}\exp\left(\frac{2(r_{i,\max}^l(z)^2 - r^2)}{\omega_i^2(z)}\right).$$

Collectively, sub-channel gain matrices are

$$\mathbf G(r,z) = \operatorname{diag}(g_{1}^{l_1}(r,z),\dots,g_{I}^{l_L}(r,z)).$$

Power-allocation vectors $\mathbf p=[P_1,\dots,P_U]^T$ can be per-subchannel or summed to a total power constraint.

2. OAM Beam Properties and Spatial Gain Proportionality

OAM beams exhibit constant link-gain ratios along “constant-$\beta$” curves. Select a reference wavelength $\lambda_a$ and OAM mode $l_m$; parameterize level sets via

$$r = \beta\, r_{a,\max}^{l_m}(z),\quad r_{a,\max}^{l_m}(z) = \omega_a(z)\sqrt{|l_m|/2}.$$

For two frequencies $(i,j)$ with the same $l$, the sub-channel gain ratio for positions with equal $\beta$ is

$$a_{i,j}^l(\beta,z) = \frac{g_i^l(\beta\,r_{a,\max}^{l_m},z)}{g_j^l(\beta\,r_{a,\max}^{l_m},z)} \approx \left( \frac{\lambda_j}{\lambda_i}\right)^{|l|-2} \exp\left\{ \beta^2|l_m| \left( \frac{\lambda_i}{\lambda_j} - 1 \right)\right\}.$$

For the same carrier $i$ and two modes $l_1,l_2$: $$a_i^{l_1, l_2}(\beta,z) \approx \left( \frac{\lambda_a}{\lambda_i} \right)^{|l_1| - |l_2|} (\beta^2 |l_m|)^{|l_1| - |l_2|} e^{|l_1| - |l_2|}.$$ These ratios are functions only of $\beta$, signifying that proportional-gain sub-channel matrices arise on constant-$\beta$ contours and not $z$.

Positions $(r_1, z_1)$ and $(r_2, z_2)$ with the same $\beta$ yield for every $(i,l)$: $$g_i^l(r_2, z_2) = \alpha^2 g_i^l(r_1, z_1), \quad \mathbf G(r_2, z_2) = \alpha^2 \mathbf G(r_1, z_1),\quad \mathbf H(r_2, z_2) = \alpha \mathbf H(r_1, z_1),$$ up to phase, establishing proportional-gain regions central to spatial isotropy (Wang et al., 2021).

3. Conditions for Spatially Isotropic Constellation Design

Spatially isotropic constellations are those for which the minimum Euclidean distance $d_{\min}$ under $\mathbf H$ is either invariant or tightly controlled within a region. For constellation $\mathcal C\subset\mathbb C^U$: $$d_{\min}(\mathbf H, \mathcal C) = \min_{x\ne x'\in\mathcal C} \|\mathbf H(x-x')\|_2.$$ Suppose $\mathbf H_2 = \alpha \mathbf H_1$, then

$$d_{\min}(\mathbf H_2, \mathcal C) = \alpha d_{\min}(\mathbf H_1, \mathcal C).$$

The optimizer for $\mathbf H_2$ and $\mathbf H_1$ are identical, i.e.,

$$\mathcal C^*(\mathbf H_2) = \mathcal C^*(\mathbf H_1).$$

Thus, a fixed optimal constellation suffices along constant-$\beta$ contours.

For channels $\mathbf H_2 = \alpha\mathbf H_1 + \Delta\mathbf H$ with small perturbation $\|\Delta\mathbf H\|\ll1$, the normalized MED drop is bounded (Theorem 1, Eq. (14)): $$1 - \frac{d_{\min}(\mathbf H_2, \mathcal C^*(\mathbf H_1))}{d_{\min}(\mathbf H_2, \mathcal C^*(\mathbf H_2))} \leq O(\|\Delta\mathbf H\|_F).$$ This establishes that near-proportional gain matrices enable shared constellations within a tolerable loss.

4. Fixed Power Vector and Performance Bounds

If the power allocation vector is fixed as $\mathbf p^{(f)}$ and the optimal is $\mathbf p^{(o)}$, let $$\mathbf A^{(f)} = \sqrt{\operatorname{diag}(\mathbf p^{(f)})}$$, and similarly for $\mathbf A^{(o)}$. For a reduced-alphabet $\mathcal S$, design $x = \mathbf A s$ and optimize: $$d_{\min}(\mathbf H, \mathbf p, \mathcal S) = \min_{s\ne s'\in\mathcal S} \|\mathbf H \mathbf A (s - s')\|_2.$$ Theorem 2 (Eq. (20)) bounds the normalized MED loss: $$\Delta = \left|1 - \frac{d_{\min}(\mathbf H, \mathbf p^{(f)}, \mathcal S^{(f)})}{d_{\min}(\mathbf H, \mathbf p^{(o)}, \mathcal S^{(o)})}\right| \leq O(\| \mathbf p^{(f)} - \mathbf p^{(o)} \|_2).$$ Fixed $\mathbf p^{(f)}$ near the optimum preserves the error-rate bound, particularly in central regions where gain matrices are nearly proportional.

5. Map-Assisted Spatial Partitioning and Algorithmic Construction

To construct spatially isotropic constellations, discretize the operation area $\mathcal Q$ into a fine grid. Partition $\mathcal Q = R_1 \cup \dots \cup R_K$ such that each region $R_k$ has a designated constellation $\mathcal C_k$. The normalized MED distortion for positions $q_1, q_2$ is

$$\Delta_{mn}(q_1, q_2) = \left| 1 - \frac{d_{\min}(\mathbf H_{q_2}, \mathcal C^*_{q_1})}{d_{\min}(\mathbf H_{q_2}, \mathcal C^*_{q_2})} \right|.$$

A distortion threshold $\tau_d$ (typically $0.10$–$0.15$) controls region assignment: $$R_k = \{q\in\mathcal Q : \Delta(q, q_k) \le \tau_d \}$$, with $q_k$ the region center.

The map-building pseudocode (see (Wang et al., 2021)) iteratively selects region centers, computes optimum constellations, and clusters points based on thresholded MED-distortion, targeting the minimal aggregate distortion and a compact $K$-region representation.

As $\tau_d\to0$, the number of regions $K\to|\mathcal Q|$ and distortion per region vanishes; practical implementations select $K=10$–$15$ for $M=32$–$64$ in $I=2, L=3$ systems under $1$–$5$ GHz spacing.

6. Observed Performance and Guideline Synthesis

Simulation in (Wang et al., 2021) demonstrates that optimal partitions follow curvilinear strips along constant-$\beta$ lines. Central regions with $\beta\approx\beta_{\max}$ admit larger $R_k$ owing to slow spatial variation of $\mathbf G(r, z)$, while boundaries demand finer partitioning. For the central OAM region, fixed-power constellations (with equal $P_n$) suffice with negligible loss, but near boundaries, particularly with high OAM order, tailored pre-allocations become necessary.

Numerical evaluation indicates that for normalized MED-drop $\Delta=0.10$ (SER rise by $\exp(2\Delta/M)$ for large $M$, e.g., $15\%$ SER increase for $M=64$ and $E_b/N_0\approx15$ dB). BER remains below $10^{-4}$ in central bands and rises to $10^{-3}$–$10^{-2}$ at boundaries unless remapped.

Design recommendations call for partitioning along constant-$\beta$ curves, distortion thresholds determined by MED→SER tradeoffs, with more constellations for larger $M$, carrier spacing, or OAM order. Offline construction with $C_d\sim100$ trials yields a compact LUT with $K\ll|\mathcal Q|$, eliminating runtime design requirements.

By leveraging inherent physical symmetries and channel gain proportionality indexed by $\beta$, spatially isotropic constellations provide robust, efficient, and universally applicable signaling in mmWave WDM+OAM short-range LOS environments. Fixed-power allocation and map-assisted partitioning enable scalable deployment with controllable error-rate performance across the spatial domain (Wang et al., 2021).