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 $\beta$0 parallel sub-channels. Denote channel input and output vectors as $\beta$1, and additive noise as $\beta$2. The channel transfer matrix is diagonal: $\beta$3 where

$\beta$4

and

$\beta$5

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

The power (link) gain is

$\beta$6

Collectively, sub-channel gain matrices are

$\beta$7

Power-allocation vectors $\beta$8 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$9” curves. Select a reference wavelength $\beta$0 and OAM mode $\beta$1; parameterize level sets via

$\beta$2

For two frequencies $\beta$3 with the same $\beta$4, the sub-channel gain ratio for positions with equal $\beta$5 is

$\beta$6

For the same carrier $\beta$7 and two modes $\beta$8: $\beta$9 These ratios are functions only of $z$0, signifying that proportional-gain sub-channel matrices arise on constant-$z$1 contours and not $z$2.

Positions $z$3 and $z$4 with the same $z$5 yield for every $z$6: $z$7 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 $z$8 under $z$9 is either invariant or tightly controlled within a region. For constellation $r$0: $r$1 Suppose $r$2, then

$r$3

The optimizer for $r$4 and $r$5 are identical, i.e.,

$r$6

Thus, a fixed optimal constellation suffices along constant-$r$7 contours.

For channels $r$8 with small perturbation $r$9, the normalized MED drop is bounded (Theorem 1, Eq. (14)): $(x,y)$0 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 $(x,y)$1 and the optimal is $(x,y)$2, let $(x,y)$3, and similarly for $(x,y)$4. For a reduced-alphabet $(x,y)$5, design $(x,y)$6 and optimize: $(x,y)$7 Theorem 2 (Eq. (20)) bounds the normalized MED loss: $(x,y)$8 Fixed $(x,y)$9 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 $I$0 into a fine grid. Partition $I$1 such that each region $I$2 has a designated constellation $I$3. The normalized MED distortion for positions $I$4 is

$I$5

A distortion threshold $I$6 (typically $I$7–$I$8) controls region assignment: $I$9, with $\{f_i=i\triangle f+f_0\}$0 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 $\{f_i=i\triangle f+f_0\}$1-region representation.

As $\{f_i=i\triangle f+f_0\}$2, the number of regions $\{f_i=i\triangle f+f_0\}$3 and distortion per region vanishes; practical implementations select $\{f_i=i\triangle f+f_0\}$4–$\{f_i=i\triangle f+f_0\}$5 for $\{f_i=i\triangle f+f_0\}$6–$\{f_i=i\triangle f+f_0\}$7 in $\{f_i=i\triangle f+f_0\}$8 systems under $\{f_i=i\triangle f+f_0\}$9–$L$0 GHz spacing.

6. Observed Performance and Guideline Synthesis

Simulation in (Wang et al., 2021) demonstrates that optimal partitions follow curvilinear strips along constant-$L$1 lines. Central regions with $L$2 admit larger $L$3 owing to slow spatial variation of $L$4, while boundaries demand finer partitioning. For the central OAM region, fixed-power constellations (with equal $L$5) 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 $L$6 (SER rise by $L$7 for large $L$8, e.g., $L$9 SER increase for $\mathcal L=\{l_1,\dots,l_L\}$0 and $\mathcal L=\{l_1,\dots,l_L\}$1 dB). BER remains below $\mathcal L=\{l_1,\dots,l_L\}$2 in central bands and rises to $\mathcal L=\{l_1,\dots,l_L\}$3–$\mathcal L=\{l_1,\dots,l_L\}$4 at boundaries unless remapped.

Design recommendations call for partitioning along constant-$\mathcal L=\{l_1,\dots,l_L\}$5 curves, distortion thresholds determined by MED→SER tradeoffs, with more constellations for larger $\mathcal L=\{l_1,\dots,l_L\}$6, carrier spacing, or OAM order. Offline construction with $\mathcal L=\{l_1,\dots,l_L\}$7 trials yields a compact LUT with $\mathcal L=\{l_1,\dots,l_L\}$8, eliminating runtime design requirements.

By leveraging inherent physical symmetries and channel gain proportionality indexed by $\mathcal L=\{l_1,\dots,l_L\}$9, 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).