Metasurface-Enabled Superheterodyne Architecture

• MSA is a novel architecture that integrates metasurfaces with superheterodyne detection, enabling robust, spatially isotropic mmWave communications.
• It exploits the cylindrical symmetry of OAM beams to maintain constant link-gain ratios across spatial regions, ensuring minimal degradation in error performance.
• The design employs fixed power allocation and map-assisted region partitioning to optimize constellation assignment and sustain high minimum Euclidean distance in LOS environments.

Spatially isotropic constellations are multi-dimensional signaling schemes for millimeter-wave (mmWave) wavelength division multiplexing (WDM) channels employing orbital angular momentum (OAM), specifically constructed to provide robust and near-uniform communication performance across a spatial region in line-of-sight (LOS) settings. By leveraging the cylindrical symmetry of OAM beams, these constellations exploit the invariance of link-gain ratios along specific geometric loci—parameterized by a normalized radial coordinate $\beta$—enabling the creation of compact sets of “universal” constellation patterns that ensure minimal worst-case degradation in minimum Euclidean distance (MED), and consequently maintain spatially uniform error rates under practical constraints (Wang et al., 2021).

1. System Model and Channel Geometry

The canonical system consists of co-axially aligned transmitter and receiver along the $z$-axis. The receiver may be positioned at a radial offset $r$ in the transverse ($x$, $y$) plane, with the signal propagation modeled in cylindrical coordinates $(r, \phi, z)$. The transmission leverages $I$ carrier frequencies $f_i = i\Delta f + f_0$ (wavelengths $\lambda_i = c/f_i$) and $L$ discrete OAM modes $\mathcal{L} = \{l_1, \dots, l_L\}$, producing $U = I \cdot L$ orthogonal sub-channels indexed either by $n = 1, \ldots, U$ or by the $(i, l)$ pair.

The complex channel response from transmitter sub-channel $(i, l)$ to a receiver at $(r, \phi, z)$ is given by

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

with the magnitude

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

where $r_{i,\max}^l(z)$ is the OAM “ring radius,” $\omega_i(z)$ is the beam-spot size, $d_{i,m}^l(z)$ is the propagation distance, and $\zeta_i^l \simeq 1$ (antenna gain). The power gain is thus $g_i^l(r, z) = |h_i^l(r, z)|^2$, forming a diagonal matrix $$\mathbf{G}(r, z) = \operatorname{diag}(g_1^{l_1}(r, z),\dotsc,g_I^{l_L}(r, z))$$ for the $U$ parallel sub-channels.

Inputs $\mathbf{x} \in \mathbb{C}^U$ and outputs $\mathbf{y} \in \mathbb{C}^U$ are related by

$\mathbf{y} = \mathbf{H} \mathbf{x} + \mathbf{n}$

with $\mathbf{H}$ diagonal as above, and $\mathbf{n}$ additive noise.

2. OAM Beam Properties and Spatial Symmetry

Fundamental to spatially isotropic constellation design is the observation that the ratios of link gains $g_i^l$ among sub-channels are constant on “constant-$\beta$” contours. Fixing a reference OAM mode $l_m \neq 0$ and reference wavelength $\lambda_a$, the normalized radial variable $\beta$ is defined via $r = \beta r_{a, \max}^{l_m}(z)$, where $r_{a, \max}^{l_m}(z)$ is the ring radius of the reference mode. Along a fixed $\beta$, the ratio of gains between different frequencies (same mode) is

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

and between modes (same frequency) is

$$a_i^{l_1,l_2}(\beta, z) \approx \left(\frac{\lambda_a}{\lambda_i}\right)^{|l_1| - |l_2|} (\beta^2 |l_m|)^{|l_1| - |l_2|} e^{|l_1| - |l_2|}$$

Both ratios depend only on $\beta$, not $z$: all points along the same $\beta$ contour share the same link-gain ratios, leading to proportional gain matrices $$\mathbf{G}(r_2, z_2) = \alpha^2 \mathbf{G}(r_1, z_1)$$ for a constant $\alpha$ whenever the $\beta$ values match.

3. Criteria for Spatially Invariant Constellation Assignment

The design goal is to choose a constellation $\mathcal{C} \subset \mathbb{C}^U$ of size $M$ that maximizes the minimum Euclidean distance (MED) at the receiver, i.e.

$$d_{\min}(\mathbf{H}, \mathcal{C}) = \min_{x \neq x' \in \mathcal{C}} \|\mathbf{H}(x - x')\|_2$$

For proportional gain matrices, optimal constellations exhibit scaling invariance: $$\mathcal{C}^*(\alpha \mathbf{H}_1) = \mathcal{C}^*(\mathbf{H}_1)$$ Therefore, all receiver locations sharing the same $\beta$ can employ a common optimum constellation. If the channel matrices are only nearly proportional ($\mathbf{H}_2 \approx \alpha \mathbf{H}_1$), the normalized MED loss is $O(\|\Delta \mathbf{H}\|_F)$ and remains bounded as long as $\Delta \mathbf{H}$ is small [(Wang et al., 2021), Theorem 1]. This supports “banding” the space into regions where a single constellation is near-optimal, with limited loss in MED.

4. Fixed Power Allocation and Robustness

In practical systems, it may be necessary to pre-assign a fixed power vector $\mathbf{p}^{(f)}$ (e.g., for hardware simplicity or fairness), whereas the true optimum is $\mathbf{p}^{(o)}$. Defining $$\mathbf{A}^{(f)} = \sqrt{\operatorname{diag}(\mathbf{p}^{(f)})}$$ and $x = \mathbf{A} s$ for some alphabet $\mathcal{S}$, the MED metric becomes

$$d_{\min}(\mathbf{H}, \mathbf{p}, \mathcal{S}) = \min_{s \neq s' \in \mathcal{S}} \|\mathbf{H} \mathbf{A}(s - s')\|_2$$

The normalized MED penalty when using $\mathbf{p}^{(f)}$ in place of the true optimum is bounded: $$\Delta \leq \|\mathbf{p}^{(f)} - \mathbf{p}^{(o)}\|_2 \max_{s_d} \frac{\|\mathbf{H} s_d\|^2}{\|\mathbf{H} \mathbf{A}^{(o)} s_d\|^2}$$ This loss is negligible in regions where equal-power allocation is near-optimal, notably at the center of the OAM beam ($\beta \approx \beta_\text{max}$), but can be significant near high-order mode boundaries.

5. Map-Assisted Partitioning and Constellation Assignment

The spatial region $(r, z)$ is discretized into a grid $\mathcal{Q}$ and partitioned into $K$ nonoverlapping regions $R_1, \ldots, R_K$, each assigned a specific constellation $\mathcal{C}_k$. The partitioning leverages the normalized MED difference

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

A threshold $\tau_d$ is set (e.g., $0.15$), and all positions with $\Delta(q, q_k) \leq \tau_d$ are grouped into the same region $R_k$. The algorithm iteratively selects centers $q^*$, generates the corresponding constellation, and accumulates regions until the space is covered, minimizing the total sum of MED distortions. Smaller $\tau_d$ yield more regions ($K \uparrow$) and tighter MED control.

In the studied scenario ($I = 2$, $\Delta f = 1$–5 GHz, $\mathcal{L} = \{0, \pm1\}$ or $\{0, \pm2\}$, $M = 32, 64$), the resulting regions $R_k$ form curvilinear strips along constant-$\beta$ contours. Central regions (high SNR) have larger $R_k$ and thus require fewer distinct constellations; peripheral or high-mode boundary regions generate more and smaller partitions.

6. Performance, Error-Rate, and Design Principles

Simulated performance demonstrates that by choosing $\tau_d = 0.15$ and $K \approx 10$–15, the aggregate MED distortion remains capped and system bit error rates (BER) in central beam regions remain below $10^{-4}$. In boundary zones, error rates may rise to $10^{-3}$–$10^{-2}$ if not remapped, especially for high-order OAM modes or large $M$. The union-bound estimate shows a normalized MED drop $\Delta = 0.10$ yields a symbol error rate (SER) increase of approximately $\exp(2\Delta/M)$, e.g., a 15% increase for 64-ary modulation at $E_b/N_0 \approx 15$ dB.

Key principles for spatially isotropic design are:

• Partition spatial regions along constant-$\beta$ contours, which respect the underlying OAM channel symmetry.
• Control region granularity $K$ based on the desired MED-to-SER loss, channel order, and constellation size.
• Employ fixed-power, equal allocation in beam centers; consider allocation-aware constellations near region boundaries.
• Construct map-based look-up tables (LUTs) offline using a small number of trials ($C_d \sim 100$), enabling efficient real-time deployment via simple spatial indexing (Wang et al., 2021).

7. Implications and Scope of Spatially Isotropic Constellations

Spatially isotropic constellations, constructed via map-assisted partitioning and informed by OAM beam properties, realize high spectral efficiency and robust bit-error performance with minimal online computation. The approach fully exploits rotational symmetry and the parameterization of channel gain by a single spatial variable $\beta$. A small library of precomputed constellations enables scalable deployment for integrated mmWave WDM+OAM communication links in short-range LOS environments.

A plausible implication is that this methodology can be generalized to other high-dimensional, spatially structured, multi-carrier MIMO systems exhibiting sufficient channel symmetries, provided the channel gain structure admits a dominant parameterization analogous to the $\beta$ bands in OAM systems. The method is particularly potent where online adaptation is infeasible and where partition-based spatial uniformity of quality-of-service is required (Wang et al., 2021).