PDNN-SSK: Planar Diffractive RF Modulation

• PDNN-SSK is a spatial modulation architecture that uses cascaded planar diffractive neural networks to perform analog RF processing via engineered phase shifts and controlled diffraction.
• It leverages a single RF chain with maximum power detection and surrogate-model training for end-to-end optimization of beamforming, modulation, and detection in a planar configuration.
• Simulation results and theoretical analysis demonstrate competitive SER performance, energy efficiency gains, and scalability for broadband and multi-user RF applications.

Planar Diffractive Neural Network-Based Space-Shift-Keying (PDNN-SSK) denotes a spatial modulation architecture in which signal processing is physically realized through wave propagation in cascaded planar RF circuits, specifically planar diffractive neural networks (PDNNs). PDNN-SSK leverages the spatial degrees of freedom available in integrated RF hardware, enabling a communication system architecture with a single radio-frequency (RF) chain and maximum power detection, supporting modulation, beamforming, and detection entirely in the analog RF domain. Unlike the three-dimensional @@@@1@@@@ (SIMs) conventionally explored for diffractive RF computation, PDNN-SSK employs a two-dimensional planar configuration amenable to integration and low-loss implementation using microstrip technology (Teng et al., 30 Nov 2025).

1. PDNN-SSK System Architecture

A PDNN consists of $L$ cascaded planar layers, each comprising a fixed coupling module (transfer matrix $\mathbf W^{(l)}$) that emulates diffraction through structures such as branch-line microstrip couplers, and a trainable diagonal phase-shifting module ($$\boldsymbol\Phi^{(l)}=\mathrm{diag}\{e^{j\beta_n^{(l)}}\}$$) with electronically tunable phase taps. Nonlinear activation $f^{(l)}(\cdot)$ is generally absent, except at the receiver's (RX) final layer, where it is realized via a power detector that outputs the magnitude squared of the signal.

System operation proceeds as follows:

• Transmitter PDNN (TX-PDNN): The transmitter emits from a single RF chain. Space-shift-keying (SSK) is implemented by selecting one of $M=2^p$ output ports, corresponding to a one-hot-modulated transmit vector $\mathbf s\in\{0,1\}^M$. The TX-PDNN receives $\mathbf s$ at its input layer and, through $L_\mathrm T$ cascaded (fixed and trainable) modules, outputs a spatial field distribution across its $M$ output ports.
• Receiver PDNN (RX-PDNN): At the receiver, incident signals from a receive antenna array feed into the RX-PDNN, which consists of $L_\mathrm R$ cascaded layers (with module order reversed for compatiblity with noncoherent detection). The RX-PDNN's $M$ output ports feed into power detectors, generating the real-valued output vector $\mathbf y\in\mathbb R^M$. Detection employs a maximum power strategy: $\hat m=\arg\max_m y_m$.

Mathematically, the operation of each layer is

$$\mathbf x^{(l)}=f^{(l)}\bigl(\boldsymbol\Phi^{(l)}\,\mathbf W^{(l)}\,\mathbf x^{(l-1)}\bigr)$$

and, in the absence of inter-layer nonlinearities,

$$\mathbf x^{(L)}=\boldsymbol\Phi^{(L)}\mathbf W^{(L)}\cdots\boldsymbol\Phi^{(1)}\mathbf W^{(1)}\,\mathbf x^{(0)}.$$

The overall system architecture enables end-to-end analog-domain spatial processing of modulated RF signals.

2. End-to-End Channel and Signal Model

The end-to-end PDNN-SSK system is represented by the transfer matrices: $$\mathbf F_\mathrm T = \prod_{l=1}^{L_T} \bigl( \boldsymbol\Phi^{(l)}_\mathrm T \mathbf W^{(l)}_\mathrm T \bigr ), \quad \mathbf F_\mathrm R = \prod_{l=1}^{L_R} \bigl( \mathbf W^{(l)}_\mathrm R \boldsymbol\Phi^{(l)}_\mathrm R \bigr).$$ The physical MIMO channel is denoted $$\mathbf H \in \mathbb C^{N^{(0)}_\mathrm R \times N^{(L_\mathrm T)}_\mathrm T}$$ and is modeled as Rayleigh fading ($\mathcal{CN}(0,1)$).

The received RF signal, prior to power detection, is

$$\mathbf r = \mathbf F_\mathrm R \, \mathbf H \, \mathbf F_\mathrm T \, \mathbf s + \mathbf n, \quad \mathbf n\sim\mathcal{CN}(\mathbf 0, 2\sigma^2 \mathbf I).$$

After noncoherent power detection,

$$\mathbf y = \bigl| \mathbf F_\mathrm R \, \mathbf H \, \mathbf F_\mathrm T \, \mathbf s + \mathbf n \bigr|^2.$$

Detection is performed by selecting the index $m$ for which $y_m$ is maximized.

3. Theoretical Performance Analysis

3.1 Conditional Correct Detection Probability

Effective channel gains are defined as

$$c_{m', m} = \mathbf f_{\mathrm R, m'}^T\,\mathbf H\,\mathbf F_\mathrm T\,\mathbf s_m, \quad y_{m'} = c_{m', m} + n_{m'}.$$

For a transmission corresponding to symbol $m$, the probability of correct detection (CCDP) is

$$P_{c,m} = P\Bigl( \bigcap_{m'\neq m} \{ |y_m| > |y_{m'}| \} \,\Bigm|\, \mathbf s_m,\, \mathbf H \Bigr ).$$

This probability is expressed, via independence and Rician statistics, as

$$P_{c,m} = \int_0^\infty f_{X_m}(r; |c_{m,m}|) \prod_{m'\neq m} F_{X_{m'}}(r; |c_{m',m}|) \, dr,$$

where

$$f_X(r;\mu) = \frac{r}{\sigma^2} \exp\left(-\frac{r^2+\mu^2}{2\sigma^2}\right) I_0\left(\frac{r\mu}{\sigma^2}\right)$$

and $F_X(r;\mu) = 1 - Q_1(\mu/\sigma, r/\sigma)$.

3.2 Maximization Condition

It is established that $P_{c,m}$ is strictly increasing in $|c_{m,m}|$ and strictly decreasing in each $|c_{m',m}|$ for $m'\neq m$. The unique global optimum is attained when all interfering gains are nulled: $c_{m', m} = 0 \quad \forall\, m'\neq m,$ i.e., perfect spatial separation or orthogonalization of output ports.

3.3 Closed-Form Symbol Error Rate (SER)

Under the optimal nulling condition, the closed-form SER is

$$\mathrm{SER}_m = 1 - P_{c,m}^* = \sum_{k=1}^{M-1} \binom{M-1}{k} \frac{(-1)^{k+1}}{k+1} \exp\left(-\frac{\gamma_m k}{k+1}\right),$$

with $\gamma_m = \frac{|c_{m,m}|^2}{2\sigma^2}$.

The high-SNR regime yields the approximation

$$\mathrm{SER}_m \approx \frac{M-1}{2} \exp\left(-\frac{\gamma_m}{2}\right).$$

4. Surrogate Model-Based Training of PDNN Phase Taps

Optimization of the continuous phase shift variables $(\beta_{n, T}^{(l)}, \beta_{n, R}^{(l)})$ is performed via a differentiable surrogate model approach. The entire PDNN-SSK chain (TX-PDNN, channel $\mathbf H$, RX-PDNN) is represented as a computational graph, enabling end-to-end gradient-based optimization.

The optimization objective is the maximization of the aggregate sum rate: $$\max_{\{\boldsymbol\Phi_\mathrm T^{(l)},\,\boldsymbol\Phi_\mathrm R^{(l)}\}} \sum_{m=1}^M \log_2\bigl(1 + \mathrm{SINR}_m\bigr),$$ subject to $|\Phi_{nn}|=1$, where

$$\mathrm{SINR}_m = \frac{|c_{m,m}|^2}{\sum_{m'\neq m} |c_{m',m}|^2 + 2\sigma^2}.$$

The negative sum-rate $\mathcal J=-\sum_{m=1}^M \ln(1+\mathrm{SINR}_m)$ is minimized via backpropagation and the Adam optimizer over $10^3$–$10^4$ epochs. Convergence is typically achieved in a few hundred epochs, and this approach proves robust against local minima and superior to projected gradient or random-phase training benchmarks.

5. Simulation Results and Architectural Trade-Offs

Comprehensive simulation studies validate the theoretical analysis and SER expressions.

• With $L_T=L_R=2$, $M=4$, and $N=8$, phase-optimized PDNNs focus a single-port excitation into a spatially structured output, which is correctly refocused by the RX-PDNN.
• Simulated SER matches the closed-form analysis for $M=4, 16, 64$ in the ideal regime; the asymptotic formula is highly accurate for $\mathrm{SER}<10^{-2}$.
• Energy efficiency is observed to improve with increased $M$: to achieve $\mathrm{SER}=10^{-3}$, SNR requirements are 8.5 dB for $M=4$ and 5 dB for $M=64$.
• Structural performance is influenced collectively by network depth ($L$), width ($N$), and coupling strength ($M_c$). Shallow or sparsely coupled networks exhibit early performance saturation, while deeper and more strongly coupled PDNNs benefit substantially from increased width.
• Asymmetry between TX and RX depths incurs negligible performance penalty, permitting greater hardware flexibility.
• Compared to free-space SIM-based architectures, coupler-based PDNNs avoid millimeter-wave propagation loss by confining energy to low-loss microstrip transmission lines. With sufficient cascaded couplers ($M_c\ge 3$), these PDNNs match or outperform SIM in terms of spatial interconnectivity and separation.

The table below summarizes comparative aspects:

Attribute	PDNN-SSK	SIM-Based SSK
Propagation Loss	Low (microstrip-confined)	Potentially high
Interconnectivity	Tunable via coupling	Complete (free-space)
Fabrication	PCB/IC-compatible planar	3D stacking required
Training	Surrogate gradient/model	Not usually end-to-end

6. Practical Design Considerations

Physical realization of PDNNs involves a choice between discrete (delay-line plus RF switch) and continuous (varactor) phase shifting, each presenting trade-offs in resolution and power consumption. The planar geometry of PDNNs eases layer registration and facilitates integration with standard PCB or IC manufacturing processes, with electromagnetic interactions well-characterized by transmission line theory.

Nonlinearity is present only at the receiver's final stage via power detection, but in principle, further nonlinear functions could be implemented inside the network for extended classification or regression tasks. Extension to broadband communications is anticipated to require multi-tone or dispersion-compensated couplers. Multi-user, MIMO, and ISAC scenarios can be addressed by expanding the optimization framework to handle multiple transmitted symbol vectors and corresponding interference.

7. Connections, Performance Benchmarks, and Future Scope

PDNN-SSK achieves performance similar to noncoherent $M$-FSK when spatial channels are orthogonalized. The SNR penalty for noncoherent (NC) detection, compared to ML-coherent detection, ranges from 1 to 3 dB, decreasing at high SNR. As $M$ increases, PDNN-SSK demonstrates superior scaling properties compared with $M$-QAM spatial modulation.

Rapid, monotonic convergence to high sum-rate is consistently reported using surrogate-model Adam optimization, unlike projected gradient or random-phase approaches that result in convergence plateaus and persistent error floors.

Planar diffractive spatial modulation architectures present a pathway to highly integrable, energy-efficient RF front ends capable of replacing traditional digital baseband modules. The modularity and adaptability of the PDNN-SSK scheme render it extensible for broadband, multi-user, and MIMO scenarios, contingent upon advances in multi-tone circuit design and large-scale gradient-driven optimization (Teng et al., 30 Nov 2025).