Two-Dimensional Direction-of-Arrival Estimation Using Stacked Intelligent Metasurfaces

• Conventional DOA methods face aperture-limited resolution, computational and storage costs, sensitivity to few snapshots, and degradation from practical array imperfections.Source: §I · pp. 2–3
• ML-based DOA methods can adapt to complex EM environments and array imperfections, but require complex training and broad datasets, especially in high-Doppler scenarios.Source: §I · p. 3
• Unlike fixed wave-based D2NNs, reconfigurable SIM layers can implement processing functions,.Source: §I · p. 4
• The paper introduces a SIM-based physical DOA estimator that manipulates wave propagation through adjustable multilayer meta-atoms to estimate a single source’s direction.Source: §I · p. 5
• The SIM performs a 2D DFT to map incident waves into angular-frequency bins, and the receiver probe with the greatest energy identifies the DOA.Source: §I · p. 5
• Propagation-based computation eliminates ADCs and adds no processing delay, simplifying receiver hardware compared with array signal processing.Source: §I · p. 5
• As waves pass through SIM layers, processing occurs faster than traditional digital calculations; a configured SIM maps radio waves to the angular domain, where each antenna corresponds to a signal direction.DOA is determined by measuring signal strength at each antenna.Source: §I · p. 6
• The authors minimize Frobenius-norm error between the ideal 2D DFT matrix and SIM response, then use gradient descent to update phase shifts.The optimization includes a constant transmission-level constraint for each meta-atom and is non-convex.Source: §I · p. 6
• Changing the zeroth-layer phase shifts across snapshots generates 2D DFT matrices with mutually orthogonal spatial-frequency bins, improving DOA accuracy at an observation-time trade-off.The SIM focuses energy onto the grid point matching the signal direction, producing the strongest magnitude there; outputs also provide coarse on-grid estimates with few probes.Source: §I · p. 6
• The estimator derives an upper bound on mean square error and uses low-complexity energy detectors to reduce hardware costs without compromising DOA accuracy.The receiver array directly observes the incident signal’s angular spectrum, unlike techniques relying on phase-sensitive receivers and array signal processing.Source: §I · p. 6
• The proposed SIM estimates DOA with an MSE of 10^-4.The introduction attributes this result to advanced wave-based computation.Source: §I · p. 7

II. SIM-BASED ARRAY SYSTEM MODEL

• The section introduces the system model for a SIM-based array used to perform DOA estimation.Source: §II · p. 8

A. Incident Signal Model

• The model places a UPA in the x-y plane and integrates it with an SIM of L + 1 metasurface layers to transform incident signals into an angular spectrum.Source: A. Incident Signal Model · p. 8
• The source DOA is specified by physical azimuth ϕ and elevation ϑ relative to the SIM's zeroth layer.Source: A. Incident Signal Model · p. 8
• The zeroth-layer steering vector is built from x- and y-direction vectors whose entries follow position-dependent phase progressions.Source: A. Incident Signal Model · pp. 8–9
• The transmitted signal is modeled as a zero-mean, unit-variance CSCG random variable, and the passage introduces the incident signal at the SIM's zeroth layer.Source: A. Incident Signal Model · p. 9

B. SIM Model

• The SIM’s overall response is the product of inter-layer propagation and layer transmission matrices, with the zeroth layer and receiver array arranged symmetrically.The intermediate layers are modeled as uniformly spaced UPAs, and each meta-atom controls the phase shift of the passing EM waves.Source: B. SIM Model · pp. 9, 11

C. Received Signal Model

• The modeled receiver is a UPA containing N = NxNy antennas, and the SIM’s transformation is presented as simplifying hardware and subsequent signal processing while reducing energy consumption.The receiver noise is modeled as a CSCG random vector with covariance IN.Source: C. Received Signal Model · p. 11
• Configuring the SIM’s phase shifts applies a controllable analog transformation so the receiver array can directly observe the incoming signal’s angular spectrum.The received signal model assumes one source, with uncorrelated signal and complex Gaussian measurement noise.Source: C. Received Signal Model · p. 11
• Wave propagation through the SIM performs the matrix operations at the speed of light, completing calculations in nanoseconds.The paper states that the inherent parallel processing makes calculation time independent of the number of meta-atoms.Source: C. Received Signal Model · pp. 11–12

A. Optimization Problem

• The optimization minimizes the Frobenius-norm fitting error between the SIM response and a 2D DFT matrix, subject to the layered propagation model and phase-shift constraints.A scaling factor normalizes the response for a fair error comparison; it is not applied to the received signal.Source: A. Optimization Problem · pp. 12–13

B. Proposed Gradient Descent Algorithm

• The phase shifts are highly coupled under a non-convex constant-modulus constraint, so the paper uses customized gradient descent to find a near-optimal solution.The algorithm iteratively adjusts the phase shifts to reduce the loss.Source: A. Optimization Problem · p. 13, B. Proposed Gradient Descent Algorithm · p. 13

1) Derivative Calculation:

• For each layer, the method calculates the loss gradient with respect to that layer’s phase-shift vector using an equivalent coefficient matrix and the illuminating signal component.The paper identifies these quantities as P_l,n and q_l,n, respectively.Source: 1) Derivative Calculation · pp. 13–14
• The derivative calculation stores and recursively updates an intermediate variable from the final layer backward, avoiding redundant multiplications when computing gradients for all layers.The passage describes this recursion as an efficiency measure.Source: 1) Derivative Calculation · p. 14

2) Parameter Update:

• The SIM’s phase shifts are updated in a loss-decreasing direction, with a learning rate that decays over time for stable convergence.The iteration stops when the fractional loss reduction falls below a threshold or the maximum iteration count is reached.Source: 2) Parameter Update · pp. 14–15
• The scaling factor β is recalculated by least squares from the tentative SIM response G and target matrix F to maintain the required normalized level.The vectors g and f are the vectorizations of G and F.Source: 2) Parameter Update · p. 14

C. The Choice for M

• A necessary condition for zero fitting loss is M ≥ N, because fewer meta-atoms per layer restrict the SIM response rank below the rank of the 2D DFT matrix.The theorem states the condition as necessary, not sufficient.Source: C. The Choice for M · p. 15
• The zeroth layer supplies an extra design degree of freedom for generating 2D DFT matrices with different frequency bins, potentially improving DOA accuracy with a moderate-size SIM.The paper notes that arbitrary-grid DFT implementation would require many meta-atoms and that a large probe count is impractical given space and cost limits.Source: C. The Choice for M · pp. 15–16

A. Estimation Protocol

• The protocol divides observation time T into Ty blocks of Tx slots, keeping the first-to-Lth layer phase shifts fixed while reconfiguring the zeroth layer across slots.This configuration generates a set of DFT matrices with mutually orthogonal spatial-frequency bins.Source: A. Estimation Protocol · pp. 16–17

B. SIM-Based DOA Estimator

• The SIM-based estimator uses energy across the receiver array to obtain DOA parameters, in contrast to conventional algorithms relying on phase-sensitive receivers and array signal processing.The SIM directly processes received RF signals without an individual RF chain and ADC at each antenna element.Source: B. SIM-Based DOA Estimator · pp. 17–18
• The estimator selects the strongest received-signal magnitude across antenna and snapshot indices, then uses the resulting peak index to estimate azimuth and elevation.The angle estimates are obtained from the corresponding electrical-angle estimates.Source: B. SIM-Based DOA Estimator · pp. 17–18
• Increasing the number of snapshots can improve estimation accuracy, but requires faster switching of the FPGA controlling the zeroth layer within a fixed period.The paper presents T as a flexible tradeoff between accuracy and the number of snapshots needed.Source: B. SIM-Based DOA Estimator · p. 18

V. PERFORMANCE ANALYSIS

• The paper derives analytical upper bounds for the MSEs of the estimated electrical angles, addressing errors that can arise when receiver noise causes the peak index to be identified incorrectly.The analysis evaluates electrical-angle MSE; azimuth and elevation MSE can be obtained by substitution.Source: §V · pp. 18–19

A. Ablation Study

• For fitting the 2D DFT matrix with (2, 2) grids, the optimal SIM configuration is (9λ, 7, 121, 2λ/4), achieving a normalized MSE of −208.78 dB.Source: A. Ablation Study · p. 21
• For the (4, 4)-grid fit, the optimal design is around (12λ, 13, 225, 4λ/9), with higher fitting NMSE than the (2, 2) case.The paper attributes the higher NMSE to the challenge of fitting a larger DFT matrix and says it has negligible effect on DOA performance.Source: A. Ablation Study · p. 21
• For fixed layer count, excessive SIM thickness can make inter-layer propagation singular, while a very thin SIM makes it nearly diagonal; too few layers may lack adequate degrees of freedom.Both extremes reduce the propagation diversity needed to fit the 2D DFT matrix.Source: A. Ablation Study · pp. 21–22
• Too few meta-atoms may violate Theorem 1, while excessive meta-atoms add unnecessary links; extreme element spacing can make propagation diagonal or rank one.The rank-one case cannot fit a rank-N 2D DFT matrix.Source: A. Ablation Study · p. 22

B. Convergence Behavior of the Proposed Gradient Descent Algorithm

• SIM fitting performance first improves and then degrades as ζ increases; about ζ = 0.80 achieves best performance under the worst-case condition, while at ζ = 0.75 loss may drop in under 50 iterations before getting stuck locally.Source: B. Convergence Behavior of the… · pp. 22–23
• For the (4, 4)-grid fit, ζ = 0.95 reaches NMSE below 10^-3 after about 200 iterations, while larger or smaller decay values worsen performance.Fitting the larger grid remains poorer than the (2, 2)-grid case despite using a larger SIM.Source: B. Convergence Behavior of the… · pp. 23–24

C. 2D DFT Capability of SIM

• The SIM produces nearly the same angular spectrum as digital 2D DFT and concentrates received signal at the true DOA, enabling estimates by locating the strongest received magnitude.The SIM’s computations occur as signals propagate through it, which the paper says reduces computing delay and simplifies hardware design.Source: C. 2D DFT Capability of SIM · pp. 24–25
• With a (4 × 4) receiver array, the spectrum has reduced leakage relative to the (2 × 2) case, supporting a more accurate noiseless DOA estimate.The SIM is reported to generate the same angular spectrum as the digital 2D DFT for the (4 × 4) array.Source: C. 2D DFT Capability of SIM · p. 25, Figure 7 · p. 27, Figure 6 · p. 27

D. Validation of Theoretical Analysis

• The analytical performance bound is an upper bound in both evaluated cases and becomes asymptotically tight at high SNR.The gap between analytical and simulated results narrows as SNR increases.Source: D. Validation of Theoretical Analysis · p. 26
• At effective SNR 10 dB, increasing snapshots from T = 16 to T = 64 reduces MSE from 1.5 × 10^-3 to 0.75 × 10^-3, a 3 dB improvement.Increasing effective SNR to 30 dB provides an additional 3 dB gain.Source: D. Validation of Theoretical Analysis · p. 26
• Doubling receiver-array aperture reduces MSE from 0.6 × 10^-2 to 1.3 × 10^-3, about a 6 dB gain.The reported comparison is between (2 × 2) and (4 × 4) receiver arrays.Source: D. Validation of Theoretical Analysis · p. 27

E. Performance Comparison with Conventional Beamforming Methods

• The proposed SIM-based estimator performs similarly well to the digital beamforming-based method under all setups, while relying on energy detection rather than phase-sensitive receivers and digital signal processing.Source: E. Performance Comparison with… · pp. 27–28
• A digital (32 × 32) array estimates DOA accurately across SNR regions but requires large aperture; SIM does so with a smaller array and a moderate number of snapshots.The (32 × 32) digital-array comparison is described as a lower bound for the SIM-based scheme.Source: E. Performance Comparison with… · p. 28

F. Effects of Receiver Array Arrangement

• For receiver-array spacing ux ≥ λ/2, seven SIM layers can accurately perform 2D DFT, whereas fewer layers cannot fit the matrix well.The corresponding (4, 4)-grid arrangement test also reports accurate fitting with 13 layers when ux ≥ λ/2.Source: F. Effects of Receiver Array Arrangement · p. 29
• A small number of meta-atoms weakens inference; with M = 121 per layer, fitting is insensitive to receiver-array rotation, though robustness may worsen near ω = 45°.The paper identifies ux = uy = λ/2 and ω = 0° as effective previous settings.Source: F. Effects of Receiver Array Arrangement · p. 29
• With adequate meta-atoms such as M = 225 per layer, fitting becomes less sensitive to receiver-array rotation, supporting an isomorphic arrangement between the array and SIM input layer.A small meta-atom count is associated with poor inference capability.Source: F. Effects of Receiver Array Arrangement · p. 29

VII. CONCLUSIONS

• The proposed SIM architecture targets 2D DOA estimation and is presented as improving computation speed and hardware complexity.The conclusion describes the estimator in a single-radiation-source setting.Source: §VII · p. 31
• As waves propagate through the SIM, they are transformed into the angular domain; orthogonal spatial-frequency bins support peak-based DOA estimation.The estimate is obtained by searching for the antenna and snapshot with the strongest magnitude.Source: §VII · p. 31
• For the (2, 2)-grid, a 9λ-thick, 7-layer SIM with 121 λ/2-spaced meta-atoms per layer minimizes the 2D DFT fitting loss.For the (4, 4)-grid, the reported design is 12λ thick, with 13 layers of 225 meta-atoms spaced 4λ/9 apart; the reported decay parameters are ζ = 0.8 and ζ = 0.95 for the (2, 2) and (4, 4) grids, respectively.Source: §VII · p. 31
• A well-trained SIM generates the incoming signal’s angular spectrum and provides a DOA estimate with an MSE of 10^-4 under moderate conditions.The paper also reports theoretical evaluation and numerical verification of the estimator’s MSE.Source: §VII · p. 31
• Open directions include super-resolution and wave-domain compressed sensing, multiple-source scenarios, and discrete-phase optimization under more realistic response models.The authors also describe amplifier-integrated, nonlinear SIMs as a possible route to DNN-like processing for tasks such as near-field positioning.Source: §VII · pp. 31–32
• The SIM’s inverse system is proposed for angular division multiplexing, with a SIM precoder sending each user’s signal directly from its corresponding antenna.The authors state this would substantially simplify wireless-communications hardware design.Source: §VII · p. 32

APPENDIX A

• The gradient of the fitting loss with respect to each layer’s phase-shift vector is assembled from its meta-atom partial derivatives using the chain rule.The appendix gathers the partial derivatives into a vector to obtain the gradient used in the derivation.Source: Appendix A · pp. 32–33

APPENDIX B

• The bound derivation approximates the relevant noncentral chi-squared distributions using a three-moment approximation and the Wilson-Hilferty transformation.The three-moment approximation is used when μ3 > 0; otherwise, the appendix applies it to the distribution of −X.Source: Appendix B · pp. 33–34
• The MSE bound depends on the probability that the corresponding index is detected as the peak of the received energy.The appendix defines the peak-detection event through a comparison of received magnitudes.Source: Appendix B · p. 34