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Bounds on Shaping Partially Coherent Microwaves with Programmable Scattering Systems

Philipp del Hougne

arXiv:2608.30729v1physics.app-pheess.SP

TL;DR

The paper addresses how programmable scattering systems can transform partially coherent microwaves despite prototype-level hardware constraints. It combines multiport network theory with semidefinite relaxation and derives architecture-independent, scalar-refined, and prototype-aware bounds for concentration and coherency synthesis. Across measured RIS-parametrized systems, the bounds are generally tight against feasible discrete optimization, while different bounding strategies remain complementary.

  • Problem

    Existing programmable-wave results largely assume coherent excitation, while practical prototypes impose constraints that make arbitrary transfer functions and coherency transformations unattainable.

  • Method

    The paper combines multiport network theory with semidefinite relaxation to derive architecture-independent and prototype-aware bounds for power concentration and coherency-matrix synthesis.

  • Results

    The bounds are generally tight compared with feasible discrete-optimization outcomes, and simpler coherency-synthesis bounds can sometimes be tighter than fully prototype-aware bounds.

  • Takeaways & Limitations

    The bounds certify near-optimality or infeasibility and distinguish algorithmic limitations from genuine hardware limitations in wave-domain processing and RF energy harvesting.

Abstract

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We derive bounds on manipulating partially coherent microwaves with programmable scattering systems. We first consider the concentration of power from a partially coherent input into a single output port and derive a prototype-aware bound by combining multiport network theory (MNT) with semidefinite relaxation (SDR). We then address general coherency-matrix synthesis, deriving architecture-independent bounds on fidelity and useful strength, prototype-specific scalar refinements thereof, and fully prototype-aware SDR bounds on the useful-strength--fidelity Pareto frontier. The prototype-aware formulations account for mutual coupling, loss, discrete tunability, and static scattering. We evaluate the bounds on four experimental RIS-parametrized MIMO systems with up to 100 1-bit-programmable elements, using proxy-MNT models estimated from measurements. The resulting bounds are generally tight compared with feasible discrete-optimization outcomes. Interestingly, for coherency synthesis, the simpler bounds can in some cases be tighter than the fully prototype-aware bounds, highlighting their complementarity. Our results provide certified limits for wave-domain processing and harvesting of partially coherent microwaves.

I. INTRODUCTION

The paper develops certified limits for programmable wave systems driven by partially coherent waves, addressing concentration and coherency transformation under concrete hardware constraints.

  • Motivation: Partially coherent excitation is important because coherence determines interference stability, while programmable-wave research has overwhelmingly focused on coherent excitation.Spatial coherence quantifies correlations among field components at distinct points or channels.
  • Motivation: Practical prototypes cannot realize arbitrary transfer functions because tunability, attenuation, static scattering, and mutual coupling constrain their responses.These effects are incorporated electromagnetically through multiport network theory, which models tunable components as virtual ports with tunable loads.
  • Why bounds matter: Prototype-aware bounds can certify near-optimality or infeasibility by distinguishing algorithmic suboptimality from genuine hardware limitations.They can guide whether to improve optimization, add tunable elements, change hardware, or relax the target transformation.
  • Related work: Existing coherency-statistics studies combine multiport models with second-order field statistics but do not provide bounds on coherency transformations.Prior prototype-aware SDR bounds primarily addressed coherently excited systems and related gain or synthesis metrics.
  • Contributions: The paper derives architecture-independent and prototype-aware bounds for concentrating partially coherent power and synthesizing coherency matrices.The prototype-aware formulations retain the feasibility set of a specific programmable prototype.

III. CONCENTRATION BOUND

The concentration problem asks how much power a concrete reciprocal programmable system can direct into one output port from a partially coherent input.

  • Problem formulation: Single-port concentration models collecting power from a multi-input system into one output, a setting relevant to spatially incoherent or partially coherent waves.The formulation uses NT > 1 input ports and NR = 1 output port.
  • Architecture-independent benchmark: The architecture-independent bound equals the largest eigenvalue of the input coherency matrix but ignores losses and the restrictions of concrete prototypes.Experimental systems generally cannot realize arbitrary unitary transformations.
  • Prototype-aware formulation: The prototype-aware objective maximizes concentrated power over binary tunable-load states using the sole row h(r) of the end-to-end transmission matrix.Each load satisfies ri ∈ {α, β}.
  • Reciprocity: Reciprocity converts the multiple-input single-output problem into a reciprocal single-input multiple-output problem, reducing the auxiliary variable from an NS × NT matrix to an NS-element vector.This avoids repetition constraints needed in the direct MIMO treatment.

B. QCQP Formulation

The concentration optimization is reformulated as a QCQP in an auxiliary vector whose quadratic objective and binary-programmability constraints encode feasible load states.

  • Objective: The end-to-end transmission expression is specialized to one output and rewritten as a quadratic objective in the auxiliary variable y.The resulting formulation is the QCQP representation of the concentration problem.
  • Binary constraints: Feasible binary loads are enforced through per-element quadratic constraints constructed from the rows of the internal coupling matrix and the two load states.The matrices and vectors defining these constraints are introduced element by element.
  • Complex constraints: Because the constraint matrices can be non-Hermitian, the complex equalities are imposed separately through their real and imaginary parts.This produces the stated QCQP constraints for every tunable element.

C. SDR Bound

The QCQP is lifted into a semidefinite relaxation whose feasible set contains every binary configuration, so its optimum certifies an upper bound on achievable concentration.

  • Lifted formulation: Lifting defines Y = yy† and converts quadratic terms into trace expressions such as tr(RiY).This makes the quadratic dependence affine in the lifted variables.
  • Semidefinite relaxation: The nonconvex rank-one equality is relaxed to Y ⪰ yy†, yielding a convex semidefinite program solvable with standard SDP solvers.The complex equality constraints remain separately imposed for their real and imaginary parts.
  • Certification: Every feasible binary configuration maps to a rank-one feasible SDP point, while higher-rank points are also permitted by the relaxation.Therefore the SDP optimum upper-bounds the achievable concentration.
  • Additional bounds: The resulting SDR bound complements a direct concentration SDR and can be combined with passivity to obtain further aggregate-gain bounds.Within the reciprocal proxy-MNT representation, the SDR bounds are insensitive to reciprocity-preserving internal-coordinate ambiguities.

IV. COHERENCY-SYNTHESIS BOUNDS

The paper frames coherency synthesis around fidelity to a target structure and useful output strength, deriving architecture-independent and prototype-aware bounds. Rank limits constrain fidelity, passivity constrains useful strength, and prototype power ceilings refine the latter.

  • Metrics: Fidelity measures structural agreement with the target coherency matrix while remaining insensitive to overall output-power scale.Useful strength instead weights total output power by fidelity, discounting power with a mismatched coherency structure.
  • Architecture-independent bounds: Exact nonzero synthesis of an output proportional to the target requires rank(ρ⋆) ≤rank(Cin).Deterministic linear transformations cannot increase coherency-matrix rank.
  • Architecture-independent bounds: The architecture-independent fidelity bound depends on the output-support rank and the dominant eigenvalues of the target coherency matrix.It applies irrespective of useful strength.
  • Architecture-independent bounds: Passivity yields an architecture-independent useful-strength bound based on the eigenvalues of the input and target coherency matrices.The bound is independent of output fidelity and reduces to the scattering-concentration bound for a rank-one target.
  • Prototype-aware refinements: A prototype-specific scalar output-power ceiling tightens the architecture-independent useful-strength bound without retaining the detailed response geometry.Fully prototype-aware bounds additionally account for how feasible transmission responses relate to the input and target coherency matrices.

C. Coherent-Mode Representation

The formulation represents the partially coherent input in its coherent-mode basis, retaining only the excited modes and exploiting diagonal modal statistics. This reduces variables for rank-deficient inputs and removes cross terms between distinct modes.

  • Modal basis: The input coherency matrix is represented using K = rank(Cin) nonzero coherent modes and their eigenvalues.The eigenvectors form mutually orthogonal modes with mutually uncorrelated modal amplitudes, while eigenvalues give average modal powers.
  • Modal basis: Diagonal modal statistics express the output coherency matrix as a sum of modal output coherencies without cross terms between distinct modes.This is one benefit of working in the coherent-mode basis rather than the physical input-port basis.
  • Modal basis: Only the K excited modes need auxiliary variables, reducing the formulation whenever Cin is rank deficient.The kth column xk represents the auxiliary vector-valued response associated with the kth coherent-mode excitation.
  • MNT representation: The modal responses are connected to the MNT model through He ≜ H0L and Be ≜ BL.These matrices encode the effective responses of the power-weighted coherent modes.

D. Prototype-Feasibility Constraints

Prototype feasibility is enforced by binary programmability and repetition constraints requiring all coherent modes to use one common physical load state. The resulting nonconvex formulation is lifted and relaxed into an SDP.

  • Repetition constraints: Per-mode binary constraints alone are insufficient when K > 1 because different modes could select different states for the same tunable element.The physical system instead supplies all modal responses through one common load vector.
  • Repetition constraints: A compact single-reference formulation couples each mode to a reference mode using 2NS(K −1) complex equalities.It enforces common load states when the reference mode excites the element, but can fail when both reference auxiliary quantities vanish.
  • Repetition constraints: The all-pairs formulation imposes NSK(K −1) complex equalities and avoids dependence on a particular reference mode.For K ≥2, it uses K/2 times as many constraints as the single-reference formulation.
  • Optimization objectives: The four optimization problems separately bound useful strength, fidelity, and the useful-strength–fidelity Pareto frontier.Threshold sweeps reveal tradeoffs that single-objective limits cannot expose.
  • SDR formulation: Lifting replaces the nonconvex rank-one equality Z = xx† with a Schur-complement relaxation while preserving affine coherency-matrix expressions.The fidelity metrics are represented through semidefinite formulations.

F. Complementary Pareto-Frontier Bounds √

Complementary SDPs parameterize the relaxed useful-strength–fidelity frontier from either a fidelity threshold or a useful-strength threshold. Charnes–Cooper removes the fractional normalization efficiently for the latter direction.

  • Fidelity-threshold formulation: USDR(Fmin) upper-bounds useful strength achievable with fidelity F ≥Fmin, and sweeping Fmin parameterizes one relaxed Pareto frontier.USDR(0) gives the useful-strength-only bound.
  • Useful-strength-threshold formulation: The complementary fidelity-constrained problem is fractional because fidelity normalizes by total output power.The Charnes–Cooper transformation removes this normalization.
  • Useful-strength-threshold formulation: The auxiliary total-output-power bound is solved once per scenario before the Pareto sweep and upper-bounds output power for both repetition formulations.The all-pairs feasible set is a subset of the single-reference feasible set.
  • Useful-strength-threshold formulation: FSDR(Umin) upper-bounds fidelity achievable subject to useful strength U ≥Umin.Setting Umin = 0 yields the fidelity-only bound.
  • Computational procedure: The direct Charnes–Cooper formulation requires one SDP per useful-strength threshold after one reusable solve, whereas bisection requires multiple SDP solves per threshold.This makes the direct formulation computationally more efficient.

G. Validity and Ambiguity Insensitivity

The SDR bounds upper-bound the corresponding binary optimization objectives, while alternative formulations differ in how they enforce common-load constraints. The relaxed Pareto frontier is invariant to reciprocity-preserving proxy-MNT representations.

  • Validity: Every admissible binary configuration produces a feasible rank-one lifted point, so the SDR objectives upper-bound the corresponding discrete optima.The SDR additionally permits higher-rank matrices through rank relaxation.
  • Validity: The all-pairs formulation avoids the reference-mode dependence that can affect the single-reference formulation when a reference mode fails to excite an element.
  • Validity: The four relaxed quantities bound the optimal objectives for useful strength and fidelity at the specified constraint levels.
  • Ambiguity Insensitivity: The complete relaxed useful-strength–fidelity Pareto frontier is independent of the chosen reciprocity-preserving proxy-MNT representation.Invertible affine and congruence transformations preserve lifted feasibility, physical output coherence, useful strength, and fidelity constraints.

V. DISCRETE-OPTIMIZATION BENCHMARKS

The study benchmarks SDR-derived bounds against exhaustive search and scalable discrete optimizers, using several procedures to obtain feasible binary configurations. Exhaustive search certifies global binary optimality for small systems, while the scalable methods provide practical comparisons at larger sizes.

  • Coherency Synthesis: For coherency synthesis, coordinate ascent and the genetic algorithm maximize a weighted combination of normalized useful strength and fidelity over 25 prescribed weights.
  • Exhaustive Search: Exhaustive search enumerates all 2^NS binary configurations for sufficiently small NS and is the only benchmark certifying global binary optimality.Gray-code ordering and Woodbury updates improve enumeration efficiency.
  • Scalable Optimizers: Coordinate ascent tests single-load flips from multiple initializations, accepting only objective-improving changes in the non-convex binary problem.
  • Scalable Optimizers: The genetic algorithm evolves binary populations through selection, crossover, mutation, and elitism, then polishes its best outcome with coordinate ascent.
  • SDR Extraction: P-SDR projects the relaxed SDR optimizer onto the two admissible binary load states to produce a feasible candidate design.The projection uses the MNT fixed-point relation and its matrix-valued analogue.
  • SDR Extraction: DE-SDR extracts a candidate from the dominant eigenvector of the augmented lifted optimizer before applying binary projection, while RP-SDR promotes low rank with a linearized penalty.

A. Experimental Setups

The experiments use four RIS-parametrized MIMO systems measured at 2.45 GHz across distinct propagation environments, with proxy-MNT models and varying programmable-element counts. Concentration bounds are compared with discrete outcomes for fully coherent, partially coherent, and incoherent inputs, and are generally tight while reflecting strong prototype restrictions.

  • Experimental Setups: The four prototypes combine an RIS, transmit array, and receive array at 2.45 GHz, with propagation conditions varied to change RIS-element coupling.
  • Experimental Setups: The environments comprise a strongly scattering reverberation chamber, an attenuated chamber, a scattering-object anechoic chamber, and an anechoic chamber without those objects.
  • Experimental Setups: Proxy-MNT models are inferred because direct MNT measurement and full-wave simulation are impractical for the 108-port system with 100 RIS elements.
  • Concentration Evaluation: The concentration study uses NT = 4 input ports, NR = 1 output port, output port 1, and NS values from 4 through 100.
  • Concentration Results: At NS = 100, prototype-aware bounds reach only 0.01%–1.03% of the architecture-independent PSCB, reflecting restrictions imposed by the experimental prototypes.
  • Concentration Results: For FC and PC inputs, the transmission-scaled SCB is looser than direct SDR, with scaled-to-direct ratios of 2.19–9.51 and 1.47–2.87, respectively, at NS = 100.
  • Concentration Results: Across 132 plotted points, the certification gap is at most 1% in 101 cases and 3% in 115 cases, with the least favorable discrete result attaining 80.89% of the SDR bound.

C. Synthesis Pareto-Frontier Results

The synthesis results compare architecture-independent, scalar-refined, and prototype-aware bounds with feasible outcomes across four environments and three transformations. The bounds are often tight, but their relative restrictiveness varies by transformation and environment.

  • Evaluation setup: The evaluated coherency-synthesis bounds and discrete outcomes are organized across four radio environments and the transformations defined in (58).The SDR bounds use single-reference formulations for non-fully-coherent inputs; all-pairs spot checks tightened the results by at most 6.4%.
  • Fidelity bounds: The rank-only fidelity bound is tighter than all prototype-aware F-only bounds for FC →IC, while the comparison reverses for IC →FC.For PC →IC, the rank-only ceiling is tighter in the first three environments, whereas the prototype-aware value 0.63 is tighter without environmental scattering.
  • Useful-strength bounds: 8.73 × 10^-4 to 2.05 × 10^-2 are the SDR-certified transmission factors used to tighten useful-strength bounds through transmission scaling.The resulting USMB improves on the direct prototype-aware U-only SDR in two of twelve cases; for FC→IC, reductions are 36.8% and 37.9% in two environments.
  • Useful-strength bounds: 2.38 × 10^-5 to 8.10 × 10^-3 is the range of U-only useful-strength values across the displayed panels, showing strong environmental dependence.The unscaled architecture-independent useful-strength bounds are 0.25, 0.65, and 0.25 for the three transformations and remain loose for the lossy prototypes.
  • Feasible frontiers: 288 of 300 matched scalarizations yield identical (F, U) pairs for CD and GA, while P-SDR tracks the realized frontier closely but can fall 8.8% below the best discrete useful strength.The largest CD–GA differences over the remaining pairs are 9.9×10^-4 in fidelity and 8.0×10^-5 in useful strength.
  • Fidelity bounds: 0.25 is the architecture-independent fidelity ceiling for FC →IC, and optimized configurations essentially attain it in all four environments.This ceiling follows because deterministic linear transformations cannot increase a coherent input’s rank beyond one.
  • Bound complementarity: The fully prototype-aware SDR is not uniformly strongest: simpler spectral or transmission-scaled bounds can be more restrictive in parts of the Pareto plane.The generally small gaps between the tightest bound and best feasible performance support distinguishing algorithmic limitations from hardware limitations.
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