Pinching-Antenna-Assisted Integrated AirComp-NOMA: Joint Transceiver and Antenna-Position Optimization

I. INTRODUCTION

• AirComp signals affect NOMA decoding conditions, while NOMA QoS requirements constrain resources for accurate computation, creating a coupled mixed continuous-discrete design problem.Source: §I · pp. 1–2
• The paper jointly optimizes the AirComp transceiver, NOMA transmit coefficients, PA positions, and SIC order to minimize AirComp MSE under QoS, power, and deployment constraints.Source: §I · p. 2
• An exact NOMA feasibility condition and optimal SIC-ordering rule eliminate NOMA power variables and decoding-order search, reducing the design to continuous optimization.Source: §I · p. 2
• The proposed iterative algorithm combines optimal block-wise AirComp transceiver updates with feasibility-preserving majorization for PA placement and guarantees a nonincreasing, convergent MSE sequence.Source: §I · p. 2
• Numerical results report AirComp-MSE improvements over benchmark schemes and quantify the effects of NOMA QoS, PA deployment, and waveguide attenuation on computation accuracy.The introduction does not provide numerical values for these comparisons or effects.Source: §I · p. 2

A. Network Architecture

• The BS uses a PAS to serve computation-oriented AirComp users and communication-oriented NOMA users over shared time-frequency resources; user locations are assumed known.Source: A. Network Architecture · pp. 2–3

B. PAS-Aided Uplink Channel Model

• The channel model assumes LoS-dominant user-to-PA propagation and combines PA contributions coherently at the feed point after guided-wave attenuation and phase accumulation.Source: B. PAS-Aided Uplink Channel Model · p. 3

• AirComp users transmit preprocessed symbols and NOMA users transmit information-bearing symbols; their symbols are mutually independent, zero-mean, and unit-power.Source: C. Uplink Signal Model · p. 3
• Each AirComp and NOMA user applies a complex transmit coefficient subject to an individual maximum-power constraint.Source: C. Uplink Signal Model · p. 3
• The PAS feed-point receiver observes the aggregate baseband signal, with effective additive Gaussian noise of power σ^2.Source: C. Uplink Signal Model · p. 3

D. NOMA Reception and Achievable Rates

• The base station decodes NOMA users in SIC order, canceling already decoded users while AirComp signals remain interference.Under ideal SIC, undecoded NOMA users also remain as intra-NOMA interference.Source: D. NOMA Reception and Achievable Rates · pp. 3–4

E. AirComp Reception and Computation MSE

• After NOMA cancellation, the base station scales the residual signal to estimate the sum of AirComp users’ data symbols.The computation accuracy is measured by the MSE between the estimate and desired sum.Source: E. AirComp Reception and Computation MSE · p. 4

F. Problem Formulation

• The joint design minimizes AirComp MSE while satisfying NOMA QoS, individual transmit-power budgets, and PA-position constraints.The optimization jointly selects transmit coefficients, receive scaling, PA positions, and SIC order.Source: F. Problem Formulation · p. 4
• The formulation is challenging because SIC order is discrete and the MSE and NOMA QoS constraints depend nonlinearly on position-dependent channels.The variables are coupled across the AirComp transceiver and PA positions.Source: F. Problem Formulation · p. 4

III. PROPOSED OPTIMIZATION FRAMEWORK

• The framework removes NOMA power variables and the decoding-order search, reducing the joint design to optimization over continuous variables.It first eliminates powers for a given order, then characterizes the order maximizing admissible interference-plus-noise.Source: §III · p. 4

A. NOMA Feasibility Characterization and Power Elimination

• For fixed PA positions, AirComp coefficients, and SIC order, NOMA power feasibility is characterized by an exact necessary-and-sufficient condition.The condition determines whether some NOMA allocation can satisfy QoS and individual power limits.Source: A. NOMA Feasibility Characterization… · p. 4
• When the feasibility condition holds, the componentwise minimum NOMA power allocation is available, allowing NOMA powers to be eliminated from the joint optimization.The resulting QoS and power constraints are represented by an upper bound on aggregate AirComp interference plus noise.Source: A. NOMA Feasibility Characterization… · pp. 4–5

B. Optimal SIC Decoding Order and Problem Reduction

• An ordering metric identifies an SIC order that maximizes NOMA interference tolerance for any fixed PA-position vector.The paper states that this avoids enumerating all KN! permutations.Source: B. Optimal SIC Decoding Order and… · p. 5
• The optimal tolerance is obtained by maximizing the order-dependent tolerance over SIC orders, with the maximizing order given by the ordering rule.For fixed positions, the objective does not depend on SIC order; the order affects feasibility through the tolerance.Source: B. Optimal SIC Decoding Order and… · p. 5
• The reduced problem is equivalent to the original in minimum achievable AirComp MSE, and the optimal order and minimum NOMA powers can be recovered afterward.The order follows from the rule at optimized PA positions, and the powers follow from the minimum-power allocation.Source: B. Optimal SIC Decoding Order and… · p. 5

C. Optimal Receive Scaling Coefficient

• For fixed AirComp transmit coefficients and PA positions, the receive-scaling subproblem is strictly convex and has a unique optimum.Strict convexity follows from positive noise variance.Source: C. Optimal Receive Scaling Coefficient · p. 5

D. Optimal AirComp Transmit Coefficients

• For fixed receive scaling and PA positions, transmit coefficients are phase-aligned with the desired AirComp coefficient and optimized through received amplitudes.The amplitude vector is uniquely optimized for nonzero receive scaling; when the interference budget is zero, the solution is specified separately.Source: D. Optimal AirComp Transmit Coefficients · pp. 5–6
• The transmit-coefficient solution has a zero-multiplier case when the unconstrained amplitudes fit the interference budget; otherwise the budget is active and its positive multiplier can be found by bisection.The optimal transmit coefficients are recovered for nonzero receive scaling and nonzero channels.Source: D. Optimal AirComp Transmit Coefficients · p. 6

E. PA-position Optimization

• Because the PA-dependent objective and feasibility constraints are generally nonconvex but continuously differentiable, quadratic upper models are constructed with backtracking-selected curvature parameters.The surrogates are tight at the current position iterate.Source: E. PA-position Optimization · p. 6
• The PA-position update solves a convex QCQP built from surrogate constraints for the current SIC order.The PA variables also obey minimum-spacing constraints.Source: E. PA-position Optimization · p. 6
• The accepted position update remains feasible for the current SIC order and does not increase the AirComp objective; reordering afterward preserves NOMA feasibility.The order is updated according to Proposition 2 after each position update.Source: E. PA-position Optimization · p. 6

F. Convergence and Complexity Analysis

• The globally solved AirComp updates and accepted PA-position update make the MSE nonincreasing; feasibility is preserved, so the nonnegative MSE sequence converges.The SIC-order update cannot decrease the admissible interference-plus-noise level.Source: F. Convergence and Complexity Analysis · p. 6

IV. NUMERICAL RESULTS

• The numerical evaluation examines the interplay among AirComp accuracy, NOMA QoS, SIC ordering, and PA deployment.The results are averaged over 500 independent user-location realizations unless otherwise stated.Source: §IV · p. 7, A. Simulation Setup · p. 7

A. Simulation Setup

• The default setup uses 4 AirComp users, 3 NOMA users, and 6 PAs, with a 20 dBm maximum transmit power and a 0.5 Mbit/s minimum NOMA rate.The carrier frequency is 28 GHz and bandwidth is 1 MHz.Source: A. Simulation Setup · p. 7

B. Benchmark Schemes

• The proposed algorithm's MSE decreases monotonically, falling rapidly in early iterations before approaching a stable value.This behavior is reported as corroborating the convergence analysis.Source: Figure 2 · p. 7, B. Benchmark Schemes · p. 7
• As the number of PAs increases, the proposed design achieves the lowest MSE; comparisons show gaps from Fixed-PA, Random-PA, Discrete-PA, and Equal-Power AirComp.The reported comparisons attribute the gaps to adaptive PA positioning, unrestricted locations, and user-specific power optimization.Source: B. Benchmark Schemes · p. 7, Figure 3 · p. 8
• As maximum transmit power increases, MSE decreases for all schemes, and the proposed design remains lowest with gains over fixed, random, and discrete PA deployments.The pronounced gap from Equal-Power AirComp highlights user-specific power optimization.Source: B. Benchmark Schemes · p. 7, Figure 4 · p. 8
• As the number of AirComp users increases, MSE generally rises, while the proposed design maintains the lowest MSE across the considered range.The passage relates the rise to signal alignment and tighter NOMA QoS constraints.Source: B. Benchmark Schemes · p. 8, Figure 5 · p. 8
• As the minimum NOMA rate requirement increases, MSE rises for all schemes, while the proposed design achieves the lowest MSE.The passage attributes the increase to reduced admissible AirComp interference under more stringent QoS constraints.Source: Figure 6 · p. 8, B. Benchmark Schemes · p. 8
• As waveguide attenuation increases, MSE generally rises; the proposed design remains lowest by adapting PA positions and the AirComp transceiver to attenuation-aware channels.The passage links higher attenuation to weaker effective channels and less flexibility for signal alignment.Source: Figure 7 · p. 8, B. Benchmark Schemes · p. 8

V. CONCLUSION

• The joint design substantially improves computation accuracy over the considered benchmarks and balances AirComp performance with NOMA QoS requirements.The conclusion identifies PAS-enabled spatial reconfigurability as promising for communication-computation coexistence.Source: §V · p. 9

APPENDIX A PROOF OF PROPOSITION 1

• Activating the QoS constraints with equality in reverse SIC order yields the componentwise minimum feasible NOMA power vector.The proof uses the fact that each user's QoS requirement is nondecreasing in the powers of subsequently decoded users.Source: Appendix A · p. 9
• A feasible NOMA power allocation exists if and only if the recursively obtained minimum powers meet every user's individual power budget.The minimum-power vector satisfies the NOMA QoS constraints with equality.Source: Appendix A · p. 9

APPENDIX B PROOF OF PROPOSITION 2

• Removing adjacent inversions in the nonincreasing order of the user metric never decreases NOMA interference tolerance, so that order maximizes it.The proof establishes the result by successive adjacent swaps.Source: Appendix B · p. 9

APPENDIX C PA-POSITION UPDATE

• Channel derivatives use the user–PA distance derivative and the nth PA contribution to form gradient components for the quadratic models.The passage defines κ = 2π/λ and βg = 2π/λg.Source: Appendix C · p. 9
• Bounded Hessians over the compact feasible PA region give Lipschitz-continuous gradients, allowing valid quadratic upper bounds for the objective and constraints.The positive user–PA separation follows from H > 0.Source: Appendix C · p. 9
• Backtracking increases curvature parameters and resolves the QCQP until a candidate satisfies the surrogate bounds; finite Lipschitz constants ensure termination.The surrogate functions are tight at the current position, so a feasible current iterate is feasible for the surrogate problem.Source: Appendix C · p. 9
• The PA-position update preserves NOMA feasibility and does not increase the AirComp objective.The update inequalities establish feasibility for the current SIC order and a nonincreasing objective sequence.Source: Appendix C · p. 9