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Robust and Secure Resource Allocation for Full-Duplex MISO Multicarrier NOMA Systems

Yan Sun, Derrick Wing Kwan Ng, Jun Zhu, Robert Schober

arXiv:1710.01391v1cs.ITmath.IT

TL;DR

Half-duplex base-station operation leaves resources underutilized even with NOMA, and the resource-allocation problem is difficult to solve. The paper applies monotonic optimization and develops a low-complexity alternative; simulations report simultaneous uplink and downlink security, higher performance than the named baselines, and robustness to imperfect CSI.

  • Problem

    Half-duplex base-station operation leaves resources underutilized even with NOMA, while the resource-allocation problem is difficult to solve.

    Source: §I · pp. 2, 4

  • Method

    The paper solves the allocation problem optimally using monotonic optimization and develops a low-complexity, polynomial-time suboptimal iterative scheme.

    Source: §I · p. 4, §VI · p. 26

  • Results

    The FD MISO MC-NOMA schemes secure downlink and uplink simultaneously, outperform FD MISO MC-OMA and two baseline systems, and are robust to imperfect CSI.

    Source: §VI · pp. 26–27

  • Takeaways & Limitations

    The polynomial-time suboptimal iterative algorithm offers a close-to-optimal alternative to the optimal resource-allocation scheme.

    Source: §I · p. 4, §VI · p. 26

Abstract

from arXiv · show

In this paper, we study the resource allocation algorithm design for multiple-input single-output (MISO) multicarrier non-orthogonal multiple access (MC-NOMA) systems, in which a full-duplex base station serves multiple half-duplex uplink and downlink users on the same subcarrier simultaneously. The resource allocation is optimized for maximization of the weighted system throughput while the information leakage is constrained and artificial noise is injected to guarantee secure communication in the presence of multiple potential eavesdroppers. To this end, we formulate a robust non-convex optimization problem taking into account the imperfect channel state information (CSI) of the eavesdropping channels and the quality-of-service (QoS) requirements of the legitimate users. Despite the non-convexity of the optimization problem, we solve it optimally by applying monotonic optimization which yields the optimal beamforming, artificial noise design, subcarrier allocation, and power allocation policy. The optimal resource allocation policy serves as a performance benchmark since the corresponding monotonic optimization based algorithm entails a high computational complexity. Hence, we also develop a low-complexity suboptimal resource allocation algorithm which converges to a locally optimal solution. Our simulation results reveal that the performance of the suboptimal algorithm closely approaches that of the optimal algorithm. Besides, the proposed optimal MISO NOMA system can not only ensure downlink and uplink communication security simultaneously but also provides a significant system secrecy rate improvement compared to traditional MISO orthogonal multiple access (OMA) systems and two other baseline schemes.

I. INTRODUCTION

Earlier MC-NOMA and full-duplex designs leave security gaps: FD NOMA increases information-leakage risk, while existing secure NOMA work does not address robust FD MISO MC-NOMA with QoS constraints. This paper formulates that resource-allocation problem and develops optimal and low-complexity suboptimal solutions.

I. INTRODUCTION

  • FD designs using orthogonal UL and DL resources can lose spectral efficiency, while simultaneous FD NOMA transmissions increase information-leakage risk.Existing FD NOMA designs cited here cannot guarantee communication security.Source: §I · pp. 2–3
  • Prior secure NOMA studies address downlink single-carrier systems with half-duplex base stations, not FD MISO MC-NOMA security.Those works also assume perfect eavesdropper CSI, which may be unavailable for idle users.Source: §I · pp. 3–4
  • Robust secure resource allocation for FD MISO MC-NOMA with QoS constraints remains open, with coupled pairing, beamforming, artificial-noise, and allocation decisions complicating optimization.Imperfect eavesdropper CSI also creates infinitely many constraints.Source: §I · p. 4
  • The authors formulate weighted-throughput maximization with imperfect eavesdropper CSI and legitimate-user QoS constraints, then solve the non-convex problem optimally using monotonic optimization.The design includes beamforming, artificial-noise, subcarrier, and power allocation.Source: §I · p. 4
  • A low-complexity suboptimal scheme is also proposed and is reported to achieve a close-to-optimal solution.It is presented as an alternative to the computationally demanding optimal approach.Source: §I · p. 4

II. NOTATION AND SYSTEM MODEL

The system model pairs a full-duplex, multi-antenna base station with half-duplex single-antenna users on orthogonal subcarriers. Unscheduled idle users may eavesdrop, so the model includes artificial noise and a cooperating equivalent eavesdropper.

A. Notation

  • The notation section establishes conventions for vectors, matrices, and common matrix operations and sets.For example, boldface denotes vectors and matrices, and X−1 denotes a matrix inverse.Source: A. Notation · pp. 4–5

B. FD MISO MC-NOMA System

  • The FD base station has multiple antennas and serves half-duplex, single-antenna users over orthogonal subcarriers, with up to two UL and two DL users scheduled per subcarrier.The user-count restriction is stated to limit hardware complexity, processing delay, and interference.Source: B. FD MISO MC-NOMA System · p. 5
  • Unscheduled idle users are legitimate system users who may eavesdrop on active users’ signals and are treated as potential eavesdroppers.The example system in Fig. 1 has two idle users.Source: B. FD MISO MC-NOMA System · p. 5, Figure 1 · p. 6

C. Channel Model

  • The channel model includes downlink data streams and artificial noise injected by the base station to degrade potential eavesdropper channels.The artificial noise is modeled as a complex Gaussian vector with a covariance matrix.Source: C. Channel Model · p. 6
  • For secure design, the model assumes all potential eavesdroppers cooperate as an equivalent multi-antenna eavesdropper.The equivalent eavesdropper has M antennas.Source: C. Channel Model · p. 7

III. RESOURCE ALLOCATION PROBLEM FORMULATION

The resource-allocation formulation uses weighted user rates, SIC-based reception, and secrecy rates against a worst-case cooperating eavesdropper. It combines these performance measures with scheduling and resource-allocation variables.

III. RESOURCE ALLOCATION PROBLEM FORMULATION

A. Weighted System Throughput and Secrecy Rate

  • The formulation serves at most two UL and two DL users on each subcarrier.The system also performs SIC to mitigate multiuser interference among users sharing a subcarrier.Source: A. Weighted System Throughput and… · p. 7
  • For uplink reception, the base station uses MRC-SIC, first decoding UL user r and then user t.The paper adopts maximum-ratio combining for UL reception.Source: A. Weighted System Throughput and… · p. 8
  • The throughput weights and scheduling variables encode the priorities of DL and UL users and which users are assigned to each subcarrier.The formulation collects subcarrier allocation, beamforming, and artificial-noise variables as optimization variables.Source: A. Weighted System Throughput and… · p. 8
  • Downlink SIC requires the paired user to meet the stated SINR condition before it can decode and cancel the other user’s signal.The rate model also accounts for the paired users’ decoding order and treats UL signals as noise at DL users.Source: A. Weighted System Throughput and… · pp. 7, 9
  • The security model assumes the equivalent eavesdropper can cancel multiuser and cross-link interference before decoding a target signal.The model then defines achievable secrecy rates from legitimate-user rates and eavesdropper capacities.Source: A. Weighted System Throughput and… · p. 9

B. Channel State Information

The base station obtains CSI for legitimate links through handshaking at each scheduling slot, but only imperfect CSI is available for the equivalent eavesdropper’s channels. The paper represents channel uncertainty using continuous sets.

B. Channel State Information

  • CSI for downlink, uplink, and UL-to-DL interference links is available through handshaking at the start of each scheduling slot.Idle users hand-shake only occasionally, so the base station has only imperfect CSI for the equivalent eavesdropper’s channels.Source: B. Channel State Information · pp. 9–10
  • The CSI model separates channel estimates from uncertainty terms and uses continuous sets to represent the eavesdropper-channel uncertainty.The model applies this treatment to the base-station-to-eavesdropper and UL-user-to-eavesdropper links.Source: B. Channel State Information · p. 10

C. Optimization Problem Formulation

The formulation maximizes weighted system throughput subject to secrecy, QoS, power, and subcarrier-allocation requirements. Its mixed non-convex and combinatorial structure motivates optimal monotonic optimization and a lower-complexity sequential convex approximation scheme.

C. Optimization Problem Formulation

  • The objective maximizes weighted system throughput while imposing secrecy and QoS requirements for downlink and uplink users.The secrecy constraints bound tolerable rates at equivalent eavesdroppers, while the QoS constraints impose minimum required rates.Source: C. Optimization Problem Formulation · pp. 10–11
  • The secrecy constraints provide flexibility to control communication security levels for different services.The thresholds represent maximum tolerable data rates at equivalent eavesdroppers for decoding downlink or uplink information.Source: C. Optimization Problem Formulation · pp. 10–11
  • Each subcarrier can be allocated to at most two uplink and two downlink users, while transmit powers and artificial-noise covariance satisfy system constraints.The formulation caps base-station and uplink-user transmit power and requires the artificial-noise covariance to be Hermitian positive semidefinite.Source: C. Optimization Problem Formulation · pp. 10–11
  • The problem is mixed non-convex and combinatorial, with binary selection, non-convex constraints and objective, and infinitely many inequalities from continuous CSI uncertainty sets.The paper identifies these features as obstacles to systematic resource-allocation algorithm design.Source: C. Optimization Problem Formulation · p. 11

IV. SOLUTION OF THE OPTIMIZATION PROBLEM

  • Monotonic optimization is applied to solve the problem optimally through an iterative algorithm using semidefinite programming relaxation in each iteration.A suboptimal sequential convex approximation scheme is also proposed to reduce computational complexity while achieving close-to-optimal performance.Source: §IV · p. 11

1) Monotonic Optimization Framework:

  • The monotonic-optimization reformulation introduces slack variables and rewrites constraints so the problem has monotonically increasing functions in canonical form.The transformed feasible set is expressed as V = G ∩ H.Source: 1) Monotonic Optimization Framework · pp. 12–13

2) Optimal Algorithm:

The optimal algorithm reformulates resource allocation as monotonic optimization and uses polyblock outer approximation to approach the feasible-set boundary. It recovers the optimal subcarrier policy from the resulting vertex.

2) Optimal Algorithm:

  • The algorithm applies polyblock outer approximation to the monotonic optimization problem, whose optimum lies on the boundary of feasible set V.The paper identifies the FD MISO MC-NOMA iterative resource-allocation design as more challenging than the cited SISO case.Source: 2) Optimal Algorithm · pp. 13–14
  • The iteration handles infinitely many inequality constraints and a rank restriction on optimization variables, adding difficulty to the resource-allocation procedure.The next algorithmic steps address non-convex constraints C5, C6, and C13.Source: 2) Optimal Algorithm · p. 14
  • Because V's boundary is not analytically available, successive enclosing polyblocks are shrunk by replacing vertices using projections onto G.At each iteration the selected vertex maximizes the objective through its projection, and the process stops when the vertex-to-projection distance meets tolerance.Source: 2) Optimal Algorithm · pp. 14–15
  • The approximation accuracy is controlled by tolerance ε, and the paper illustrates the procedure with a one-element z and t example.Figure 2 depicts an optimal point on the boundary of feasible set V.Source: Figure 2 · p. 14, 2) Optimal Algorithm · p. 15
  • The optimal subcarrier allocation is recovered from the optimal vertex because the associated variables exceed one only when the corresponding users are multiplexed on that subcarrier.The optimal allocation policy is obtained after restoring the auxiliary variables from the vertex.Source: 2) Optimal Algorithm · p. 15

3) Computation of Projection:

The projection computation converts robust constraints into tractable forms and solves a relaxed convex SDP within an iterative projection procedure. Under the stated power condition, the resulting beamforming matrix is rank one.

3) Computation of Projection:

  • The projection algorithm uses a Dinkelbach-type procedure and bisection search to obtain the resource-allocation policy at each iteration.The projection scales the vertex using the smaller of two bounds, with β* obtained from a bounded optimization.Source: 3) Computation of Projection · pp. 15–16
  • Robust constraints C5 and C6 are non-convex, involve inverse matrices, and contain infinitely many inequalities, so they are transformed into linear matrix inequalities.The generalized S-Procedure is used to simplify the resulting robust constraints.Source: 3) Computation of Projection · p. 16
  • Removing rank-one constraint C13 yields a convex semidefinite program that can be solved by standard convex-program solvers.The paper verifies the relaxation's tightness with a theorem.Source: 3) Computation of Projection · p. 17
  • When P_DL_max > 0, the optimal beamforming matrix from the relaxed problem is rank one.Source: 3) Computation of Projection · p. 17

4) Summary:

The monotonic-optimization resource-allocation algorithm provides globally optimal precoding and subcarrier allocation, but its complexity grows exponentially. A penalized sequential convex approximation scheme offers a polynomial-time locally optimal alternative.

4) Summary:

  • The monotonic-optimization algorithm yields globally optimal precoding and subcarrier allocation, providing a performance benchmark for suboptimal algorithms.Its computational complexity grows exponentially with the number of vertices, 5D, used in each iteration.Source: 4) Summary · p. 17

B. Suboptimal Resource Allocation Scheme

  • The proposed suboptimal scheme targets a locally optimal solution with low computational complexity.It uses penalized sequential convex approximation and can start from an arbitrary initial point.Source: B. Suboptimal Resource Allocation Scheme · pp. 18–19
  • Products of integer variables and continuous variables impede efficient optimization, motivating the use of the big-M method.The method decomposes the product terms using additional constraints.Source: B. Suboptimal Resource Allocation Scheme · p. 18
  • The suboptimal formulation relaxes subcarrier-selection variables to the interval from zero to one and penalizes non-binary values.For a sufficiently large penalty factor, the paper states that the penalized and original problems are equivalent.Source: B. Suboptimal Resource Allocation Scheme · p. 18
  • Sequential convex approximation obtains a locally optimal solution, while the traditional approach requires favorable initialization for good performance and incurs high computational complexity.Source: B. Suboptimal Resource Allocation Scheme · p. 19
  • The penalized sequential convex approximation algorithm converges to a locally optimal solution with polynomial-time computational complexity.The obtained beamforming matrix is rank one when P_DL_max > 0.Source: B. Suboptimal Resource Allocation Scheme · p. 20

V. SIMULATION RESULTS

The simulations assess the proposed resource-allocation schemes under modeled channel and user settings. Both algorithms converge to the optimal solution across tested system sizes, while the suboptimal scheme needs fewer iterations in the reported K = J = 2 case.

V. SIMULATION RESULTS

  • User weights equal normalized distance from the FD BS to support resource-allocation fairness, particularly for cell-edge users.Source: §V · p. 20
  • Simulations are averaged over 1,000 different path-loss and multipath-fading realizations; equivalent eavesdroppers are modeled as independent and identically Rayleigh distributed, while each subcarrier’s multipath fading self-interference channel is independent and identically Rician.Source: §V · pp. 20–21

A. Convergence of Proposed Optimal and Suboptimal Schemes

B. Average System Throughput versus Maximum Transmit Power

The proposed optimal and suboptimal schemes outperform the considered baselines in system throughput, with the suboptimal scheme closely approaching the optimal one. Secrecy throughput improves with more antennas and users, although antenna-related gains diminish at large antenna counts and stricter QoS requirements reduce average user secrecy throughput.

B. Average System Throughput versus Maximum Transmit Power

  • As maximum downlink transmit power rises, system throughput increases, but its growth slows beyond 36 dBm as greater information leakage requires more artificial-noise power and stronger self-interference reduces uplink throughput.The uplink reduction partially offsets the downlink-throughput improvement.Source: B. Average System Throughput versus… · p. 22
  • The proposed schemes achieve roughly 22%, 42%, and 70% higher average system throughput than baseline schemes 1, 2, and 3, respectively.The suboptimal algorithm closely approaches the optimal resource allocation scheme.Source: B. Average System Throughput versus… · p. 23

C. Average System Secrecy Throughput versus Number of Antennas

  • Average system secrecy throughput grows faster with antenna count for the proposed schemes than for the baselines, but gains diminish at large NT; additional antennas aid beamforming and artificial-noise injection.The added degrees of freedom also help limit the equivalent eavesdropper’s achievable data rate.Source: C. Average System Secrecy Throughput… · p. 23
  • The proposed schemes retain considerably higher average system secrecy throughput than all baselines even at relatively large NT.Baseline scheme 1 quickly saturates as NT increases, while schemes 2 and 3 cannot exploit the full benefits of additional antennas.Source: C. Average System Secrecy Throughput… · pp. 23–24

D. Average User Secrecy Throughput versus Total Number of Users

  • Average user secrecy throughput for baseline schemes 1 and 2 increases with total user count K + J, whereas baseline scheme 3 does not scale with K + J; the proposed suboptimal scheme performs similarly to the optimal scheme even at relatively large user counts.Source: D. Average User Secrecy Throughput… · p. 24
  • More stringent downlink and uplink QoS requirements lower average user secrecy throughput because more radio resources go to users with poor channel conditions.Source: D. Average User Secrecy Throughput… · p. 25

E. Average System Secrecy Throughput versus Maximum Channel Estimation Error

The proposed FD MISO MC-NOMA schemes secure downlink and uplink transmission simultaneously and outperform FD MISO MC-OMA and two baseline systems. Simulations also show robustness to imperfect channel information, alongside throughput-security trade-offs as eavesdropper leakage tolerance changes.

E. Average System Secrecy Throughput versus Maximum Channel Estimation Error

VI. CONCLUSIONS

  • The design maximizes weighted throughput subject to leakage and QoS constraints, accounting for imperfect eavesdropping-channel CSI, and monotonic optimization solves it optimally.Source: §VI · p. 26
  • A polynomial-time suboptimal iterative resource-allocation algorithm was also developed.Source: §VI · p. 26
  • The proposed schemes secure downlink and uplink transmission simultaneously while outperforming FD MISO MC-OMA and two baseline systems.Source: §VI · pp. 26–27

APPENDIX

The appendix proves equivalences used to reformulate constraints and establishes properties of the relaxed semidefinite program. Its theorem proof uses convexity, strong duality, and KKT conditions to characterize the optimal downlink covariance structure.

A. Proof of Proposition 1

  • The Proposition 1 proof uses determinant identities and rank conditions to relate constraints C5 and C6 to reformulated expressions.Source: A. Proof of Proposition 1 · p. 27
  • For positive semidefinite matrices, a determinant bound by one plus the trace supports the constraint reformulation in the proof.Source: A. Proof of Proposition 1 · p. 27

B. Proof of Theorem 1

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