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Simultaneous Wireless Information Power Transfer for MISO Secrecy Channel
Zheng Chu, Martin Johnston, Stephane Le Goff
TL;DR
The paper considers secrecy-rate optimization for SWIPT in MISO channels with multiple eavesdroppers and energy-harvesting receivers. It develops beamforming methods with and without artificial noise, including robust formulations for channel uncertainty. Simulations report that the robust scheme outperforms the non-robust scheme, while the proposed scheme matches the rank-relaxed SDP performance in the reported comparison.
Problem
The paper addresses nonconvex secrecy-rate and harvested-energy optimization for MISO SWIPT with multiple eavesdroppers and energy-harvesting receivers.
Source: 1) Secrecy rate optimization without AN · p. 2, §I · p. 2Method
The paper reformulates beamforming optimization with and without artificial noise using bisection, one-dimensional search, and successive convex approximation, and extends schemes to channel uncertainty.
Source: 1) Secrecy rate optimization without AN · p. 2, §V · p. 5, C. Robust Secrecy Rate Optimization · p. 8Results
The robust scheme outperforms the non-robust scheme in achieved secrecy rate, and the proposed scheme achieves the same performance as the SDP scheme via rank relaxation.
Source: §VI · p. 10Takeaways & Limitations
The paper reports a rank-one optimal transmit covariance matrix for the artificial-noise secrecy-rate problem.
Source: §VII · p. 11
Abstract
from arXiv · showhide
This paper investigates simultaneous wireless information and power transfer (SWIPT) for multiuser multiple-input-single-output (MISO) secrecy channel. First, transmit beamfoming without artificial noise (AN) design is considered, where two secrecy rate optimization frameworks (i.e., secrecy rate maximization and harvested energy maximization) are investigated. These two optimization problems are not convex, and cannot be solved directly. For secrecy rate maximization problem, we employ bisection method to optimize the associated power minimization problem, and first-order Taylor series expansion is consider to approximate the energy harvesting (EH) constraint and the harvested energy maximization problem. Moreover, we extend our proposed algorithm to the associated robust schemes by incorporating with channel uncertainties, where two-level method is proposed for the harvested energy maximization problem. Then, transmit beamforming with AN design is studied for the same secrecy rate maximization problem, which are reformulated into semidefinite programming (SDP) based on one-dimensional search and successive convex approximation (SCA), respectively. Moreover, tightness analysis of rank relaxation is provided to show the optimal transmit covariance matrix exactly returns rank-one. Simulation results is provided to validate the performance of the proposed algorithm.
I. INTRODUCTION
The paper studies secure SWIPT beamforming for a MISO secrecy channel with multiple eavesdroppers and energy-harvesting receivers. It develops optimization approaches with and without artificial noise, including robust variants.
I. INTRODUCTION
- The paper addresses secrecy-rate and harvested-energy optimization without artificial noise, and joint beamforming and artificial-noise design for secrecy-rate maximization.It also extends the proposed schemes to channel uncertainty and to multiantenna eavesdroppers and energy-harvesting receivers.Source: 1) Secrecy rate optimization without AN · p. 2, §I · p. 2
A. Notation
- The notation defines Vec(A) as stacking a matrix’s columns into a vector, and A ⪰ 0 as indicating that A is positive semidefinite.It also defines Tr(·) as matrix trace and E{·} as statistical expectation.Source: A. Notation · p. 2
II. SYSTEM MODEL
- The system model includes one multiantenna transmitter, one legitimate user, multiple single-antenna eavesdroppers, and multiple single-antenna energy-harvesting receivers.The study focuses on secrecy optimization under reliable transmission; how harvested power is subsequently used is outside its scope.Source: §II · p. 2
1) Secrecy rate optimization without AN:
- The proposed relaxation is intended to avoid rank relaxation, which prior work used but could not show yielded a rank-one solution and therefore provided a suboptimal secrecy-rate solution.The paper states that its schemes can achieve the same secrecy performance as the SDR approach.Source: 1) Secrecy rate optimization without AN · p. 2, §II · p. 3
A. Secrecy Rate Maximization
- For secrecy-rate maximization without artificial noise, bisection searches target rates by solving a corresponding power-minimization problem at each rate.The power-minimization formulation uses an SDP secrecy-rate constraint and a first-order Taylor approximation of the nonconvex energy-harvesting constraint.Source: A. Secrecy Rate Maximization · p. 3
B. Harvested Energy Maximization
The harvested-energy formulation remains nonconvex because of its energy constraint. Under channel uncertainty, the paper models bounded channel errors and reformulates robust secrecy-rate optimization into iterative convex subproblems.
B. Harvested Energy Maximization
- The harvested-energy constraint keeps the relaxed maximization problem nonconvex, motivating an iterative approximation.The formulation uses t as a slack variable for the minimum energy-harvesting constraint.Source: B. Harvested Energy Maximization · p. 3
IV. ROBUST SECRECY RATE OPTIMIZATION WITH CHANNEL UNCERTAINTY
- Channel uncertainty is represented by estimated channels plus errors bounded in norm, and the robust scheme uses a worst-case secrecy rate.The stated motivation includes limited cooperation and channel estimation or quantization error.Source: §IV · p. 4, A. Channel Uncertainty · p. 4
B. Robust Secrecy Rate Optimization with Channel Uncertainty
- With uncertain CSI for the legitimate user, eavesdroppers, and energy-harvesting receivers, the robust power-minimization problem is formulated using secrecy-rate constraints.The secrecy-rate constraint is reformulated, including a first-order Taylor approximation and a lemma-based reformulation.Source: B. Robust Secrecy Rate Optimization… · p. 4
- The robust power-minimization subproblem is convex for a fixed iterate and is solved iteratively with an interior-point method.An initialized value is updated until convergence.Source: B. Robust Secrecy Rate Optimization… · p. 4
C. Robust Harvested Energy Maximization with Channel Uncertainty
Robust harvested-energy maximization incorporates channel uncertainty while requiring achieved secrecy rate and transmit-power constraints. The paper addresses the nonconvex formulation with a two-level SCA method and a one-dimensional search over a bounded slack variable.
C. Robust Harvested Energy Maximization with Channel Uncertainty
- The robust objective maximizes harvested energy subject to achieved secrecy-rate and transmit-power constraints under channel uncertainty.The original problem is nonconvex in the channel uncertainties and secrecy-rate constraint.Source: C. Robust Harvested Energy… · pp. 4–5
- The proposed two-level SCA algorithm uses an equivalent reformulation and optimizes the slack variable τ through an outer search.For a given τ, the reformulated problem is convex at an initialized value of t.Source: C. Robust Harvested Energy… · p. 5
- For the fixed-τ subproblem, SCA approximates the energy constraint; a line search is restricted to 0 ≤ τ ≤ P_s(∥h̄_s∥² + ε_s)².The upper bound follows from Cauchy–Schwarz and triangle inequalities, the transmit-power bound, and the legitimate-user channel-error bound.Source: C. Robust Harvested Energy… · p. 5
V. SECURE SWIPT OF JOINT BEAMFORMING AND ARTIFICIAL NOISE DESIGN
The paper studies joint information-signal beamforming and artificial-noise design for SWIPT secrecy-rate maximization. Its SDP relaxation is reported to return an optimal transmit covariance matrix of rank one.
V. SECURE SWIPT OF JOINT BEAMFORMING AND ARTIFICIAL NOISE DESIGN
- Joint signal beamforming and artificial noise are designed for secrecy-rate maximization, with artificial noise also available for energy harvesting.The formulation includes transmit-power and minimum energy-harvesting constraints.Source: §V · p. 5, A. Problem Formulation · p. 6
- The proposed SDP relaxation returns an optimal solution that is exactly rank one, improving on the cited line-search method's rank-at-most-two result.The relaxation is stated to also suit multiple-antenna eavesdroppers and energy-harvesting receivers.Source: §V · p. 6, A. Problem Formulation · p. 6
A. Problem Formulation
- The formulation defines Q_s as the transmit covariance and W as the artificial-noise covariance, with per-antenna power constraints.It also expresses mutual information for the legitimate user and each eavesdropper.Source: A. Problem Formulation · p. 6
B. Secrecy Rate Optimization
The secrecy-rate problem jointly optimizes the transmit covariance matrix and artificial-noise beamforming. The one-dimensional line-search method converts its fixed-parameter inner problem into a convex SDP.
B. Secrecy Rate Optimization
- Two methods are proposed for joint covariance and artificial-noise optimization: SDP-based one-dimensional line search and SDP based on SCA.The secrecy-rate formulation includes per-antenna power constraints and minimum energy-harvesting constraints.Source: B. Secrecy Rate Optimization · p. 6, 1) SDP-Based One-Dimensional Line… · p. 6
1) SDP-Based One-Dimensional Line Search Method:
- The method introduces slack variable t and optimizes an outer objective as a function of t, with a lower bound t_min derived for the search.The cited formulation gives the outer objective as t log(1 + f(t)) + log(t).Source: 1) SDP-Based One-Dimensional Line… · p. 6
- For a fixed t, the inner problem is quasi-convex and a Charnes–Cooper transformation yields a convex problem solvable by an interior-point method.The solution of the transformed problem recovers the solution of the original inner problem.Source: 1) SDP-Based One-Dimensional Line… · pp. 6–7
2) Tightness Analyses of Rank Relaxation:
The rank relaxation admits a rank-one transmit covariance solution, and the paper develops an SDP-based SCA method for jointly optimizing transmit beamforming and artificial noise (AN).
2) Tightness Analyses of Rank Relaxation:
- For every feasible t, problem (32) has a rank-one solution, establishing tightness of its rank relaxation.Theorem 2 states that an optimal Qs satisfies rank(Qs) ≤1.Source: 2) Tightness Analyses of Rank Relaxation · p. 7
3) SDP Based Successive Convex Approximation:
- The SDP-based SCA algorithm jointly optimizes transmit beamforming and AN by iteratively linearizing a nonconvex constraint.The approximation is updated until constraints (39c) and (39d) hold with equality.Source: 3) SDP Based Successive Convex… · p. 7
- When problem (37) is feasible for a positive secrecy rate, its optimal solution can be rank-one despite the rank relaxation.The tightness argument establishes a rank-one optimal solution for the beamforming covariance matrix.Source: 3) SDP Based Successive Convex… · p. 7
C. Robust Secrecy Rate Optimization
The robust secrecy-rate framework accounts for channel uncertainty when perfect channel state information is unavailable. For its one-dimensional-search formulation, the associated power-minimization problem has a rank-one optimum whenever feasible.
C. Robust Secrecy Rate Optimization
- Because channel estimation and quantization can prevent perfect CSI, the robust design incorporates channel uncertainties.The robust power constraint also accounts for uncertainty in the per-antenna PSD matrix, modeled with a norm bound εi.Source: C. Robust Secrecy Rate Optimization · pp. 7–8
1) Robust SDP Based One Dimension Search:
- For a fixed search value t, the robust inner problem is reformulated as a power-minimization problem using the S-Procedure and Charnes–Cooper transformation.The resulting problem (46) yields the optimal value f(t).Source: 1) Robust SDP Based One Dimension Search · p. 8
- Whenever problem (45) is feasible, its optimal solution is rank-one and can be obtained by solving problem (46).This establishes tightness of the rank relaxation for the robust one-dimensional-search formulation.Source: 1) Robust SDP Based One Dimension Search · p. 8
2) Robust SDP Based Successive Convex Optimization:
The paper formulates robust secrecy-rate maximization with beamforming and AN as an SDP-based SCA problem under channel uncertainty. It also states that rank-one solutions persist when eavesdroppers and energy-harvesting receivers have multiple antennas.
2) Robust SDP Based Successive Convex Optimization:
- The robust secrecy-rate problem jointly optimizes beamforming and AN using an SDP-based SCA reformulation under channel uncertainty.S-Procedure and first-order Taylor approximation are used to form constraints; each iteration solves a convex problem for fixed slack variables.Source: 2) Robust SDP Based Successive Convex… · pp. 8–9
D. Extension to The Case of Multiantenna Eavesdroppers and EH Receivers
- With multiple-antenna eavesdroppers and EH receivers, the optimal solutions of problems (32) and (45) remain rank-one under perfect and imperfect CSI.The extension models channel errors with Frobenius-norm bounds and uses relaxed constraints in the rank analysis.Source: D. Extension to The Case of… · p. 9
- The robust problem (45) retains a rank-one solution for the multiantenna case through constraints incorporated into its dual function.The passage describes replacing corresponding dual-function constraints using the relaxed constraints and S-Procedure.Source: D. Extension to The Case of… · p. 9
VI. NUMERICAL RESULTS
Simulations examine secrecy rate and harvested energy in a MISO SWIPT setting, including robust designs and beamforming with AN. The reported comparisons favor robust designs over non-robust ones and show higher secrecy performance for AN and the SCA scheme in the stated comparisons.
VI. NUMERICAL RESULTS
- Robust secrecy-rate optimization outperforms the non-robust scheme, while the proposed scheme matches the SDP rank-relaxation scheme.The reported secrecy rate increases with transmit power.Source: §VI · p. 10, Figure 1 · p. 10
- Harvested power increases with transmit power and decreases as the error bound ε increases.The comparisons are presented in Figs. 2 and 3.Source: §VI · p. 10, Figure 2 · p. 10, Figure 3 · p. 10
B. Secrecy Rate Optimization with AN
- The SCA-based scheme outperforms the one-dimensional-search, two-dimensional-search, and cited schemes in the reported secrecy-rate comparison.The reported comparison also says AN-based beamforming performs better than beamforming without AN.Source: B. Secrecy Rate Optimization with AN · p. 10, Figure 6 · p. 11
- The SCA algorithm converges more slowly as transmit power increases.The paper reports this convergence comparison for the SCA-based scheme.Source: B. Secrecy Rate Optimization with AN · p. 10, Figure 5 · p. 11
VII. CONCLUSIONS
- The conclusion reports that the proposed secrecy-rate optimization with AN has a rank-one optimal solution and that simulations validate the proposed schemes.The conclusion describes reformulations based on one-dimensional line search and SCA.Source: §VII · p. 11
APPENDIX
The appendix converts secrecy and energy-harvesting constraints into tractable forms and establishes equivalence between two formulations of robust energy maximization.
A. Proof of Lemma 1
- Lemma 1 uses the Schur complement to reformulate the secrecy-rate constraint and a first-order Taylor approximation to upper-bound the nonconvex energy-harvesting constraint.The energy-harvesting constraint is split using two real variables; one resulting constraint is linear while the other remains nonconvex before approximation.Source: A. Proof of Lemma 1 · pp. 11–12
B. Proof of Lemma 2
- The robust secrecy-rate constraint is reformulated using Nemirovski’s lemma (Lemma 4): the derivation rewrites the second constraint in (11), then uses the lemma to rewrite constraint (61).Source: B. Proof of Lemma 2 · p. 12
C. Proof of Theorem 1
- Theorem 1 proves that problems (16) and (18) have equal optimal values.The proof bounds each value by the other, establishing equality.Source: C. Proof of Theorem 1 · p. 12
D. Proof of Theorem 2
The proofs use KKT and rank arguments to establish rank-one structure for the transmit covariance matrix and develop a rank relation for the associated relaxation.
D. Proof of Theorem 2
- Theorem 2 shows that the optimal transmit covariance matrix Qs has rank one.The proof derives rank(Y)=NT−1, placing Qs in Y’s null space; the alternative full-rank case would violate the secrecy-rate constraint.Source: D. Proof of Theorem 2 · p. 13
E. Proof of Theorem 3
- Theorem 3’s proof uses KKT conditions and positive-semidefinite matrices to derive a rank relation for Qs.It states that the main diagonal matrices P1 and P4 are positive semidefinite and that nonsingular pre- or post-multiplication preserves matrix rank.Source: E. Proof of Theorem 3 · p. 13