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Secure Beamforming For MIMO Broadcasting With Wireless Information And Power Transfer

Qingjiang Shi, Weiqiang Xu, Jinsong Wu, Enbin Song, Yaming Wang

arXiv:1403.3196v2cs.IT

TL;DR

Secure beamforming in MIMO information-energy broadcasting must balance secrecy with energy harvesting over an open wireless medium. The paper derives global solutions for special cases and an iterative general-case algorithm, proving monotonic convergence to a KKT solution and showing artificial noise improves secrecy rate.

  • Problem

    Secure information transmission while maintaining efficient energy harvesting is an important challenge in MIMO wireless information-and-power-transfer broadcasting.

  • Method

    The paper uses semidefinite relaxation for global solutions in special cases and develops an IBCD algorithm for arbitrary streams, extending it to artificial-noise design.

  • Results

    The IBCD algorithm converges monotonically, with every limit point satisfying the secrecy-rate problem's KKT conditions, while artificial noise yields better secrecy rates in simulations.

  • Takeaways & Limitations

    The study supplies globally optimal beamforming for selected cases and a convergent beamforming solution for the general arbitrary-stream case.

Abstract

from arXiv · show

This paper considers a basic MIMO information-energy (I-E) broadcast system, where a multi-antenna transmitter transmits information and energy simultaneously to a multi-antenna information receiver and a dual-functional multi-antenna energy receiver which is also capable of decoding information. Due to the open nature of wireless medium and the dual purpose of information and energy transmission, secure information transmission while ensuring efficient energy harvesting is a critical issue for such a broadcast system. Assuming that physical layer security techniques are applied to the system to ensure secure transmission from the transmitter to the information receiver, we study beamforming design to maximize the achievable secrecy rate subject to a total power constraint and an energy harvesting constraint. First, based on semidefinite relaxation, we propose global optimal solutions to the secrecy rate maximization (SRM) problem in the single-stream case and a specific full-stream case where the difference of Gram matrices of the channel matrices is positive semidefinite. Then, we propose a simple iterative algorithm named inexact block coordinate descent (IBCD) algorithm to tackle the SRM problem of general case with arbitrary number of streams. We proves that the IBCD algorithm can monotonically converge to a Karush-Kuhn-Tucker (KKT) solution to the SRM problem. Furthermore, we extend the IBCD algorithm to the joint beamforming and artificial noise design problem. Finally, simulations are performed to validate the performance of the proposed beamforming algorithms.

I. INTRODUCTION

The paper addresses secure wireless information and power transfer in a MIMO information-energy broadcast system, where a dual-functional energy receiver can eavesdrop. It develops beamforming methods to maximize secrecy rate under power and energy-harvesting constraints, including global solutions for special cases and an IBCD approach for the general case.

  • Motivation: Wireless information and power transfer can simultaneously support communication and radio-based energy harvesting, but battery limitations and environmental concerns motivate energy harvesting.Conventional battery-powered systems have short lifetimes and require frequent recharging, while information and communication technologies consume substantial energy.
  • Security motivation: Security is critical because wireless transmissions are exposed, and a dual-functional energy harvester capable of information decoding may act as an eavesdropper.Earlier WIPT research had not considered security issues in MIMO WIPT systems.
  • System and problem: The considered system has a multi-antenna transmitter sending information and energy to multi-antenna information and energy receivers, maximizing secrecy rate under total-power and energy-harvesting constraints.The energy receiver can switch from energy-harvesting mode to information-decoding mode and potentially eavesdrop the information receiver’s transmission.
  • General-case algorithm: For arbitrary stream numbers, the secrecy rate maximization problem is reformulated and addressed with an inexact block coordinate descent algorithm whose convergence is studied.The resulting secrecy rate maximization problem is difficult because the secrecy-rate function is generally non-concave and the energy-harvesting constraint is nonconvex.
  • Special-case solutions: Global optimal solutions are proposed using semidefinite relaxation for the single-stream case and a full-stream case with a positive-semidefinite Gram-matrix difference.These are identified as two special cases of the secrecy rate maximization problem.
  • Extensions and evaluation: The IBCD algorithm is extended to joint beamforming and artificial-noise design, and simulations evaluate the effectiveness of the proposed beamforming algorithms.The paper’s stated contributions include both the artificial-noise extension and numerical validation.

II. SYSTEM MODEL AND PROBLEM FORMULATION

The paper models a MIMO information-energy broadcast system in which an energy receiver can also eavesdrop, and formulates secrecy-rate-maximizing beamforming under energy-harvesting and power constraints. The resulting problem is generally nonconvex, may yield negative secrecy rates under stringent energy requirements, and extends beyond previously considered intermediate-stream formulations.

  • System model: A multi-antenna transmitter simultaneously sends information and power to an information receiver and a dual-function energy receiver operating in information-decoding or energy-harvesting mode.The energy receiver may switch to information decoding and eavesdrop on the information receiver’s transmission.
  • Problem formulation: The beamforming design maximizes achievable secrecy rate subject to harvested power E(V) ≥ PE and total transmit power Tr(VV^H) ≤ PT.Physical-layer security is assumed to protect transmission from the transmitter to the information receiver.
  • Problem characteristics: The secrecy-rate objective is generally nonconcave and the energy-harvesting constraint is nonconvex, making the secrecy-rate maximization problem nonconvex and hard to solve.Removing the energy-harvesting constraint yields the established power-constrained MIMO wiretap beamforming problem PnoEH.
  • Problem characteristics: Under the energy-harvesting constraint, the problem can have a negative maximum secrecy rate even when the corresponding unconstrained problem has a positive maximum secrecy rate.In a single-stream example, the feasible solution is unique up to phase rotation and achieves negative secrecy rate for some information channels.
  • Problem characteristics: The formulation addresses PnoEH with an intermediate number of streams, 1 < d < NT, which had not yet been considered in the literature and is not covered by existing algorithms.The existing algorithms for d = 1 or d = NT do not apply because of the nonconvex energy-harvesting constraint.

III. SECURE BEAMFORMING DESIGN: GLOBAL SOLUTION TO TWO SPECIAL CASES · A. Single-stream case: d = 1

This section develops global solutions for two special cases of the secure beamforming problem, focusing here on the single-stream case. When d = 1, the beamforming design is reduced to a vector problem that can be reformulated and globally solved through semidefinite optimization.

  • III. SECURE BEAMFORMING DESIGN: GLOBAL SOLUTION TO TWO SPECIAL CASES: The section studies global solutions for the single-stream case and a full-stream case with H_H^H − H_E^H H_E ⪰ 0.
  • A. Single-stream case: d = 1: For d = 1, the beamforming matrix V reduces to a vector, denoted by v.
  • A. Single-stream case: d = 1: Using a determinant identity, the single-stream problem is equivalently transformed into a quadratically constrained quadratic fractional program.
  • A. Single-stream case: d = 1: A direct Charnes-Cooper transformation and semidefinite relaxation convert the fractional problem into an SDP with three linear constraints and one additional variable.
  • A. Single-stream case: d = 1: The proposed more efficient reformulation produces an SDP with only two linear constraints.
  • A. Single-stream case: d = 1: Lemma 3.1 removes the fractional objective form without introducing extra quadratic constraints, while solving problem (7) yields the optimal solution to problem (6).
  • A. Single-stream case: d = 1: When applicable, an eigenvector of Q^-1 corresponding to its minimum eigenvalue, with proper scaling, is an optimal solution to problem (8).
  • A. Single-stream case: d = 1: The remaining two-constraint quadratic optimization is globally solved by semidefinite relaxation, followed by rank reduction and eigen-decomposition when the SDP solution has rank greater than one.

B. Full-stream case: d = NT and HH · IV. SECURE BEAMFORMING DESIGN: KKT SOLUTION TO GENERAL CASE · A. Inexact Block Coordinate Algorithm For Problem (5)

The paper obtains an exact convex reformulation for the full-stream case through semidefinite relaxation, then develops an IBCD framework for the general arbitrary-stream problem. The framework handles nonconvexity by reformulating the Shannon-capacity objective with auxiliary variables and extends to artificial-noise-aided transmission.

  • B. Full-stream case: d = NT and HH: For d = NT, semidefinite relaxation defines X = VVH by dropping the rank constraint.Here, V is square, enabling the relaxation used for problem (5).
  • B. Full-stream case: d = NT and HH: The semidefinite relaxation is tight when V is square, so the relaxed problem is equivalent to problem (5).An optimal solution to the relaxed problem can therefore recover a solution to the original problem through eigen-decomposition.
  • B. Full-stream case: d = NT and HH: Problem (10) is equivalently reformulated as an explicit convex problem with a linear matrix inequality.The convex formulation can be solved using off-the-shelf tools such as CVX.
  • IV. SECURE BEAMFORMING DESIGN: KKT SOLUTION TO GENERAL CASE: For arbitrary stream counts d, the paper proposes an inexact block coordinate descent algorithm to solve problem (5).This section addresses the general case after the full-stream analysis.
  • IV. SECURE BEAMFORMING DESIGN: KKT SOLUTION TO GENERAL CASE: The IBCD framework is extended to a more general design that employs artificial noise to jam the energy harvester.The extension jointly considers beamforming and artificial-noise transmission.
  • A. Inexact Block Coordinate Algorithm For Problem (5): Problem (5) is generally harder than problem (10) because its objective function and constraint are nonconvex.The proposed approach first derives an equivalent formulation before applying inexact block coordinate descent.
  • A. Inexact Block Coordinate Algorithm For Problem (5): The reformulation extends the WMMSE idea by introducing auxiliary variables, converting rate maximization into an equivalent problem suitable for block coordinate descent.The construction relies on the facts summarized in Lemma 4.1 and the identity log det(I + AB) = log det(I + BA).

I + HEVVHHH … E HE ˜V) + Tr( ˜VHHH

The paper develops an IBCD method for problem (20) by iteratively updating variable blocks without globally solving every subproblem. It replaces the difficult nonconvex energy-harvesting update with an efficiently solvable convex approximation that preserves a non-descending objective.

  • I + HEVVHHH: Problem (20) is handled with block coordinate descent, which updates one variable or variable group while fixing the others.The standard method requires solving three or four subproblems at each iteration.
  • I U) + Tr(VHHH: The most difficult block-coordinate subproblem fixes WI, WE, and U and remains nonconvex because of the energy-harvesting constraint.Global solution requires semidefinite relaxation and rank-one reduction.
  • I U) + Tr(VHHH: Repeated semidefinite programs make standard BCD less efficient, motivating the proposed inexact block coordinate descent method for problem (20).IBCD is introduced for improved efficiency and easier implementation.
  • I U) + Tr(VHHH: IBCD updates one variable or variable group at a time, solves some subproblems inexactly, and maintains a non-descending objective function.Each iteration consists of three sub-iterations.
  • I U) + Tr(VHHH: In sub-iteration 1, IBCD optimizes U with V, WI, and WE fixed, minimizing Tr(WIE(U, V)) to obtain the optimal U.The update follows Fact 2) in Lemma 4.1.
  • I + HIVVHHH: In sub-iteration 2, IBCD updates WI and WE with U and V fixed, exploiting separability and Fact 1) in Lemma 4.1.The two weight-matrix updates are obtained separately.
  • I + HEVVHHH: In sub-iteration 3, IBCD updates V using a replacement subproblem rather than the original BCD subproblem.This update is constructed for the V block while WI, WE, and U remain fixed.
  • E HE ˜V) + Tr( ˜VHHH: The replacement V subproblem linearizes the quadratic energy-harvesting term at the previous iterate, making the problem convex and efficiently solvable.Although its solution is only feasible for the original V subproblem, it keeps problem (20)'s objective non-descending.

I U) + Tr(VHHH … B. Extension To Joint Artificial Noise and Beamforming Design

The proposed IBCD framework uses closed-form convex subproblem solutions and bisection to solve the beamforming design efficiently, with monotonic convergence to KKT or stationary solutions. It is further extended to jointly design beamforming and artificial noise for improved secrecy against the energy harvester.

  • EHE ˜V: Problem (28) is formulated as a linearly constrained convex quadratic optimization problem and solved in closed form using Lagrange multipliers.For λ > 0, its solution is summarized in Proposition 4.1 using an eigen-decomposition-based expression.
  • I UWIUHHI +: The solution uses the eigen-decomposition of HH and defines Θ(λ) ≜ P(λI + Σ)−1P^H.If the λ = 0 solution satisfies the total power constraint, the optimal λ is zero.
  • I UWI: The dual problem, equivalently problem (25), can be efficiently solved using the bisection method because the relevant derivative has an analytic form.The bisection procedure is summarized in TABLE I.
  • I UWI: Each IBCD iteration has complexity O(N^3) when NT ≥ max(NI, NE), whereas direct SDR solution of problem (21) requires at least O(d^3.5N^3.5).The proposed algorithm’s complexity is dominated by the eigen-decomposition operation.
  • I UWI: The IBCD algorithm produces a non-descending objective-value sequence, and every limit point is a KKT point of problem (20).The corresponding V* is a KKT point of problem (5), so the method monotonically converges to a stationary point.
  • I UWI: Monotonic convergence guarantees an improved objective value with arbitrary random initialization.This property is identified as attractive in the paper and is further examined numerically later.
  • B. Extension To Joint Artificial Noise and Beamforming Design: The IBCD algorithm is extended to joint artificial-noise and beamforming design, transmitting AN with covariance matrix Z to jam the energy harvester and improve secrecy rate.The transmitted signal is x ≜ Vs + n, where n denotes zero-mean artificial noise.

I + HIVVHHH … V. NUMERICAL EXAMPLES

The paper reformulates the artificial-noise secrecy-rate maximization problem and extends IBCD to jointly update beamforming and artificial-noise variables. Numerical examples evaluate the proposed algorithms under Rayleigh-fading channels using feasible warm-start initialization.

  • I + HIVVHHH: The secrecy-rate maximization problem is stated for the considered transmission setting.
  • I (I + HIZHH: An equivalent formulation is derived for the secrecy-rate maximization problem.
  • E + HEVVHHH: The artificial-noise secrecy-rate maximization problem is shown to be equivalent to a reformulated optimization problem.
  • E + HEVVHHH: IBCD generalizes to the artificial-noise case by alternately updating auxiliary variables in closed form and beamforming variables after linearizing the energy-harvesting constraint.
  • HE(VVH + VEVH: The generalized algorithm uses bisection to solve the resulting beamforming subproblem.
  • E + HEVVHHH: The generalized IBCD algorithm monotonically converges to a KKT point of problem (32).
  • V. NUMERICAL EXAMPLES: The numerical examples illustrate the performance of the proposed beamforming algorithms under 50dB signal attenuation and approximately 5 meters of transmitter-receiver distance.
  • V. NUMERICAL EXAMPLES: Channels are generated using i.i.d. Rayleigh fading with average power 1e-5, while warm-start procedures obtain feasible initial points for IBCD and artificial-noise beamforming.

A. Convergence performance · B. Secrecy rate performance · VI. CONCLUSIONS

The IBCD algorithms converge reliably across the studied nonconvex beamforming problems, while artificial-noise-aided designs improve secrecy rates. The paper provides global solutions for special cases and an iterative KKT-convergent method for general stream configurations and joint artificial-noise design.

  • A. Convergence performance: A. Convergence performance: IBCD converges to the global optimum irrespective of initialization in the single-stream case.The comparison uses the SDR-derived optimal value under PT = 10dBm, PE = −40dBm, and NT = 4.
  • A. Convergence performance: A. Convergence performance: IBCD also achieves global convergence in the full-stream case with a positive-semidefinite channel Gram-matrix difference.The optimal value is obtained using the proposed full-stream method.
  • A. Convergence performance: A. Convergence performance: The generalized IBCD algorithm reaches the same objective value regardless of initialization in the artificial-noise problem.The example uses PT = 15dBm, PE = −35dBm, and NT = 4.
  • A. Convergence performance: A. Convergence performance: Numerical examples indicate good IBCD convergence despite the high nonconvexity of both optimization problems.This summary covers both the secrecy-rate maximization and joint artificial-noise design problems.
  • B. Secrecy rate performance: B. Secrecy rate performance: Achieved secrecy rate increases with total transmission power, and artificial noise provides better secrecy-rate performance.The comparison fixes PE = −30dBm and averages each data point over 100 random channel realizations.
  • B. Secrecy rate performance: B. Secrecy rate performance: Secrecy rate decreases as the harvested-power target increases, while AN-aided beamforming outperforms beamforming without artificial noise.This experiment fixes PT = 25dBm and NT = 4.
  • VI. CONCLUSIONS: VI. CONCLUSIONS: The paper offers global beamforming solutions for single-stream and suitable full-stream cases, plus IBCD for arbitrary streams and joint artificial-noise design.The IBCD algorithm has monotonic convergence, and every limit point is a KKT solution to the secrecy-rate maximization problem; simulations show improved secrecy rate with artificial noise.

APPENDIX A … I + HEXHH

The appendices establish equivalences among reformulated optimization problems, using variable substitution, auxiliary-variable elimination, determinant identities, and linear-matrix-inequality reformulation. They conclude that the transformed problems preserve feasible sets, optimal values, or optimal solutions under the stated constructions.

  • THE PROOF OF LEMMA 3.1: Variable substitution v = u reformulates problem (41) as problem (7).
  • THE PROOF OF LEMMA 3.1: Eliminating t from problem (42) and combining its constraints yields problem (43).
  • THE PROOF OF LEMMA 3.1: Problems (42) and (43) have the same feasible solution set regarding u and therefore the same optimal solution set.
  • THE PROOF OF LEMMA 3.1: Restricting u^H I u to 1 rewrites problem (43) equivalently as problem (7).
  • THE PROOF OF LEMMA 3.1: The optimal solution u* of problem (7) is also optimal for problem (42), and the associated v* is optimal for problem (6).
  • I + HIXHH: The subsection I + HIXHH uses determinant manipulations involving I + F(I + XHH.
  • I + HEXHH: Using det(I + AB) = det(I + BA), the objective of problem (10) is rewritten with auxiliary variable Y as problem (45).
  • I + HEXHH: Problems (45) and (46) are equivalent because the relaxation preserves the optimal value under determinant monotonicity and feasibility.

E HEX … THE PROOF OF PROPOSITION 4.2

The appendix transforms a constraint using matrix identities and the Schur complement, then concludes convexity after replacing the original constraint with an LMI. It also establishes notation for analyzing the IBCD algorithm’s iterates, solution sets, constraint sets, and objective function.

  • E HEX: A matrix identity is applied to the right-hand side of an earlier equation to derive an equivalent constraint representation.The derivation uses (I + AB)−1A = A(I + BA)−1.
  • I + HEXHH: The first constraint of problem (46) is shown to be equivalent to a transformed expression.This equivalence is stated as an intermediate step before the Schur-complement reformulation.
  • I + HEXHH: The transformed constraint is rewritten as an LMI using the Schur complement.The Schur complement is cited as the basis for the equivalent formulation.
  • I + HEXHH: Replacing the first constraint in problem (46) with the derived LMI yields a convex problem.The proof notes convexity after the constraint replacement.
  • THE PROOF OF PROPOSITION 4.2: Problem (25) is denoted by P(˜V, U, WI, WE), with S(˜V, U, WI, WE) denoting its solution set.The proof introduces this notation for the subsequent analysis.
  • THE PROOF OF PROPOSITION 4.2: The proof defines C≤(˜V) as the constraint set and considers iterates generated by the IBCD algorithm in Table II.The iterates include Vk, Uk, and Wk, with updates obtained through specified algorithm steps.
  • THE PROOF OF PROPOSITION 4.2: The objective function of problem (25) is denoted by f(V, U, WI, WE).This notation is introduced alongside the problem’s solution and constraint sets.

E HE ˜V) + Tr( ˜VHHH

The proof establishes that the IBCD iterates remain feasible, their objective values converge monotonically, and every limit point satisfies the KKT conditions of the original problem.

  • Feasibility: Each iterate V_k remains feasible for problem (20) whenever V_k is feasible.The proof establishes feasibility inductively through the successive subproblems.
  • Monotonic convergence: The objective value sequence {C(V_k)} converges monotonically because it is nondecreasing and upper bounded.The upper bound follows from compactness of the iterate set and continuity of C(V).
  • KKT convergence: Every limit point (V*, U*, W*) of the iterates is a KKT point of problem (20).The argument first verifies the KKT condition for V* and then extends it to the complete limit point using Slater’s condition.
  • KKT convergence: The KKT conditions of the surrogate problem reduce to the KKT conditions of problem (5), proving that V* is a KKT point of the original problem.This reduction uses the stated matrix identity and substitutions into the first-order and feasibility conditions.
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