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Wireless Information and Energy Transfer in Multi-Antenna Interference Channel

Chao Shen, Wei-Chiang Li, Tsung-Hui Chang

arXiv:1308.2838v1cs.ITmath.IT

TL;DR

The paper studies transmitter design for wireless information and energy transfer in a multi-user MISO interference channel with energy-harvesting and power constraints. It proposes ideal, time-splitting, and power-splitting schemes; simulations show stronger cross-link power can improve sum rate, while no scheme always dominates.

  • Problem

    WIET transmitter design must maximize weighted sum rate under individual energy-harvesting and power constraints, while cross-link signals affect information detection and energy harvesting in opposite ways.

    Source: §I · p. 2

  • Method

    The paper proposes TDMS, TDMA, and PS schemes, using successive convex approximation for the K-user ideal and PS problems and convex formulations for TDMS and TDMA.

    Source: §VII · p. 10

  • Results

    Stronger cross-link channel power can improve achievable sum rate under energy-harvesting constraints, and none of the considered schemes always dominates the others.

    Source: §VII · p. 10

  • Takeaways & Limitations

    Cross-link signals can degrade information sum rate while boosting receiver energy harvesting, creating an information–energy tradeoff in the considered interference channel.

    Source: §I · p. 2

Abstract

from arXiv · show

This paper considers the transmitter design for wireless information and energy transfer (WIET) in a multiple-input single-output (MISO) interference channel (IFC). The design problem is to maximize the system throughput (i.e., the weighted sum rate) subject to individual energy harvesting constraints and power constraints. Different from the conventional IFCs without energy harvesting, the cross-link signals in the considered scenario play two opposite roles in information detection (ID) and energy harvesting (EH). It is observed that the ideal scheme, where the receivers can simultaneously perform ID and EH from the received signal, may not always achieve the best tradeoff between information transfer and energy harvesting, but simple practical schemes based on time splitting may perform better. We therefore propose two practical time splitting schemes, namely time division mode switching (TDMS) and time division multiple access (TDMA), in addition to a power splitting (PS) scheme which separates the received signal into two parts for ID and EH, respectively. In the two-user scenario, we show that beamforming is optimal to all the schemes. Moreover, the design problems associated with the TDMS and TDMA schemes admit semi-analytical solutions. In the general K-user scenario, a successive convex approximation method is proposed to handle the WIET problems associated with the ideal scheme and the PS scheme, which are known to be NP-hard in general. The K-user TDMS and TDMA schemes are shown efficiently solvable as convex problems. Simulation results show that stronger cross-link channel powers actually improve the information sum rate under energy harvesting constraints. Moreover, none of the schemes under consideration can dominate another in terms of the sum rate performance.

I. INTRODUCTION

The paper studies wireless information and energy transfer in a MISO interference channel, where cross-link signals can both degrade information rates and boost harvested energy. It formulates practical transmission schemes and shows why time sharing can improve on simultaneous information detection and energy harvesting in some two-user settings.

I. INTRODUCTION

  • The paper proposes TDMS and TDMA time-splitting schemes alongside power splitting because the ideal simultaneous ID/EH scheme may not maximize sum rate in complex interference settings.TDMS assigns all receivers EH and then ID slots; TDMA assigns one receiver ID while others harvest energy in each slot.Source: §I · p. 2, §III · p. 4
  • Stronger cross-link signals can boost energy harvesting while degrading information rates, creating a tradeoff that motivates transmission strategies balancing both objectives.The design maximizes weighted sum rate subject to individual energy-harvesting and power constraints.Source: §I · p. 2, §III · pp. 3–4

II. SIGNAL MODEL AND PROBLEM STATEMENT

  • The K-user MISO interference-channel model seeks transmission strategies that maximize information throughput while meeting each receiver’s energy-harvesting requirement.Receivers operate in information-detection or energy-harvesting modes; the study considers an ideal simultaneous-mode case as well as practical mode-separated schemes.Source: §II · pp. 2–3

III. OPTIMAL WIET DESIGN FOR IDEAL SCHEME

  • For two users, beamforming is optimal for the ideal scheme, with each transmitter’s beam direction in the span of its direct- and cross-link channels.The beamforming coefficients must account for both energy harvesting and information transfer.Source: §III · p. 4
  • Time sharing between EH and ID modes or between users can yield a higher information sum rate than the ideal scheme in some two-user cases.The paper’s examples motivate the practical TDMS and TDMA schemes, whose receivers perform only one mode at a time.Source: §III · p. 4

A. Time Division Mode Switching (TDMS) Scheme

TDMS separates energy harvesting and information detection into two shared time slots. Its energy-harvesting phase minimizes the time needed to meet both users’ requirements, while its design admits a semi-analytical solution.

A. Time Division Mode Switching (TDMS) Scheme

  • TDMS assigns one shared slot to energy harvesting and the remaining slot to information detection for both receivers.The ID slot maximizes information throughput; the EH slot fulfills both energy requirements.Source: A. Time Division Mode Switching (TDMS)… · pp. 4–5
  • The EH phase minimizes its time fraction, and the scheme is infeasible if requirements cannot be met even when the entire interval is devoted to harvesting.The optimization uses β = 1/α; an optimal β below one implies α > 1 and infeasibility.Source: A. Time Division Mode Switching (TDMS)… · p. 5
  • The EH design problem admits a semi-analytical solution under the stated linear-independence and nonorthogonality assumptions on each user’s channels.The solution uses a dual variable and principal eigenvectors; the optimal dual variable can be found by bisection.Source: A. Time Division Mode Switching (TDMS)… · p. 5
  • Beamforming is optimal for the EH mode of TDMS.The stated channel assumptions hold with probability one for random continuous fading channels.Source: A. Time Division Mode Switching (TDMS)… · p. 5

B. TDMA Scheme

The two-user TDMA scheme alternates which receiver detects information and which harvests energy. For feasible time allocations, its design admits semi-analytical solutions, and the best allocation can be selected by line search.

B. TDMA Scheme

  • Each TDMA slot serves one receiver in ID mode and the other in EH mode, with the roles exchanged in the second slot.The first slot maximizes one user’s information rate while satisfying the other user’s harvesting requirement.Source: B. TDMA Scheme · p. 5
  • TDMA may be of interest when users have asymmetric energy requirements and cross-link powers, and may outperform spectrum sharing in interference-dominated scenarios.These are stated as conditions under which the scheme may be attractive, not guaranteed outcomes.Source: B. TDMA Scheme · p. 5
  • The TDMA scheme is feasible exactly when its two slot-specific design problems can both be satisfied by a common feasible time allocation.The resulting feasible allocation lies in an interval determined by the two users’ energy requirements and channel powers.Source: B. TDMA Scheme · p. 5
  • For a feasible time fraction, both slot designs can be solved semi-analytically, and line search selects the fraction maximizing the two-user sum rate.Beamforming is optimal for the TDMA scheme.Source: B. TDMA Scheme · pp. 5–6

C. TDMA via Deterministic Signal for Energy Harvesting

The paper extends practical WIET designs to K users and develops an SCA approach for the ideal-receiver problem, which is NP-hard in general. Its deterministic-signal TDMA variant allows receivers to remove cross-link interference, subject to channel-knowledge requirements.

C. TDMA via Deterministic Signal for Energy Harvesting

  • Deterministic signals from a transmitter in EH mode can be removed by the other receiver before information detection, eliminating that cross-link interference.The receivers need perfect knowledge of the cross-link channels; imperfect knowledge may degrade performance through imperfect cancellation.Source: C. TDMA via Deterministic Signal for… · p. 6

V. WIET DESIGN FOR K-USER MISO IFC

A. Transmitter Optimization for Ideal Receivers

  • The K-user ideal-receiver WIET problem is NP-hard in general, motivating an efficient approximate solution.The problem includes individual energy requirements for users.Source: A. Transmitter Optimization for Ideal… · p. 6
  • The proposed ideal-receiver method uses log-exponential reformulation and successive convex approximation, with a convex SDP solved at each iteration.The iterative approximation conservatively replaces a nonconvex constraint; the resulting algorithm can converge to a stationary point.Source: A. Transmitter Optimization for Ideal… · pp. 6–7

B. Practical K-User WIET Schemes

The K-user TDMS and TDMA designs extend the two-user time-sharing schemes by optimizing signal covariances and time fractions. The TDMA formulation is convex, while the TDMS ID phase retains a classical sum-rate problem.

B. Practical K-User WIET Schemes

  • K-user TDMS uses a shared EH slot followed by a shared ID slot, with the EH allocation obtained from a convex problem.The subsequent ID phase is the classical sum-rate maximization problem and can be handled by existing methods or block coordinate descent methods.Source: B. Practical K-User WIET Schemes · p. 7
  • K-user TDMA divides transmission into K slots, each assigning one user to ID and the others to EH, with deterministic EH signals enabling interference removal.The design jointly optimizes slot covariances and time fractions under energy-harvesting constraints.Source: B. Practical K-User WIET Schemes · p. 7
  • The K-user TDMA design problem is a convex optimization problem because its constraints are linear and its objective is concave.The formulation includes optimization over the time fractions as well as signal covariance matrices.Source: B. Practical K-User WIET Schemes · pp. 7–8

C. Practical Scheme by Power Splitting

The power-splitting (PS) scheme divides received-signal power between information detection and energy harvesting, while jointly designing transmit covariances and receiver splitting fractions. For two users beamforming is optimal; for general K, the design is handled with successive convex approximation.

C. Practical Scheme by Power Splitting

  • PS assigns fraction ρ_i of received-signal power to information detection and 1−ρ_i to energy harvesting at receiver i.The design optimizes both transmit signal covariance matrices and receiver power-splitting fractions.Source: C. Practical Scheme by Power Splitting · p. 8
  • For K = 2, transmit beamforming is optimal for the PS design problem.Source: C. Practical Scheme by Power Splitting · p. 8
  • For general K, introducing θ_i = 1/ρ_i and applying log-exponential reformulation with successive convex approximation handles the PS design problem.The transformed energy-harvesting constraints are convex; the remaining nonconvexity is mainly due to the sum-rate function.Source: C. Practical Scheme by Power Splitting · p. 8

VI. SIMULATION RESULTS AND DISCUSSIONS

The simulations compare WIET schemes across cross-link powers, energy-harvesting requirements, and user settings. Their relative sum-rate performance depends on these conditions: stronger cross-links can improve rates, and no scheme always dominates.

VI. SIMULATION RESULTS AND DISCUSSIONS

  • Simulations use unit transmitter power, receiver noise power 0.1, and 500 channel realizations; η sets relative cross-link channel power.Source: §VI · p. 8
  • In the K = 2, N_t = 4 simulations, all schemes’ sum rates rise with η; PS leads TDMS for η ≤ 2.2, while TDMS can lead for η > 2.2.The ideal, TDMS, and PS schemes outperform TDMA and TDMA (D) in this setting.Source: §VI · p. 9, Figure 4 · p. 9
  • For symmetric energy requirements with η = 4, sum rate falls as E rises; the ideal scheme performs best for E ≤ 2, but TDMS leads for E > 2.With asymmetric requirements and E_1 = 2, TDMA and TDMA (D) can perform best when E_2 is large.Source: §VI · p. 9, Figure 5 · p. 9
  • For K = 4 and N_t = 2, TDMA (D) outperforms TDMS and PS for E ≥ 1.3 at η = 1, and yields the highest sum rates for E ≥ 2 at η = 4.At η = 4, TDMS also performs better than the ideal and PS schemes when E ≥ 1.8.Source: §VI · p. 9, Figure 6 · p. 10

VII. CONCLUSIONS AND FUTURE WORKS

  • The study presents ideal and practical schemes, with two-user beamforming optimality and general-K SCA methods for ideal and PS designs.The TDMA (D) design is solved as a convex problem.Source: §VII · p. 10
  • Stronger cross-link power can improve sum rate under energy-harvesting constraints, and none of the considered schemes always dominates.Source: §VII · p. 10
  • With N_t ≥ K, PS tends to perform better when η and E are not large, while TDMS generally performs best when both are large and can surpass the ideal scheme.Source: §VII · p. 10
  • When N_t < K and E is large, TDMA can generally yield the highest sum rate.Source: §VII · p. 10

APPENDIX

The proof establishes that an optimal two-user solution can be restricted to the channel vectors’ range spaces and has rank-one transmit covariances. It therefore supports beamforming as an optimal structure.

A. Proof of Proposition 1

  • An optimal covariance lies in the range space spanned by that user’s direct- and cross-link channel vectors.Projecting a covariance into this range preserves optimality; the proof argues that an optimal solution must lie there.Source: A. Proof of Proposition 1 · p. 10
  • The optimal transmit covariances can be chosen rank one, with each beamforming vector formed from the user’s direct- and cross-link channel vectors.The proof uses the effective constraint count to establish rank at most one, then writes the covariance as an outer product.Source: A. Proof of Proposition 1 · p. 11
  • The rank-one structure follows by reducing the effective number of inequality constraints in the auxiliary problem to three.The proof invokes [30, Theorem 3.2] for the resulting rank bound.Source: A. Proof of Proposition 1 · p. 11

B. Proof of Proposition 2

The proof derives a semi-analytical solution by solving a Lagrange dual problem and constructing transmit covariances from principal eigenvectors. Under the stated channel assumptions, those eigenvectors are unique, and the dual variable can be found by bisection.

B. Proof of Proposition 2

  • Given the optimal transmit covariances, the optimal β is specified by equation (13b).The proof obtains β after determining the optimal S1 and S2.Source: B. Proof of Proposition 2 · p. 11
  • Because problem (12) satisfies Slater’s condition, it can be solved by handling its Lagrange dual; the dual variables µ ≥ 0 and η ≥ 0 correspond to constraints (12b) and (12c), respectively.Source: B. Proof of Proposition 2 · p. 11
  • The optimal transmit covariances are constructed from principal eigenvectors of Ψ_i(µ), with the optimal dual variable obtainable by bisection.Under linearly independent, nonorthogonal channel vectors, the principal eigenvectors are unique.Source: B. Proof of Proposition 2 · pp. 11–12

C. Proof of Proposition 3

The proof converts a quasi-convex design problem into an equivalent convex semidefinite program using the Charnes-Cooper transformation. It then reduces the remaining search to a scalar variable that can be optimized by bisection.

C. Proof of Proposition 3

  • The Charnes-Cooper transformation recasts problem (14) as a convex semidefinite program, equivalent to the original design problem.The proof establishes correspondence between feasible solutions and preservation of the objective value.Source: C. Proof of Proposition 3 · p. 12
  • For a feasible scalar y, the associated optimal X1(y) and X2(y) are obtained by solving their respective subproblems.The proof uses equality of constraint (A.14b) at the optimum to characterize the subproblems.Source: C. Proof of Proposition 3 · p. 12
  • The optimal scalar y is found through a one-dimensional optimization that can be solved by bisection.The proof establishes concavity of h11^H X1(y) h11 in y.Source: C. Proof of Proposition 3 · p. 12

D. Proof of Proposition 4

The proof shows that Algorithm 1 is a successive upper-bound minimization (SUM) method. Its limit points are stationary points of the original problem.

D. Proof of Proposition 4

  • At the optimum, inequalities (26b) and (26c) hold with equality, enabling an equivalent expression of problem (26).The proof uses this equivalence to identify Algorithm 1 with the SUM method.Source: D. Proof of Proposition 4 · p. 13
  • The exponential approximation used by the algorithm provides a universal lower bound on the original objective function.The proof also verifies equality of the bound and objective at the current iterate.Source: D. Proof of Proposition 4 · p. 13
  • Any limit point generated by Algorithm 1 is a stationary point of the original problem.The proof identifies Algorithm 1 as a SUM method and applies the stated convergence property.Source: D. Proof of Proposition 4 · p. 13

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