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Wireless Information and Power Transfer: A Dynamic Power Splitting Approach

Liang Liu, Rui Zhang, Kee-Chaing Chua

arXiv:1302.0585v2cs.IT

TL;DR

The paper addresses wireless information and energy transfer when a receiver cannot independently decode and harvest from the same signal. It proposes dynamic power splitting, jointly considers transmitter power control with CSIT, and extends the approach to SIMO systems through uniform splitting and antenna switching.

  • Problem

    A SISO receiver cannot independently decode information and harvest energy from the same received signal, motivating rate-energy trade-off analysis under wireless energy harvesting.

  • Method

    The paper derives CSI-based dynamic receiver power splitting, jointly optimizes transmitter power control when CSIT is available, and extends the design to SIMO systems using uniform power splitting and antenna switching.

  • Results

    The derived DPS rules characterize rate-energy trade-offs; uniform power splitting is optimal for SIMO, while antenna switching approaches optimal UPS as receiving antennas increase.

  • Takeaways & Limitations

    DPS uses favorable fading states for both information decoding and energy harvesting, while antenna switching offers a lower-complexity practical implementation.

Abstract

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Scavenging energy from ambient radio signals, namely wireless energy harvesting (WEH), has recently drawn significant attention. In this paper, we consider a point-to-point wireless link over the flat-fading channel, where the receiver replenishes energy via WEH from the signals sent by the transmitter. We consider a SISO (single-input single-output) system where the single-antenna receiver cannot decode information and harvest energy independently from the same signal received. Under this practical constraint, we propose a dynamic power splitting (DPS) scheme, where the received signal is split into two streams with adjustable power levels for information decoding and energy harvesting separately based on the instantaneous channel condition that is assumed to be known at the receiver. We derive the optimal power splitting rule at the receiver to achieve various trade-offs between the maximum ergodic capacity for information transfer and the maximum average harvested energy for power transfer. Moreover, for the case when the channel state information is also known at the transmitter, we investigate the joint optimization of transmitter power control and receiver power splitting. Finally, we extend the result for DPS to the SIMO (single-input multiple-output) system and investigate a low-complexity power splitting scheme termed antenna switching.

I. INTRODUCTION

The paper develops dynamic power splitting for wireless information and power transfer under practical receiver constraints, then extends the approach from SISO to SIMO systems. It characterizes rate-energy trade-offs with and without transmitter CSI and examines lower-complexity antenna switching.

  • Motivation: Wireless energy harvesting can support information transfer but practical receivers cannot independently decode and harvest from the same received signal.The paper addresses this receiver constraint rather than the idealized independent-processing assumption.
  • SISO approach: Dynamic power splitting adjusts the received-signal allocation between information decoding and energy harvesting according to the instantaneous channel state.The receiver knows the channel state; transmitter CSI may be unavailable or available for joint power-control optimization.
  • SISO results: For SISO fading channels, optimal DPS allocates a fixed received-power amount to information decoding and the remainder to harvesting above a channel-gain threshold, while allocating all power to decoding below it.This rule applies with or without CSIT and uses good fading states for both information and energy transfer.
  • SIMO extension: For SIMO systems, uniform power splitting is optimal, while antenna switching provides a lower-complexity alternative whose performance approaches optimal UPS as antennas increase.With two receiving antennas, optimal antenna switching is already close to optimal UPS in R-E performance.
  • Optimization framework: The paper formulates rate-energy regions and uses Lagrangian duality to solve the transmitter-power-control and receiver-splitting optimization globally despite general non-convexity.The stated zero-duality-gap property supports global optimization of the CSIT case.
  • SISO results: DPS achieves substantially improved rate-energy trade-offs over dynamic time switching and is compared with an ideal receiver upper bound.At 90% of maximum harvested energy, ergodic capacity increases by 64% with CSIT and 120% without CSIT versus time switching.

IV. OPTIMAL POLICY FOR THE SISO FADING CHANNEL

The paper studies optimal receiver power splitting and, when CSIT is available, transmitter power control for rate-energy trade-offs in SISO fading channels.

  • The SISO fading-channel analysis optimizes receiver power splitting with or without CSIT, and jointly optimizes transmitter power control when CSIT is available.The objective is to achieve various rate-energy trade-offs.

A. The Case Without CSIT

Without CSIT, the optimal dynamic power-splitting policy depends on the fading state: weak states favor information decoding, while strong states split received power between information and energy.

  • The optimization is decomposed into one subproblem per fading state and solved iteratively by updating the dual variable through bisection.The harvested-energy constraint is met with equality at the solution.
  • The optimal solution is characterized by a threshold rule that allocates a fixed information-receiver power above the threshold and all received power to information below it.The fixed allocation is constant across fading states above the threshold.
  • For poor fading states, both time switching and power splitting allocate all received power to information decoding.For power splitting, this applies when 0 < h(ν) ≤ 1/P.
  • For good fading states, power splitting allocates a constant received-power amount to information decoding and the remainder to energy harvesting.The information allocation is 1/λ_PS−σ^2, while the remaining received power goes to energy harvesting.
  • Power splitting lets both information decoding and energy harvesting benefit from good fading states, whereas time switching benefits only energy harvesting there.This is the principal distinction between the optimal policies without CSIT.

B. The Case With CSIT

With CSIT, transmitter power control is jointly optimized with receiver power splitting. Poor states are suppressed, moderate states use water-filling, and good states transmit at peak power while power splitting preserves information decoding.

  • The CSIT case jointly optimizes transmit power p(ν) and receiver splitting α(ν) for every fading state.The optimization is solved through per-state subproblems and dual-variable updates using the ellipsoid method.
  • In poor fading states, both power splitting and time switching turn off transmission to save transmit power.This occurs below the respective threshold β_PSσ^2 or β_TSσ^2.
  • In moderate fading states, both schemes transmit information using water-filling power allocation capped by P_peak and allocate all received power to information decoding.The moderate-state range differs between power splitting and time switching.
  • In good fading states, both schemes transmit at peak power P_peak, but only power splitting sends a constant received-power amount to information decoding.Time switching sends all received power to energy harvesting, whereas power splitting allocates the remainder there.
  • As without CSIT, power splitting allows both information decoding and energy harvesting to benefit from good fading states, unlike time switching.The difference concerns receiver resource allocation in good fading states.

C. Performance Upper Bound

The paper derives an ideal-receiver upper bound for the rate–energy region and characterizes how transmitter power allocation creates information–energy trade-offs with CSIT.

  • Performance Upper Bound: An ideal receiver that independently decodes and harvests from the same signal provides an upper bound for DPS and other practical receiver designs.The bound sets the information and energy splitting ratios simultaneously to their respective extremes.
  • Performance Upper Bound: Without CSIT, constant transmit power eliminates the information–energy trade-off, so the upper-bound rate–energy region is a box.Here p(ν) = P for every fading state.
  • Performance Upper Bound: Fig. 5 compares transmit power allocation and received power allocation for power splitting and time switching across fading states with CSIT.The comparison focuses on how the two receiver designs allocate resources over different channel conditions.
  • Performance Upper Bound: With CSIT, transmitter power allocation creates a trade-off between per-state information rate r(ν) and harvested energy Q(ν).The optimal allocation is obtained by solving the constrained optimization problem using dual variables for harvested energy and average power.

V. EXTENSION AND APPLICATION: DYNAMIC POWER SPLITTING FOR THE SIMO FADING CHANNEL

The paper extends dynamic power splitting to SIMO fading channels, models antenna-level information and energy allocation, and proves that uniform power splitting is optimal through an equivalent virtual-antenna formulation.

  • SIMO Extension: The SIMO extension studies dynamic power splitting and antenna switching when the receiver has multiple antennas.The achievable rate uses maximal ratio combining, while harvested energy aggregates contributions from the receiving antennas.
  • SIMO Extension: Each receiving antenna can independently split its received power between information decoding and energy harvesting at each fading state.The antenna-specific information fractions satisfy 0 ≤ α_m(ν) ≤ 1, with the remaining fraction sent to the energy receiver.
  • Without CSIT: Uniform power splitting is optimal for the SIMO fading channel without CSIT, so all receiving antennas use the same splitting ratio.The proof establishes equivalence between the SIMO optimization and a SISO problem based on an equivalent channel sum-power.
  • SIMO Extension: The SIMO and SISO optimal DPS policies are equivalent when all receiving antennas are treated as one virtual antenna.This equivalence follows from the corresponding optimization problems having the same optimal value.

2) The Case With CSIT:

For SIMO channels with CSIT, the paper preserves the uniform power-splitting and virtual-antenna structure, then considers antenna switching as a lower-complexity alternative to multiple power splitters.

  • The Case With CSIT: With CSIT, the SIMO optimization is decoupled into per-fading-state subproblems involving transmitter power control and receiver power splitting.The formulation introduces dual variables for harvested energy and average transmit power.
  • The Case With CSIT: Uniform power splitting remains optimal with CSIT, and the SIMO problem is equivalent to the SISO problem under a virtual-antenna representation.The optimal transmitter power and common receiver splitting ratio can therefore be obtained from the SISO result.
  • Antenna Switching: Optimal uniform power splitting requires multiple antenna-connected power splitters, which may be costly to implement.This hardware requirement motivates the study of antenna switching.
  • Antenna Switching: Antenna switching assigns one subset of antennas to information decoding and the remaining subset to energy harvesting at each fading state.It uses time switchers at the antennas instead of the more costly power splitters required by uniform power splitting.

1) The Case Without CSIT:

For SIMO without CSIT, the paper develops optimal and low-complexity antenna-switching policies to approximate uniform power splitting while reducing implementation complexity.

  • Optimal Antenna Switching: Antenna switching can be optimized by searching over 2^M antenna combinations at each fading state.This exhaustive search maximizes the resulting rate-energy objective for the selected antenna partition.
  • Low-Complexity Antenna Switching: The proposed algorithm instead approximates the optimal uniform power-splitting policy by efficiently selecting information-decoding and energy-harvesting antenna subsets.It first solves the equivalent SISO uniform power-splitting problem, then finds antenna partitions close to that solution.
  • Performance and Complexity: For any positive ϵ, the algorithm provides a solution with arbitrarily high accuracy and worst-case complexity O(M^2).Setting ϵ to zero recovers exhaustive search with complexity O(2^M).

2) The Case With CSIT:

With CSIT, the paper combines transmitter power control with receiver power splitting and evaluates optimal and low-complexity antenna switching for SIMO channels.

  • Optimal Uniform Power Splitting: The optimal SIMO uniform power-splitting policy is obtained by treating all receiving antennas as one virtual antenna with an equivalent channel sum-power.The same SISO-derived rule determines receiver splitting and, with CSIT, transmitter power control.
  • Numerical Results: The numerical comparison includes optimal dynamic power splitting, exhaustive-search antenna switching, and low-complexity antenna switching.The schemes are evaluated for SIMO systems without and with CSIT.
  • Numerical Results: Two receiving antennas significantly enlarge the achievable rate-energy region compared with the SISO case, even with low-complexity antenna switching.As M increases, exhaustive-search antenna switching approaches optimal uniform power splitting.

APPENDIX

The appendix establishes structural properties of optimal solutions by comparing feasible policies over infinitesimal channel intervals and analyzing their Lagrangian objectives.

  • Proof Structure: Within an infinitesimal channel interval, optimal power-splitting and power-control policies can be taken as constant under shared KKT conditions.The channel gain and its density are treated as constant over the interval.
  • Proof Structure: Combining two constant policies across subintervals preserves the relevant average constraints while producing the required averaged harvested-energy and rate relations.The constructed policy divides the interval according to a fraction θ and assigns each original policy to one subinterval.
  • Proof Structure: The appendix concludes the intermediate lemma after showing the constructed policy satisfies the average-power constraint and the desired energy-rate inequalities.This supports the structural reduction used in the optimization analysis.

C. Proof of Proposition 4.2

The proof of Proposition 4.2 partitions the CSIT optimization into channel-dependent subproblems and compares their optimal values to derive the transmitter power and receiver splitting choices.

  • Feasibility Boundary: The proof also identifies a parameter range in which the derivative with respect to α is nonpositive for every feasible p, making the optimization problem infeasible under the stated constraint.This range is excluded from the subsequent analysis.
  • Subproblem Decomposition: For a fixed channel power h, the optimal transmitter power is found by comparing two subproblems corresponding to different power-splitting regimes.The feasible sets S1 and S2 depend on the relationship between h and the relevant thresholds.
  • Case Analysis: When the channel falls into one regime, the optimal solution uses peak transmitter power with full information allocation, p*=Ppeak and α*=1.The proof establishes this by showing the second subproblem has the larger optimal value.
  • Case Analysis: In another regime, the first subproblem dominates and the optimal transmitter power is selected according to the channel-dependent expression derived in the proof.The associated receiver splitting ratio remains α*=1 in the stated case.
  • Case Analysis: For sufficiently weak channels, the feasible second regime is empty and the optimal power-splitting ratio is α*=1.The transmitter power is then determined solely by the first subproblem.

D. Proof of Proposition 5.1

The proof bounds the per-iteration multiplicative error and uses the growth of the sets S_i to establish the algorithm’s worst-case complexity.

  • Each iteration introduces a multiplicative error factor of at most ǫ.
  • The first and second parts of Proposition 5.1 are concluded after establishing the respective bounds and complexity result.
  • The proof tracks the smallest positive element s_min in each set S_i to bound set-size growth across iterations.
  • The sets S_i grow linearly with M, and at most M iterations yield worst-case complexity O(M^2).
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