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On Capacity-Achieving Distributions for Complex AWGN Channels Under Nonlinear Power Constraints and their Applications to SWIPT

Morteza Varasteh, Borzoo Rassouli, Bruno Clerckx

arXiv:1712.01226v3cs.IT

TL;DR

The paper studies capacity and SWIPT design for complex AWGN channels when delivered power is constrained through nonlinear even-moment effects. It derives capacity results and two SWIPT rate-power inner bounds, finding that Gaussian/OOK signaling and restricted input distributions can outperform Gaussian-only designs within the stated settings.

  • Problem

    Existing AWGN analyses largely omit nonlinearities in devices such as energy harvesters, limiting fundamental results for nonlinear power constraints and SWIPT.

  • Method

    The paper models delivered power as a linear combination of even input moments, analyzes capacity under average-power, amplitude, and delivered-power constraints, and derives two SWIPT rate-power inner bounds.

  • Results

    The constrained capacity equals ordinary AWGN capacity; depending on the constraints it is achieved by Gaussian input or approached through Gaussian/OOK time sharing, while nonlinear SWIPT designs yield larger restricted-input rate-power regions than Gaussian counterparts.

  • Takeaways & Limitations

    Nonlinear rectenna behavior makes higher-order input moments and signaling distributions central to SWIPT design, with asymmetric Gaussian power allocation producing an information-power tradeoff.

  • Takeaways & Limitations

    The analysis imposes iid channel-input samples; allowing correlation among samples makes the problem cumbersome, although correlation appears beneficial for power harvesting.

Abstract

from arXiv · show

The capacity of a complex and discrete-time memoryless additive white Gaussian noise (AWGN) channel under three constraints, namely, input average power, input amplitude and output delivered power is studied. The output delivered power constraint is modelled as the average of linear combination of even moments of the channel input being larger than a threshold. It is shown that the capacity of an AWGN channel under transmit average power and receiver delivered power constraints is the same as the capacity of an AWGN channel under an average power constraint. However, depending on the two constraints, the capacity can be either achieved by a Gaussian distribution or arbitrarily approached by using time-sharing between a Gaussian distribution and On-Off Keying. As an application, a simultaneous wireless information and power transfer (SWIPT) problem is studied, where an experimentally-validated nonlinear model of the harvester is used. It is shown that the delivered power depends on higher order moments of the channel input. Two inner bounds, one based on complex Gaussian inputs and the other based on further restricting the delivered power are obtained for the Rate-Power (RP) region. For Gaussian inputs, the optimal inputs are zero mean and a tradeoff between transmitted information and delivered power is recognized by considering asymmetric power allocations between inphase and quadrature subchannels. Through numerical algorithms, it is observed that input distributions (obtained by restricting the delivered power) attain larger RP region compared to Gaussian input counterparts.

I. INTRODUCTION

The paper addresses missing nonlinear AWGN capacity results by modeling harvested power through even moments and applies them to nonlinear SWIPT design. It shows that nonlinear rectenna behavior changes optimal signaling, capacity attainment, and the achievable rate-power tradeoff.

  • Motivation: Nonlinear AWGN models are underrepresented, while common analyses linearize nonlinear effects or assume Gaussian statistics for approximations and bounds.This gap is increasingly relevant for devices with nonlinear responses, including power amplifiers and energy harvesters.
  • Problem formulation: The paper studies a complex discrete-time memoryless AWGN channel with transmit average-power, amplitude, and receiver delivered-power constraints.Delivered power is modeled as an average linear combination of even moments of the channel input.
  • Main results: The constrained capacity remains equal to ordinary AWGN capacity, achieved uniquely by CSCG input when feasible or approached arbitrarily through Gaussian/OOK time sharing otherwise.The OOK component uses low-probability high-amplitude signals.
  • Main results: With amplitude constraints, the capacity-achieving input is discrete in amplitude with finitely many mass points and uniformly distributed independent phase.This extends related constrained-AWGN results to the complex setting.
  • SWIPT application: For nonlinear SWIPT, delivered power depends on multiple input moments; Gaussian inputs are zero mean, and asymmetric inphase/quadrature power allocation creates an information-power tradeoff.The paper develops two rate-power inner bounds, including one based on Gaussian inputs and another based on restricted delivered-power optimization.
  • SWIPT implications: Restricted input distributions enlarge the rate-power region relative to Gaussian counterparts, while nonlinear harvesting can be exploited through signal design and validated with realistic circuit simulations.The paper presents energy modulation for single-carrier wireless power transfer as an alternative to multicarrier energy waveforms.
  • Additional result: The paper also derives a tight upper bound for the modified Bessel function of the first kind of order zero, which frequently appears in complex AWGN analysis.This is presented as an independent mathematical result with potential future analytical use.

III. MAIN RESULTS

The main results characterize AWGN capacity under delivered-power and amplitude constraints. With no amplitude limit, Gaussian inputs achieve capacity when feasible, while Gaussian–OOK time sharing can approach the same capacity otherwise; finite amplitude yields a unique finite-support amplitude distribution.

  • When Pd is no greater than the Gaussian delivered power PG, the unique capacity-achieving input is x ∼ CN(0, Pa).
  • When Pd > PG, capacity is not attained, but it can be approached arbitrarily through time sharing between a Gaussian input and low-probability OOK.A representative approaching sequence is explicitly described as CSCG–OOK time sharing.
  • Theorem 1 establishes that, with no amplitude constraint, capacity is independent of the delivered-power threshold Pd.The capacity remains the standard AWGN capacity for every finite Pd.
  • With finite amplitude constraint rp < ∞, the unique optimal input has a finite number of amplitude mass points and satisfies necessary and sufficient optimality conditions.The result supports numerical computation of the optimal distribution.
  • As rp grows, capacity becomes less dependent on the amplitude constraint, with the associated multiplier μ tending to zero as rp → ∞.This follows because capacity approaches the amplitude-unconstrained value, which is unchanged by finite Pd.
  • The paper disputes a prior claim that capacity remains achievable by a finite discrete distribution when Pd exceeds Gaussian feasibility, stating that it is only arbitrarily approachable.The correction applies to the cited real-AWGN claim as rp → ∞.

IV. APPLICATION

The application studies SWIPT over a complex AWGN channel with a nonlinear energy harvester. It models iid information-power symbols through RF transmission, rectenna harvesting, downconversion, and sampling to obtain a baseband channel representation.

  • The SWIPT application considers a complex AWGN channel whose receiver includes a nonlinear energy harvester.The section develops the transmission process and two inner bounds for the rate-power region.
  • The transmitter generates iid information-power symbols, upconverts them to carrier frequency, and sends the resulting waveform over the channel.The iid assumption is imposed for the random-coding achievability analysis and tractable delivered-power representation.
  • At the receiver, the filtered RF waveform is processed by the rectenna for power harvesting, while downconversion and sampling produce the baseband channel y = x + n.Narrowband transmission assumes fc ≫ 2fw.
  • The nonlinear rectenna model is based on a small-signal approximation, whereas the linear delivered-power model depends only on the second moment of the received RF signal.The approximation is motivated by the rectifier diode model and its fourth-order Taylor truncation.

A. Delivered power in the baseband

This section derives a baseband representation of harvested power for iid channel inputs. The resulting delivered power depends on multiple input moments, motivating a convex lower-bound restriction to even moments for applying the capacity results.

  • For iid channel inputs, Lemma 1 expresses the receiver delivered power in terms of system baseband parameters.The representation is derived from the nonlinear rectenna output model.
  • The delivered power depends on different odd and even moments of the channel input, including Q = E[|x|4], T = E[|x|3], P = E[|x|2], and the mean μ = E[x].The expression also separates in-phase and quadrature moments such as Qr, Qi, μr, and μi.
  • The rectenna model uses a fourth-moment truncation of the diode Taylor expansion, with first and third time-averaged moments equal to zero.The constants k2 and k4 parameterize the delivered-power expression.
  • Restricting delivered power to even moments gives a convex lower bound that permits use of the earlier capacity theorem.The restricted constraint is adopted for subsequent analysis.
  • A closed-form delivered-power expression for non-iid inputs is cumbersome because fourth-order received-signal statistics couple different time indices.This is a scope limitation of the baseband derivation.

B. Rate-Power (RP) region

The RP region is bounded using Gaussian inputs and a restricted optimization space, revealing how asymmetric subchannel power allocation trades transmitted information against delivered power.

  • B. Rate-Power (RP) region): Two inner bounds are considered: one assumes Gaussian inputs, while the other restricts the delivered-power optimization space to apply analytic capacity results.For Gaussian inputs, the optimal distributions are zero mean.
  • 1) Complex Gaussian Inputs:: The Gaussian-input problem exhibits a rate-delivered-power tradeoff determined by asymmetric power allocation between the inphase and quadrature subchannels.The delivered power depends on both second and fourth moments of the channel input, reflecting the rectenna's nonlinear output model.
  • 1) Complex Gaussian Inputs:: For Gaussian inputs, zero-mean real and imaginary components with powers Pr and Pi, satisfying Pr + Pi = Pa, attain the supremum.The maximum delivered power is denoted Pdel,max, while the minimum is Pdel,min.
  • 1) Complex Gaussian Inputs:: If Pd > Pdel,max, no solution exists; at Pd = Pdel,max, all transmit power is allocated to one subchannel, whereas intermediate Pd values require an optimal allocation P_i* between the subchannels.For Pd ≤ Pdel,min, equal allocation gives P_r* = P_a/2 and delivered power remains Pdel,min.
  • 1) Complex Gaussian Inputs:: The Gaussian-input tradeoff is therefore generated by redistributing fixed transmit power between the inphase and quadrature dimensions rather than by changing the total average power.The resulting RP region is evaluated numerically later in the paper.

2) Restricted optimization probability space:

The restricted optimization approach converts the nonlinear delivered-power constraint into a tractable moment-constrained problem, then numerically evaluates its RP region and input distributions.

  • 2) Restricted optimization probability space:: The restricted problem imposes E[r^2] ≤ Pa, Pd ≤ E[gNL(r)], and r ≤ rp, then obtains rate and delivered power from its optimal solutions.The resulting inner bound is evaluated through I(x;y) and E[gNL(r)].
  • Complex Gaussian inputs:: For Gaussian inputs, symmetric power allocation gives delivered power Pdel,min and information ln(1 + Pa/2), while concentrating power in one subchannel gives Pdel,max and 1/2 ln(1 + Pa).The Gaussian RP region is generated by varying the inphase and quadrature power allocation.
  • The following steps are summarized:: The numerical procedure fixes average power, increments Pd, and uses an interior-point algorithm to optimize mass-point positions and probabilities.The optimization is initialized from a random guess and repeated as the delivered-power constraint changes.
  • Illustration of the numerical results:: The restricted optimization yields a larger RP region than Gaussian asymmetric power allocation, while increasing the amplitude constraint changes the resulting region.The comparison uses Pa = 5 and the nonlinear model gNL(r) = 0.01(r^4 + r^2 + 1).
  • Illustration of the numerical results:: As Pd increases, the number of optimal amplitude mass points decreases, and one mass point remains at the amplitude limit rp.This behavior is illustrated for rp = 4, 5, and 6.
  • Illustration of the numerical results:: The numerical input-distribution optimization is sensitive to its initial guess as the number of mass points grows, because fixed-mass-point capacity optimization is nonconcave.This makes the computation demanding for larger m.

B. Realistic Circuit Simulations for WPT

Circuit simulations validate that rectenna nonlinearity makes delivered power depend on input distribution and higher-order moments, while the study connects these effects to broader SWIPT design limitations and future directions.

  • Circuit design: The simulated rectenna uses a conventional single-series circuit with a Schottky diode, impedance-matching network, and low-pass filter, driven by a 4-tone 2.45 GHz multisine.The input has −20 dBm average power and 2.5 MHz inter-carrier spacing; the load impedance is 10 kΩ.
  • Measured delivered power: Equal second moments across CW, CG, RG, and OOK inputs nevertheless produce substantially different harvested DC powers because rectenna nonlinearity favors larger fourth moments.The fourth moments scale differently across the tested distributions, linking waveform shape to delivered power.
  • Model validation: The polynomial second- and fourth-order model predicts the rectenna’s dependence on the input signal and confirms that a linear second-order model misses this effect.The measured comparison supports the model’s higher-order-moment interpretation.
  • Measured delivered power: The highest delivered power for the tested OOK family occurs at l = 4, while finite low-pass-filter RC dynamics reduce power for l > 4.This identifies an experimentally observed optimum within the tested signalling family rather than a general optimum over all inputs.
  • Future work: The paper identifies open boundaries involving odd moments, bounded nonlinear harvester models, phase dependence, sample correlation, computational cost, and extensions to vector or multiple-access Gaussian channels.These directions qualify the scope of the presented circularly symmetric, iid-sample analysis.
  • Conclusions: The conclusions report that the capacity result can be achieved or approached arbitrarily, and that restricting the delivered-power optimization yields significant RP-region improvements over complex Gaussian inputs.The Gaussian inner bound uses zero-mean inputs and asymmetric real/imaginary power allocation to expose the information–delivered-power tradeoff.

APPENDIX A LEMMAS

The appendix establishes analytical properties needed for the capacity proofs, including constraint-set geometry, continuity and concavity, monotonicity, and regularity of the induced output distributions and kernel functions.

  • Appendix A: The appendix develops lemmas used to prove the paper’s main capacity theorems.The supporting results address both optimization structure and analytic regularity.
  • Capacity properties: The capacity C(Pa, Pd, rp) is concave in transmit and delivered-power parameters, nondecreasing in Pa, and nonincreasing in Pd.These properties support continuity and monotonicity arguments in the capacity analysis.
  • Achievability: The achievability proof uses typical-sequence coding and shows that every rate below C(Pa, Pd, rp) is achievable after letting slack parameters vanish.Atypical sequences are replaced by randomly chosen typical sequences to satisfy the constraints.
  • Constraint-set geometry: The constraint space is convex and compact when the delivered-power threshold parameter rp is finite, but compactness can fail when rp is infinite.The counterexample uses fourth-moment constraints whose limiting distribution violates the constraint.
  • Analytic regularity: The appendix establishes boundedness and continuity for fR and K, including the bound K(R, r) < 1, while the entropy functional H(Fr) is continuous and strictly concave.These regularity properties support existence and optimization arguments for the capacity problem.
  • Complex-channel lemmas: The complex-channel lemma characterizes the output density generated by adding CSCG noise to an independent complex input, extending a corresponding real-channel result.The proof uses characteristic functions, continuity, and Hardy’s theorem.

APPENDIX B PROOF OF THEOREM 1

The proof establishes that receiver delivered-power constraints preserve the unconstrained-delivered-power capacity, although the supremum may require time-sharing and may not be attained. It also derives the finite-amplitude-support structure for the constrained optimization.

  • Capacity under delivered-power constraints: C(Pa, Pd, ∞) is non-increasing in Pd, while C(Pa, 0, ∞) = ln(1 + Pa/2) is uniquely achieved by a CSCG input.The unique Gaussian achiever cannot satisfy Pd > PG, so the same capacity is not attained above that delivered-power threshold.
  • Capacity under delivered-power constraints: For Pd > PG, time-sharing between the Gaussian distribution and increasingly high-power OOK distributions approaches C(Pa, 0, ∞) arbitrarily closely.The mixing probability tends to zero as the OOK delivered power grows, but no distribution achieves the supremum.
  • Time-sharing construction: The time-sharing construction preserves the average-power and delivered-power constraints by selecting the mixture weight from the target Pd and the OOK delivered power.The proof uses distributions with average amplitude-square Pa and delivered power diverging with the OOK parameter.
  • Finite-amplitude-support structure: Under finite amplitude, the KKT conditions and analyticity argument rule out continuous or infinitely supported optimal amplitudes in the relevant multiplier cases.The argument uses analytic continuation and the identity theorem when the support has a limit point.
  • Finite-amplitude-support structure: The optimal amplitude is therefore discrete with finitely many mass points, while the phase remains uniformly distributed and makes the complex input continuous.This conclusion follows after candidate analytic forms are shown either non-legitimate or incompatible with the optimality conditions.

APPENDIX E PROOF OF LEMMA 2

The lemma analyzes Gaussian inputs under the SWIPT rate-power constraints by reducing the optimization to power allocation between inphase and quadrature components. It shows that rate and delivered power respond oppositely to asymmetric allocation.

  • Gaussian-input optimization: For Gaussian inputs, the delivered power is Pdel = 2αPa^2 + βPa + γ when the total average input power is Pa.The nonlinearity enters through the quadratic dependence on total power and through the allocation-dependent fourth-order term.
  • Gaussian-input optimization: When the delivered-power constraint is inactive, the optimum is zero-mean with Pr = Pi = Pa.This is the usual symmetric CSCG allocation under the unconstrained rate objective.
  • Gaussian-input optimization: With Pr + Pi = Pa, rate is concave in Pi and is maximized at symmetric allocation Pi = Pa/2.The KKT conditions establish the total-power equality for positive delivered-power multiplier λ2.
  • Gaussian-input optimization: Delivered power is convex in Pi and is maximized at the asymmetric extremes Pi = 0 or Pi = Pa.Thus, symmetric allocation maximizes rate while one-dimensional allocation maximizes delivered power under the same total power.

APPENDIX G PROOF OF LEMMA 6

The proof of Lemma 6 establishes polynomial and transform properties used to characterize the nonlinear delivered-power function. It combines special-function identities with invertibility of the relevant integral transform.

  • Special-function representation: The functions Φr(i, k) are generated recursively and are polynomials whose degree is 2(i − k).In particular, Φr(i, 1) has degree 2(i − 1), supporting finite polynomial representations of the relevant expressions.
  • Transform invertibility: The integral transform used to recover coefficients is invertible, so the coefficients ci are unique.The proof shows that a zero transformed function implies the underlying polynomial function is zero, and conversely.
  • Analytic bounds: The proof derives bounds using I0(x) < ex and related inequalities to control the transformed expressions and establish integrability.These bounds are applied across the small- and large-radius regimes.

APPENDIX I PROOF OF LEMMA 10

The lemma proves that the amplitude-entropy functional is well-defined, continuous, and strictly concave over the feasible distribution set. These properties support existence and uniqueness of the optimal amplitude distribution.

  • Moment and entropy properties: For every feasible amplitude distribution, E[R^α] exists and is bounded for 0 ≤ α < 1.This moment bound provides the integrability needed for the entropy analysis.
  • Continuity: The entropy functional H(Fr) is continuous under weak convergence because its entropy and logarithmic terms are controlled by the established integrability bounds.The proof separates the integral into regions and applies dominated-convergence arguments.
  • Concavity and uniqueness: H(Fr) is concave, and strict concavity follows from invertibility of the integral transform relating amplitude distributions to the induced output density.Strict concavity ensures that two distinct feasible distributions cannot share the same optimum.

APPENDIX K UPPERBOUND FOR K(R, r)

The appendix upperbounds K(R, r) by separately bounding terms in inequality (187), analyzing parameter ranges and monotonicity properties. It also uses Taylor expansion and summation bounds to complete the derivation.

  • The proof bounds each term on the right-hand side of inequality (187) separately, beginning with the first, second, and third terms.
  • The first term is maximized at R = 1 after differentiation with respect to R.
  • For the second term, monotonicity of erf(x)/x and erf(√x)/√x yields suprema at x = 0, supporting the resulting bound.
  • The third term is handled separately over parameter ranges such as rR ≤ 1, rR ≥ 1 with r ≥ 1, and rR ≥ 1 with r ≤ 1.
  • The derivation uses Taylor expansion, exponential inequalities, and the fact that a decreasing function composed with an increasing function remains decreasing.
  • The appendix also specifies summation conventions and derives auxiliary bounds involving powers of (2l + 1) and (2 + l).
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