Lagrange Inversion and Lambert W

Jeffrey, David J.; Kalugin, G. A.; Murdoch, Nick

doi:10.1109/synasc.2015.16

Cited by 4 publications

(5 citation statements)

References 5 publications

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“…Consider next the case when A ≥ A T,sat l . For x 0 ∈ (0, A T,sat l ), since P l (x)/x 2 increases monotonically in 0 ≤ x < A T,sat l , similar to (35), the contribution of the mass points at x = 0 and x = A T,sat l to the average harvested power is higher than the contribution of any other point x 0 ∈ (0, A T,sat l ), since P l (x0)…”

Section: Appendix F − Proof Of Theoremmentioning

confidence: 77%

“…In (31), the quadratic function, the exponential function, and their compositions are analytic functions in the whole complex domain z ∈ C [22]. In P l (z) = min 1 (15), the min(·) function, the quadratic function (·) 2 , the modified Bessel function I 0 (·), and their compositions are analytic on the whole complex domain z ∈ C. The principal branch of the LambertW function W 0 (·) is analytic everywhere in the complex domain with the exception of the branch cut along the negative real axis, i.e., on (−∞, −1/e) [29], [35]. Hence, the function s(z) is analytic in the domain D defined by D…”

Section: Appendix D − Proof Of Theoremmentioning

confidence: 99%

“…For the linear EH model, P l (x)/x 2 is constant ∀ x. From (35) in Appendix F, any distribution satisfying the AP and PP constraints maximizes the harvested energy. Consequently, the optimal distribution for maximum WIT is also optimal for maximum WPT.…”

Section: B Maximum Energy Transfermentioning

confidence: 99%

“…where p n (•) is a polynomial with coefficients given in [35,Table I.]. This series holds for some radius of convergence (ROC) |x − x 0 | < x ROC and can be expanded to a polynomial in x defined as W 0 (x) = ∞ n=0 d n (x 0 )x n .…”

Section: Appendix D − Proof Of Theoremmentioning

confidence: 99%

“…Condition (35) holds for the harvested power function in (15) since (a) P l (0) = 0, as I 0 (0) = 1 and W 0 (ae a ) = a and (b) P l (x)/x 2 increases monotonically in 0 < x < A. We derive this monotonicity by proving that xP l (x) − 2P l (x) > 0 holds for 0 < x < A, where P l (x) is the first-order derivative of P l (x).…”

Section: Appendix F − Proof Of Theoremmentioning

confidence: 99%

See 4 more Smart Citations

Conditional Capacity and Transmit Signal Design for SWIPT Systems With Multiple Nonlinear Energy Harvesting Receivers

Morsi¹,

Jamali²,

Hagelauer³

et al. 2020

IEEE Trans. Commun.

121

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In this paper, we study information-theoretic limits for simultaneous wireless information and power transfer (SWIPT) systems employing practical nonlinear radio frequency (RF) energy harvesting (EH) receivers (Rxs). In particular, we consider a SWIPT system with one transmitter that broadcasts a common signal to an information decoding (ID) Rx and multiple EH Rxs. Owing to the nonlinearity of the EH Rxs' circuitry, the efficiency of wireless power transfer depends on the waveform of the transmitted signal. We aim to answer the following fundamental question: What is the optimal input distribution of the transmit signal waveform that maximizes the information transfer rate at the ID Rx conditioned on individual minimum required direct-current (DC) powers to be harvested at the EH Rxs? Specifically, we study the conditional capacity problem of a SWIPT system impaired by additive white Gaussian noise subject to average-power (AP) and peakpower (PP) constraints at the transmitter and nonlinear EH constraints at the EH Rxs. To this end, we develop a novel nonlinear EH model that captures the saturation of the harvested DC power by taking into account not only the forward current of the rectifying diode but also the reverse breakdown current. Then, we derive a novel semiclosed-form expression for the harvested DC power, which simplifies to closed form for low input RF powers. The derived analytical expressions are shown to closely match circuit simulation results. We solve the conditional capacity problem for real-and complex-valued signalling and prove that the optimal input distribution that maximizes the rate-energy (R-E) region is unique and discrete with a finite number of mass points. Furthermore, we show that, for the considered nonlinear EH model and a given AP constraint, the boundary of the R-E region saturates for high PP constraints due to the saturation of the harvested

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Section: Appendix F − Proof Of Theoremmentioning

confidence: 77%

Section: Appendix D − Proof Of Theoremmentioning

confidence: 99%

Section: B Maximum Energy Transfermentioning

confidence: 99%

Section: Appendix D − Proof Of Theoremmentioning

confidence: 99%

Section: Appendix F − Proof Of Theoremmentioning

confidence: 99%

See 3 more Smart Citations

Conditional Capacity and Transmit Signal Design for SWIPT Systems With Multiple Nonlinear Energy Harvesting Receivers

Morsi¹,

Jamali²,

Hagelauer³

et al. 2020

IEEE Trans. Commun.

121

View full text Add to dashboard Cite

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Stirling Numbers, Lambert W and the Gamma Function

Jeffrey

Murdoch

2017

Mathematical Aspects of Computer and Information Sciences

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All the trinomial roots, their powers and logarithms from the Lambert series, Bell polynomials and Fox–Wright function: illustration for genome multiplicity in survival of irradiated cells

Belkić

2018

J Math Chem

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All the roots of the general nth degree trinomial admit certain convenient representations in terms of the Lambert and Euler series for the asymmetric and symmetric cases of the trinomial equation, respectively. Previously, various methods have been used to provide the proofs for the general terms of these two series. Taking n to be any real or complex number, we presently give an alternative proof using the Bell (or exponential) polynomials. The ensuing series is summed up yielding a single, compact, explicit, analytical formula for all the trinomial roots as the confluent Fox-Wright function 1 1. Moreover, we also derive a slightly different, single formula of the trinomial root raised to any power (real or complex number) as another 1 1 function. Further, in this study, the logarithm of the trinomial root is likewise expressed through a single, concise series with the binomial expansion coefficients or the Pochhammer symbols. These findings are anticipated to be of considerable help in various applications of trinomial roots. Namely, several properties of the 1 1 function can advantageously be employed for its implementations in practice. For example, the simple expressions for the asymptotic limits of the 1 1 function at both small and large values of the independent variable can be used to readily predict, by analytical means, the critical behaviors of the studied system in the two extreme conditions. Such limiting situations can be e.g. at the beginning of the time evolution of a system, and in the distant future, if the independent variable is time, or at low and high doses when the independent variable is radiation dose, etc. The present analytical solutions for the trinomial roots are numerically illustrated in the genome multiplicity corrections for survival of synchronous cell populations after irradiation.

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Lagrange Inversion and Lambert W

Cited by 4 publications

References 5 publications

Conditional Capacity and Transmit Signal Design for SWIPT Systems With Multiple Nonlinear Energy Harvesting Receivers

Conditional Capacity and Transmit Signal Design for SWIPT Systems With Multiple Nonlinear Energy Harvesting Receivers

Stirling Numbers, Lambert W and the Gamma Function

All the trinomial roots, their powers and logarithms from the Lambert series, Bell polynomials and Fox–Wright function: illustration for genome multiplicity in survival of irradiated cells

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