Simple Closed-form Formulas Research Articles

A simple closed-form formula for pricing discretely-sampled variance swaps under the heston model

A simple closed-form formula for pricing discretely-sampled variance swaps under the heston model

Asymptotic Error Probability Analysis of M-ary Modulation Schemes in Hoyt Fading Channels

This paper derives simple asymptotic closed-form formulas for the average error probabilities of various M-ary modulation schemes (i.e., M-ary differentially coherent phase shift-keying (DPSK) signals, noncoherent correlated binary (NCB) signals, M-ary coherent phase shift-keying (PSK) signals, and M-ary square quadrature amplitude modulation (QAM)), especially in Hoyt (or Nakagami-q) fading channels, since the formula can be effectively used to study the asymptotically achievable error performance. The derived expressions for the representative modulation schemes are much more concise than the existing conventional formulas, showing the very tightness to the error probability results from the existing exact formulas in the high signal-to-noise ratio (SNR) region. Through the rigorous numerical results, we clearly justify the validity of our derivations and analyses.

Bending Stiffness Requirement for Closed-Section Longitudinal Stiffeners of Isotropic Material Plates under Uniaxial Compression

Longitudinally stiffened plates with closed-section ribs are supposed to be an effective system for axially compressed members. However, current design specifications on closed-section ribs, particularly for the minimum required stiffness, are not sufficient, and thus, excessive designs are quite common because of the absence of suitable design guides. In this paper, simple closed-form formulas for the minimum required stiffness of compressively loaded isotropic plates braced by U-shaped longitudinal stiffeners (U-ribs) are derived. The effects of the sectional stiffness of U-ribs on the buckling modes and strengths of stiffened plates are examined by numerical analyses. Three-dimensional finite-element models of U-rib stiffened plates were obtained, and a series of eigenvalue analyses was conducted. By parametric study, thresholds of the sectional stiffness of U-ribs that may be adopted as the minimum requirement were collected, and these were used for regression analysis to obtain a simple and practical design equation for determining a suitable sectional stiffness after conducting a comparative study using a theoretically derived formula. The findings of this study would contribute to improving the optimum design of U-rib-stiffened plates.

Optimal Feedback Control for Linear Systems with Input Delays Revisited

The design problem of optimal feedback control for linear systems with input delays is very important in many engineering applications. Usually, the linear systems with input delays are firstly converted into linear systems without delays, and then all the design procedures are based on the delay-free linear systems. In this way, the feedback controllers are not designed in terms of the original states. This paper presents some new closed-form formula in terms of the original states for the delayed optimal feedback control of linear systems with input delays. We firstly reveal the essential role of the input delay in the optimal control design of the linear system with a single input delay: the input delay postpones the action of the optimal control only. Based on this fact, we calculate the delayed optimal control and find that the optimal state can be represented by a simple closed-form formula, so that the delayed optimal feedback control can be obtained in a simple way. We show that the delayed feedback gain matrix can be "smaller" than that for the controlled system with zero input delay, which implies that the input delay can be considered as a positive factor. In addition, we give a general formula for the delayed optimal feedback control of time-variant linear systems with multiple input delays. To show the effectiveness and advantages of the main results, we present five illustrative examples with detailed numerical simulation and comparison.

Novel last passage time based jitter model with application to low slew rate/high noise ring oscillator

In a ring oscillator, the behaviour in which the output voltage ramp (up/down) of its individual delay cell crosses the threshold, which triggers the next stage delay cell, is crucial in determining the timing jitter, and hence phase noise. Specifically, as the slew rate of the ramp decreases and/or the amount of noise contribution from the transistors in the delay cell increases, the voltage ramp (up or down) has a higher probability of crossing the threshold multiple times, before finally passing it at a time denoted as last passage time, which is more accurate than the conventional first passage time model. This multiple crossing results in a higher jitter. In the past, investigation in last passage time jitter model results in jitter expression that can only be calculated numerically and thus no design guidelines/insights are apparent. In this paper, a novel model is presented with a simple closed form formula, which shows the extra jitter, due to multiple crossing, adds a term that increases as a function of the fourth power of the noise strength/slew rate ratio. The formula is applied to a real life practical low slow rate/high noise ring oscillator which finds application, for example, in random number generator implementation. Corresponding transistor level simulation results agree reasonably well with the model. Furthermore, it is shown that, on example designs, the last passage time approach in this paper can lead to time jitter that is 100 % larger than that due to conventional first passage time model.

Kinetostatics of S-(nS)PU-SPU and S-(nS)PU-2SPU Nonholonomic Parallel Wrists

S-(nS)PU-SPU and S-(nS)PU-2SPU are two types of nonholonomic wrists that are generated from the “ordinary” wrists of type S-3SPU (fully parallel wrists (FPW)), by replacing a spherical pair (S) with a nonholonomic spherical pair (nS) according to the rules stated by Grosch et al. (2010, “Generation of Under-Actuated Manipulators With Nonholonomic Joints From Ordinary Manipulators,” ASME J. Mech. Rob., 2(1), p. 011005). Position analysis, controllability, and path planning of these two wrist types have been addressed and solved in two previous papers (Di Gregorio, R., 2012, “Type Synthesis of Underactuated Wrists Generated From Fully-Parallel Wrists,” ASME J. Mech. Des., 134(12), p. 124501 and Di Gregorio, R., 2012, “Position Analysis and Path Planning of the S-(nS)PU-SPU and S-(nS)PU-2SPU Underactuated Wrists,” ASME J. Mech. Rob., 4(2), p. 021006) of this author, which demonstrated that simple closed-form formulas are sufficient to control their configuration and to implement their path planning. Their kinetostatics and singularity analysis have not been addressed, yet; and they are studied in this paper. Here, the singularity analysis will reveal, for the first time, the existence of a somehow novel type of singularities, here named “jamming singularity,” that jams the platform motion in some directions and that is also present in all the parallel manipulators with SPU limbs (e.g., Gough-Stewart platforms) where it can be considered a particular type of “leg singularity.” Moreover, the static analysis will demonstrate that the reaction forces due to the static friction, in the nonholonomic constraint, can be controlled in the same way as the generalized forces exerted by the actuators, and that the possible slippage, in the same constraint, can be easily monitored and compensated.

Performance Analysis of Massive MIMO for Cell-Boundary Users

In this paper, we consider massive multiple-input multiple-output (MIMO) systems for both downlink and uplink scenarios, where three radio units (RUs) connected via one digital unit (DU) support multiple user equipments (UEs) at the cell-boundary through the same radio resource, i.e., the same time-frequency slot. For downlink transmitter options, the study considers zero-forcing (ZF) and maximum ratio transmission (MRT), while for uplink receiver options it considers ZF and maximum ratio combining (MRC). For the sum rate of each of these, we derive simple closed-form formulas. In the simple but practically relevant case where uniform power is allocated to all downlink data streams, we observe that, for the downlink, vector normalization is better for ZF while matrix normalization is better for MRT. For a given antenna and user configuration, we also derive analytically the signal-to-noise-ratio (SNR) level below which MRC should be used instead of ZF. Numerical simulations confirm our analytical results.

Invertible Exponential-Type Approximations for the Gaussian Probability Integral Q(x) with Applications

In this article, we present very tight exponential-type approximations for the Gaussian probability integral Q(.) and/or the complementary error function erfc(.) which are subsequently used to devise the conditional symbol error probability (CEP) formulas for several classes of digital modulation schemes that are both invertible and in a desirable exponential form. These invertible CEP formulas are of interest in the optimization of discrete-rate adaptive modulation designs and for computation of the symbol/bit error outage performance metric. Our framework also facilitates the derivation of simple and tight closed-form approximation formulas for the average symbol error rate performance metric for a wide range of digital modulation schemes over generalized fading channels via the moment generating function approach. The accuracies of our new closed-form approximations have been validated with that of the exact expressions in an integral form.

Optimum power and efficiency of piezoelectric vibration energy harvesters with sinusoidal and random vibrations

Assuming a sinusoidal vibration as input, an inertial piezoelectric harvester designed for maximum efficiency of the electromechanical energy conversion does not always lead to maximum power generation. In this case, what can be gained by optimizing the efficiency of the device? Detailing an answer to this question is the backbone of this paper. It is shown that, while the maximum efficiency operating condition does not always lead to maximum power generation, it corresponds always to maximum power per square unit deflection of the piezoelectric harvester. This understanding allows better optimization of the generated power when the deflection of the device is limited by hard stops. This is illustrated by experimental measurements on vacuum-packaged MEMS harvesters based on AlN as piezoelectric material. The results obtained for a sinusoidal vibration are extended to random vibrations. In this case, we demonstrate that the optimum generated power is directly proportional to the efficiency of the harvester, thus answering the initial question. For both types of studied vibrations, simple closed-form formulas describing the generated power and efficiency in optimum operating conditions are elaborated. These formulas are based on parameters that are easily measured or modeled. Therefore, they are useful performance metrics for existing piezoelectric harvesters.

Direct and closed form analytical model for the prediction of reaction kinetic of EPDM accelerated sulphur vulcanization

A novel direct model with kinetic base for the prediction of the final vulcanization level of EPDM cured with sulphur is presented. The model bases on a preliminary characterization of rubber through standard rheometer tests and allows an accurate prediction of the crosslinking degree at both successive curing times and different controlled temperatures. Both the case of indefinite increase of the torque and reversion can be handled. The approach proposed bases on a previously presented exponential model, where a calibration of three kinetic constants at fixed temperature by means of non-linear least square fitting was required. Here the exponential model is superseded and kinetic constants are evaluated through simple closed form formulas. The applicability of the approach is immediate and makes the model extremely appealing when fast and reliable estimates of crosslinking density of cured EPDM are required. To show the capabilities of the approach proposed, a comprehensive comparison with both available experimental data and results obtained numerically with the exponential model for real compounds at different temperatures is finally provided.

Kinematic Analysis of the (nS)-2SPU Underactuated Parallel Wrist

The majority of the manipulation tasks require the motion of the end-effector from an initial pose to a final one without any particular condition on the path. Thus, the reduction of the practicable paths between any two poses is a possible choice, exploitable at the design stage, for simplifying the manipulator hardware. This choice is adopted in underactuated manipulators. The (nS)-2SPU wrist is one out of the underactuated parallel wrists this author proposed in a previous paper. Here, the kinematic analysis of this wrist is studied: both its finite and its elementary kinematics are considered. It is shown that its control algorithms can be written by using simple closed-form formulas, which can take advantage from the wide literature on the spherical four-bar linkages. Moreover, the demonstration that its singular configurations can be avoided more easily than the ones of the fully parallel wrist is provided.

Onset of size effects in lattice thermal conductivity and lifetime of low-frequency thermal phonons

ABSTRACTThe onset of size effects in phonon-mediated thermal transport along a thin film at temperatures comparable or greater than the Debye temperature is analyzed theoretically. Assuming a quadratic frequency dependence of phonon relaxation rates in the low-frequency limit, a simple closed-form formula for the reduction of the in-plane thermal conductivity of thin films is derived. The effect scales as the square root of the film thickness, which leads to the prediction of measurable size-effects even at “macroscopic” distances ~100 μm. However, this prediction needs to be corrected to account for the deviation from the ω−2 dependence of phonon lifetimes at sub-THz frequencies due to the transition from Landau-Rumer to Akhiezer mechanism of phonon dissipation.

Latency of FAST TCP for HTTP Transactions

For HTTP over FAST TCP scenario, this letter presents a simple closed-form formula to estimate the latency, taking into account the network parameters such as packet size, link capacity, propagation delay and FAST TCP parameter settings. Analyzes supported by simulation results reveal the effects of network parameters and FAST TCP parameters on the FAST TCP/HTTP latency.

Analytical Modeling of Nonlinear Propagation in Uncompensated Optical Transmission Links

We present analytical results on the impact of nonlinear propagation in uncompensated links. We test the accuracy of our model in the context of ultradense wavelength-division-multiplexing polarization-multiplexed quadrature phase-shift keying (PM-QPSK) systems, at the Nyquist spectral efficiency limit. We show that the predicted system performance matches simulation results very accurately over a broad range of system scenarios. A simple closed-form analytical formula provides an effective tool for the quick and accurate prediction of system performance.

Modulation Selection from a Battery Power Efficiency Perspective

In this paper, we compare the battery power efficiencies of various pulse-based modulations widely adopted for their low complexity. Taking into account circuit modules and battery imperfectness, we establish simple closed-form analytical formulas which can be used to conveniently determine the relative preference between arbitrary pulse-based modulation pairs in terms of their actual average battery energy consumption.

A new approach to Sheppard’s corrections

A very simple closed-form formula for Sheppard’s corrections is recovered by means of the classical umbral calculus. Using this symbolic method, a more general closed-form formula for discrete parent distributions is provided and the generalization to the multivariate case turns out to be straightforward. All these new formulas are particularly suited to be implemented in any symbolic package.

Simple Formulas for Stopband Attenuation Characteristics of Asymmetric Helical Slow-Wave Structures of Traveling-Wave Tubes

Simple closed-form formulas for the estimation of the π-mode stopband and the stopband attenuation in an azimuthally asymmetric helical slow-wave structure (SWS) are developed, following the coupled-mode analysis of multiple reflections. The formulas are simple and amenable to easy computation, and also allow the use of the dispersion characteristics of the structure obtainable from any standard electromagnetic modeling, thereby accruing the accuracy of 3-D electromagnetic analysis. The analysis is benchmarked against published results with close agreement. Based on the analysis, SWS configurations that would reduce the deleterious effects of asymmetry and also would improve TWT efficiency are suggested.

Fermi-Dirac and Bose-Einstein functions of negative integer order

We find simple explicit closed-form formulas for the Fermi-Dirac function \( \mathcal{F}_{ - n} (z) \) and Bose-Einstein function \( \mathcal{B}_{ - n} (z) \) for arbitrary n e ℕ. The obtained formulas involve the higher tangent numbers defined by Carlitz and Scoville. We present some examples and direct consequences of applying the main results.

The Lerch zeta and related functions of non-positive integer order

It is known that the Lerch (or periodic) zeta function of non-positive integer order, ℓ − n ( ξ ) \ell _{-n}(\xi ) , n ∈ N 0 := { 0 , 1 , 2 , 3 , … } n\in \mathbb {N}_{0}:= \{0, 1, 2, 3, \ldots \} , is a polynomial in cot ⁡ ( π ξ ) \cot (\pi \xi ) of degree n + 1 n + 1 . In this paper, a very simple explicit closed-form formula for this polynomial valid for any degree is derived. In addition, novel analogous explicit closed-form formulae for the Legendre chi function, the alternating Lerch zeta function and the alternating Legendre chi function are established. The obtained formulae involve the Carlitz-Scoville tangent and secant numbers of higher order, and the derivative polynomials for tangent and secant are used in their derivation. Several special cases and consequences are pointed out, and some examples are also given.

An Investigation of Earth-Return Impedance Between Overhead and Underground Conductors and Its Approximation

This paper investigates numerical instability issues in the mutual impedance formula between an overhead conductor and a buried conductor proposed by Pollaczek with special reference to the integrand of Pollaczek's infinite integral. The integrand shows a negative real part and positive imaginary part at a high frequency, which result in negative resistance and inductance. The reason for this is the assumption of TEM-mode propagation in Pollaczek's formula as is well known. The fact suggests that numerical instability of the infinite integral is caused by incorrect integrand in a high-frequency region. Based on the observations, the applicable limit of Pollacezk's formula is analytically derived. Finally, a simple closed-form formula is developed, which approximates Pollaczek's formula with satisfactory accuracy.