Compact Closed-form Expression Research Articles

Linear sweeps and the characterisation of linearly time-varying acoustical systems.

Sine sweeps are widely used as test signals in the experimental characterisation of linear and non-linear acoustical systems' responses. One of the reasons for their popularity is that their immunity against time variance is assumed to be superior to that of other types of test signals. This paper explores and quantifies the consequences of linear time variance on the characterisation of acoustical systems with linear sweeps. The first result is a novel compact closed-form expression for the spectrum of a linear sweep. Second, a similar expression for the frequency response of a linearly time-varying delay to a linear sweep is derived. Additionally, this work demonstrates the application of these expressions through a numerical investigation into the frequency response of a time-invariant bandpass filter when the measurement chain exhibits linear time variance and the attenuation between a ground-borne point source and a receiver in front of a façade under the assumption of a homogeneous atmosphere subject to realistic temperature changes.

Coverage Probability and Area Potential Spectral Efficiency Analysis of 3D Dense SCMA Cellular Networks

This paper investigates the dense multi-user sparse code multiple access (SCMA) cellular networks, where base stations (BSs) and users are randomly deployed in a 3D space by using stochastic geometry tools. First, a method increasing the dimension of the assignment matrix is presented to provide massive connectivity in each cell. The random pattern of arrangement of non-zero elements access (RPA) policy is utilized to allocate the available SCMA resources. Then the coverage probability is characterized under the nearest-BS (NBS) association mechanism by developing a multi-user connectivity model based on the RPA policy and average connectivity ratio (ACR). A compact closed-form expression of the coverage probability and its approximation are derived by utilizing the Toeplitz matrix and finite sum of power series, respectively. We then extract the area potential spectral efficiency (APSE) based on the coverage probability, and provide new insights into the 3D dense multi-cell SCMA networks. Simulation results confirm the precision of the theoretical results and demonstrate the advantage of the SCMA scheme.

A Closed-Form Correction for the Spalart–Allmaras Turbulence Model for Separated Flows

The field inversion and machine learning (FIML) approach is leveraged to obtain a closed-form correction for the Spalart–Allmaras turbulence model to improve predictions of separated flows. Based on field inversion results obtained using the first-generation FIML Classic approach, a simple and compact closed-form expression is chosen to be used as correction model. The thus obtained correction model is optimized using the second-generation FIML Direct approach. Training and validation cases consist of a selection of airfoils in a wide range of flow conditions as well as the flat plate. The correction model and results for the training and validation cases obtained with the augmented turbulence model are presented, demonstrating the improved flow predictions.

Dynamic polarizability of low-dimensional excitons

Excitons in low-dimensional materials behave mathematically as confined hydrogen atoms. An appealing unified description of confinement in quantum wells or wires, etc., is found by restricting space to a fractional dimension 1 < D <= 3 serving as an adjustable parameter. We compute the dynamic polarizability of D-dimensional excitons in terms of discrete and continuum oscillator strengths. Analyzing exact sum rules, we show that continuum contributions are increasingly important in low dimensions. The dynamical responses of excitons in various dimensions are compared. Finally, an exact and compact closed-form expression for the dynamic polarizability is found. This completely general formula takes D as input and provides exact results for arbitrary frequency.

Associated Conformable Fractional Legendre Polynomials

Along with the work of Abul-Ez et al. [37], we introduce the associated conformable fractional Legendre polynomials (ACFLPs), from which the fractional differential equation of ACFLPs is established. Subsequently, some of interesting properties are derived such as generating function, hypergeometric representation, analytical formula, besides various of recurrence relations. Also, orthogonal properties of ACFLPs are developed in conformable context. We append our study by presenting the shifted ACFLPs and driving some of important properties such as Rodrigues’ type representation formula of fractional order derivative and explicit formula. An interesting compact closed-form expression is derived from the definite integral using a convenient analytical formula for the shifted ACFLPs. This result is easily generalized for integrands involving products of an arbitrary number of shifted associated Legendre polynomials in conformable sense.

Modern Random Access: An Age of Information Perspective on Irregular Repetition Slotted ALOHA

Age of information (AoI) is gaining attention as a valuable performance metric for many IoT systems, in which a large number of devices report time-stamped updates to a central gateway. This is the case, for instance, of remote sensing, monitoring, or tracking, with broad applications in the industrial, vehicular, and environmental domain. In these settings, AoI provides insights that are complementary to those offered by throughput or latency, capturing the ability of the system to maintain an up-to-date view of the status of each transmitting device. From this standpoint, while a good understanding of the metric has been reached for point-to-point links, relatively little attention has been devoted to the impact that link layer solutions employed in IoT systems may have on AoI. In particular, no result is available for modern random access protocols, which have recently emerged as promising solutions to support massive machine-type communications. To start addressing this gap we provide in this paper the first study of the AoI of a scheme in this family, namely irregular repetition slotted ALOHA (IRSA). By means of a Markovian analysis, we track the AoI evolution at the gateway, prove that the process is ergodic, and derive a compact closed form expression for its stationary distribution. Leaning on this, we compute exact formulations for the average AoI and the age violation probability. The study reveals non-trivial design trade-offs for IRSA and highlights the key role played by the protocol operating frame size. Moreover, a comparison with the performance of a simpler slotted ALOHA strategy highlights a remarkable potential for modern random access schemes in terms of information freshness.

Analytical Expression for Resonances of an Inhomogeneous, Radial, Lossless LC Network

This paper is a sequel to the authors’ previous work wherein an analytical expression was derived for the harmonic sum of squares of natural frequencies of a lossless LC ladder network, which is a widely accepted means to model a single, isolated winding for analyzing impulse behavior. The present paper extends the previous result, so that it is applicable to multiple windings in actual transformers. The complexities of the previous work, viz., inhomogeneity and an exhaustive consideration of inductive coupling are retained, and additionally all possible capacitive couplings are now considered. Furthermore, the topology considered in this paper is generic enough to accommodate inductive branches emerging from any node. To the best of authors’ knowledge, there exists no compact closed-form expression to relate the natural frequencies to the inductances and capacitances for a network having this extent of arbitrariness in topology. The success of this method can be ascribed to careful inspection of the inductance matrix to identify hidden pattern and/or structure, followed by their deft manipulation guided by the basic law of mutual coupling between inductors. Interestingly, even a physical meaning can be assigned to each term of the derived expression, and this is done with the intention of assisting its use to conjure new applications.

Direct Derivation of the Stochastic CRB of DOA Estimation for Rectilinear Sources

Several direction of arrival (DOA) estimation algorithms have been proposed to exploit the structure of rectilinear or strictly second-order noncircular signals. But until now, only the compact closed-form expressions of the corresponding deterministic Cramer–Rao bound (DCRB) have been derived because it is much easier to derive than the stochastic CRB (SCRB). As this latter bound is asymptotically achievable by the maximum likelihood estimator, while the DCRB is unattainable, it is important to have a compact closed-form expression for this SCRB to assess the performance of DOA estimation algorithms for rectilinear signals. The aim of this letter is to derive this expression directly from the Slepian–Bangs formula including in particular the case of prior knowledge of uncorrelated or coherent sources. Some properties of these SCRBs are proved and numerical illustrations are given.

Estimation Performance for the Bayesian Hierarchical Linear Model

Bayesian hierarchical modelling is a well-established branch of Bayesian inference. In this letter, we derive and study the estimation performance for the Bayesian hierarchical linear model (BHLM). Specifically, we consider a linear model with hierarchical priors for the involved amplitude and noise vectors. We provide closed-form expressions of the Bayesian Cramer-Rao bound (BCRB) for the following settings: (i) an arbitrary prior and hyperprior and (ii) a Gaussian- $\mathcal{Y}$ prior for the amplitudes, while the prior of noise is a Gaussian- $\mathcal{X}$ in both cases. Gaussian- $\mathcal{X}$ means that the conditional prior given the hyperparameter is Gaussian and $\mathcal{X}$ is the hyperprior. For the hierarchical distribution associated with spherical invariant random variables, the BCRB has a compact closed-form expression and enjoys several interesting properties that are discussed. Finally, we provide a theoretical analysis of the statistical efficiency of the linear minimum mean square error (MMSE) estimator in the low- and high-noise variance regimes when the hyperparameters are stochastically dominant.

A Novel Analytical Model for Service Delay in IEEE 802.11 Networks

In this paper, we present a novel analytical model for the service delay in IEEE 802.11 networks. This model is capable of closely tracking service delay's behavior in real wireless local area networks. We find a compact closed-form expression for the probability-generating function (PGF) of service delay using which all higher-order moments are achievable. We begin with finding the service-delay PGF; then we determine the probability density (mass) function of the delay metric by taking the inverse z-transform using numerical methods. At last, by comparing the first two statistical moments of this PGF with the mean delay and standard deviation from simulation results, the accuracy of the proposed analytical model is proved.

High-Rate Vector Quantization for the Neyman–Pearson Detection of Correlated Processes

This paper investigates the effect of quantization on the performance of the Neyman-Pearson test. It is assumed that a sensing unit observes samples of a correlated stationary ergodic multivariate process. Each sample is passed through an <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> -point quantizer and transmitted to a decision device which performs a binary hypothesis test. For any false alarm level, it is shown that the miss probability of the Neyman-Pearson test converges to zero exponentially as the number of samples tends to infinity, assuming that the observed process satisfies certain mixing conditions. The main contribution of this paper is to provide a compact closed-form expression of the error exponent in the high-rate regime, i.e., when the number <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> of quantization levels tends to infinity, generalizing previous results of Gupta and Hero to the case of nonindependent observations. If <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> represents the dimension of one sample, it is proved that the error exponent converges at rate <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2/d</sup> to the one obtained in the absence of quantization. As an application, relevant high-rate quantization strategies which lead to a large error exponent are determined. Numerical results indicate that the proposed quantization rule can yield better performance than existing ones in terms of detection error.

On the Secrecy Rate of Multi-Antenna Wiretap Channel under Finite-Alphabet Input

This work investigates the effect of limiting source signals to finite-alphabet on the secrecy rate of a multi-antenna wiretap system. Existing works have characterized maximum achievable secrecy rate or secrecy capacity for single- and multi-antenna systems based on Gaussian source signals and secrecy code. Despite the impracticality of Gaussian source, its compact closed-form expression of mutual information motivated broad use of Gaussian input assumption. For practical consideration, we study the effect of finite discrete-constellation on instantaneous and ergodic secrecy rate of multiple-antenna wire-tap channels. Our results demonstrate substantial difference between systems involving finite-alphabet inputs and systems with Gaussian inputs. Results from Gaussian inputs serve as upper-bound of secrecy rate for practical systems with finite alphabet inputs.

The Deflation-Based FastICA Estimator: Statistical Analysis Revisited

This paper provides a rigorous statistical analysis of the deflation-based FastICA estimator, where the independent components (ICs) are extracted sequentially. The focus is on two aspects of the estimator: robustness against outliers as measured by the influence function (IF) and on its asymptotic relative efficiency (ARE) as measured by the ratio of the asymptotic variance of the FastICA w.r.t. the optimal maximum likelihood estimator (MLE). The derived compact closed-form expression of the IF reveals the vulnerability of the FastICA estimator to outliers regardless of the used nonlinearity. A cautionary finding is that even a moderate observation towards certain directions can render the estimator deficient in the sense that its separation performance degrades worse than a plain guess. The IF allows the derivation of a compact closed-form expression for the asymptotic covariance matrix of the FastICA estimator and subsequently its asymptotic relative efficiencies (AREs). The ARE figures calculated for some selected source distributions illustrate the fact that the order which the ICs are found is crucial as the accuracy of the previously extracted components can dominantly affect the accuracy of the successive deflation stages.

Multiuser performance of M-ary orthogonal coded/balanced UWB transmitted-reference systems

A tractable and compact closed-form expression on the channel-averaged signal-to-interference-plus-noise ratio (SINR) is derived for M-ary orthogonal coded/balanced transmitted-reference (BTR) systems, taking into consideration both inter-pulse interference (IPI) and multiple-access interference (MAI) in dense multipath ultra-wideband (UWB) channels. The UWB channel here is a realistic and standard one with lognormal channel gain distribution and double independent Poisson arrival distribution of cluster and ray. Hence, the analytical framework developed here can be applied to typical UWB channel models, especially considering the channel sparseness and cluster overlapping observed in realistic UWB channels. Based on the channel-averaged SINR, the effect of inter-pulse distance (between the reference and data pulses in BTR) on the multiuser performance is fully investigated. A proper selection of user-specific inter-pulse distances in multiuser scenario is then determined to maximize the user capacity for a given UWB channel.

Geometric characterization and parametric representation of the singularity manifold of a 6–6 Stewart platform manipulator

In this paper, we present a compact closed-form expression for the singularity manifold of a class of 6–6 Stewart platform manipulators most commonly used in research and industry. The singularity manifold is obtained as the hyper-surface in the task-space, SE(3), on which the wrench transformation matrix for the top platform degenerates. This condition leads to an extremely large expression containing algebraic and trigonometric functions of the architecture, position and orientation variables. We present algorithms for efficient symbolic simplification of such large expressions. Using these algorithms, for a given architecture and orientation, the singularity manifold is obtained as a cubic surface in R 3 . The symbolic computations yield a simple parametric expression for the surface in terms of the architectural and orientation parameters of the manipulator, and allows us to completely characterize and visualize the singularity manifold. We show that, in general, the cubic surface is a one-parameter family of hyperbolas in planes parallel to the base of the manipulator. It is further shown that the hyperbola degenerates to a parabola in a unique plane, and to a pair of straight lines in four other planes. The explicit parameterization allows us to obtain the location of each of these special planes analytically. For a given architecture and position, the singularity manifold is a surface in SO(3), which can be, in general, algebraically described by a 6th degree polynomial in the Rodrigue’s parameters. In this paper, we present explicit expressions for the polynomial defining the orientation singularity manifold in terms architecture and orientation parameters. The theoretical results are illustrated with several numerical examples.

On the overlap integral of associated Legendre polynomials

A simple compact closed-form expression is derived from the definite integral using a convenient analytical formula for the associated Legendre function . This result is easily generalized for integrands involving products of an arbitrary number of associated Legendre polynomials.

Transmission-line analysis of a capacitively coupled microstrip-ring resonator

The resonant frequencies of a microstrip-ring resonator capacitively coupled to a feed line are accurately analyzed using a transmission-line model. By making use of ABCD- and Y-admittance matrices, a compact closed-form expression for the input impedance of the ring alone is analytically derived and shows that the ring ran be equivalently viewed as a frequency-dependent capacitor. The coupling gap is then modeled by an equivalent L-network comprising a parallel and a series gap capacitance obtained by modifying Garg and Bahl's closed-form expressions for an end-to-end microstrip gap. By simplifying the parallel and series combinations of the overall equivalent circuit, the total input impedance looking from the feed line to the gap is analytically derived to predict the resonant frequencies. To verify the analysis, the resonant frequencies of the capacitively coupled ring resonator have been accurately measured, with the experimental results showing very good agreement with the theoretical predictions.

Rotational rainbow structures in collisions of electrons with highly symmetric polyatomic molecules

Following the studies of diatomic and highly symmetric planar molecules in collision-induced rotational excitation (discussed in a spectator model) an extension to highly symmetric spatial targets is developed. For the calculated cases all molecular atoms are treated as equivalent (with the liberty of a different atom at the centre of mass). A compact closed-form expression for the state-to-state differential cross sections is derived. Selection rules and rotational rainbow structures are discussed.