Characteristic Function Approach Research Articles

A General Model for Performance Evaluation in DS-CDMA Systems with Variable Spreading Factors

This paper extends previous analytical approaches for the study of CDMA systems to the relevant case of multipath environments where users can operate at different bit rates. This scenario is of interest for the Wideband CDMA strategy employed in UMTS, and the model permits the performance comparison of classic and more innovative spreading signals. The method is based on the characteristic function approach, that allows to model accurately the various kinds of interferences. Some numerical examples are given with reference to the ITU-R M. 1225 Recommendations, but the analysis could be extended to different channel descriptions.

Precise BER Formulas for Asynchronous QPSK-Modulated DS-CDMA Systems Using Random Quaternary Spreading Over Rayleigh Channels

Precise bit-error-ratio (BER) analysis of an asynchronous QPSK-modulated direct-sequence code-division multiple-access system using random quaternary spreading sequences for transmission over Rayleigh channels is performed based on the characteristic-function approach. Its accuracy is verified by our numerical simulation results and also compared with those of the Gaussian approximation.

Approximate distribution of demerit statistic—A bounding approach

The traditional classical demerit control chart is used to plot the demerit statistic, a weighted sum of the number of defects in each category, on a control chart. The approximate normal method is usually used to obtain control limits though the distribution that depends on the values of the weights and the parameters of the Poisson distribution which may not always be normal. [Jones, L.A., Woodall, W.H., Conerly, M.D., 1999. Exact properties of demerit control charts. Journal of Quality Technology 31 (2), 207–216] used the characteristic function approach to determine the distribution of the demerit statistic. Unfortunately, the process that they used to determine the distribution needs complex integral evaluation via mathematical software packages or using the approximate truncated infinite series. Moreover, the characteristic function does not provide an accurate result easily. In this paper, a bounding approach is proposed to determine the approximate distribution of the demerit statistic. It is easy to implement and also the approximate error can be controlled to meet the desired accuracy. In addition, an example is demonstrated to illustrate the proposed method. The results indicate that the proposed approach is efficient and accurate. Finally, the performance among the approximate normal method, the characteristic function approach, and the proposed bounding approach are discussed.

Qubit quantum channels: A characteristic function approach

A characterization of qubit quantum channels is introduced. In analogy to what happens in the context of bosonic channels we exploit the possibility of representing the states of the system in terms of characteristic function. The latter are functions of noncommuting variables (Grassmann variables) and are defined in terms of generalized displacement operators. In this context we introduce the set of Gaussian channels and show that they share similar properties with the corresponding bosonic counterpart.

A characteristic function approach to the biased sampling model, with application to robust logistic regression

Many sampling problems from multiple populations can be considered under the semiparametric framework of the biased, or weighted, sampling model. Included under this framework is logistic regression under case–control sampling. For any model, atypical observations can greatly influence the maximum likelihood estimate of the parameters. Several robust alternatives have been proposed for the special case of logistic regression. However, some current techniques can exhibit poor behavior in many common situations. In this paper a new family of procedures are constructed to estimate the parameters in the semiparametric biased sampling model. The procedures incorporate a minimum distance approach, but are instead based on characteristic functions. The estimators can also be represented as the minimizers of quadratic forms in simple residuals, thus yielding straightforward computation. For the case of logistic regression, the resulting estimators are shown to be competitive with the existing robust approaches in terms of both robustness and efficiency, while maintaining affine equivariance. The approach is developed under the case–control sampling scheme, yet is shown to be applicable under prospective sampling logistic regression as well.

Goodness-of-fit tests for symmetric stable distributions—Empirical characteristic function approach

We consider goodness-of-fit tests of symmetric stable distributions based on weighted integrals of the squared distance between the empirical characteristic function of the standardized data and the characteristic function of the standard symmetric stable distribution with the characteristic exponent $\alpha$ estimated from the data. We treat $\alpha$ as an unknown parameter, but for theoretical simplicity we also consider the case that $\alpha$ is fixed. For estimation of parameters and the standardization of data we use maximum likelihood estimator (MLE) and an equivariant integrated squared error estimator (EISE) which minimizes the weighted integral. We derive the asymptotic covariance function of the characteristic function process with parameters estimated by MLE and EISE. For the case of MLE, the eigenvalues of the covariance function are numerically evaluated and asymptotic distribution of the test statistic is obtained using complex integration. Simulation studies show that the asymptotic distribution of the test statistics is very accurate. We also present a formula of the asymptotic covariance function of the characteristic function process with parameters estimated by an efficient estimator for general distributions.

Exact and approximate error-rate analysis of BPSK in Weibull fading with cochannel interference

The performance of binary phase shift keying in a Weibull fading environment is analysed in the presence of cochannel interference (CCI) in terms of the bit error rate (BER). We derive an exact expression for the BER in this case on the basis of a characteristic function approach. Furthermore, since it is difficult to obtain the distribution of the CCI and because of the complexity of the obtained exact BER expression, we derive an approximate expression for the BER by invoking the central limit theorem. The validity of the exact expression and the accuracy of the approximate one are verified by comparing their results with those obtained via Monte Carlo simulations. The effect of such factors as the number of interferers and the fading severity of both the desired as well as the interfering signals on performance is also investigated.

Performance study of linear and nonlinear diversity-combining techniques in synchronous FFH∕MA communication systems over fading channels

Based on the Fourier-Bessel series and exponential characteristic function approach, the paper presents analytical bit-error-rate expressions for synchronous fast frequency-hopped spread-spectrum multiple-access systems over Rician fading channels with M-ary frequency-shift-keying modulation. Linear-combining, product-combining and clipper receivers are studied. Numerical results show that both the clipper and product-combining receivers perform significantly better than the linear-combining receiver. The clipper receiver with an optimum threshold level slightly outperforms the product-combining receiver. However, the product-combining receiver is the preferred diversity-combining method because it is completely nonparametric. In addition, the optimum diversity levels for various diversity-combining receivers over Rayleigh fading channels are higher than that of the Rician fading channels.

Wigner distribution for operators at different times

The paper considers the problem of finding the joint probability distribution of position, P(q 0,q 1,…, qN ), at the different times t 0,t 1, …, tN . The problem is addressed both classically and quantum mechanically. The characteristic function approach is used and it is shown that for certain cases the quantum mechanical answer is the same as the classical answer with replacement of the initial classical probability distribution by the Wigner distribution of the initial wave function. This is the case for taking a certain form of the quantum characteristic function. Other choices are also discussed.

A novel analytical approach to parameter extraction for on-chip spiral inductors taking into account high-order parasitic effect

We present an analytical approach to extract model parameters for silicon on-chip spiral inductors taking into account various high-order parasitic effects such as skin effect and substrate coupling. The parameters for the series branch were extracted using iterative analytical calculations, and parameters for the substrate branch were extracted using a modified linear characteristic-function approach via the introduction of a partition rule. As demonstrated by a series of inductors fabricated using 0.18μm CMOS process, simulations can provide a high-precision fit with experimental data over a broad frequency range between 0.1 and 10GHz.

Statistical properties of the band-broadening function

Statistical properties of the band-broadening function calculated from the equilibrium and kinetic theories of Giddings and Eyring are compared with real liquid-chromatography elution curves of low-molecular-weight substances and with light-scattering elution curves of a high-molecular-weight polystyrene standard injected at several concentrations. The curves obtained by liquid chromatography agree in a large span of elution times with the theoretical ones and can be described by one value of the plate-height. The size exclusion chromatography (SEC) elution curves seem to approach the predicted statistical properties with decreasing concentration of injected polymer but, in general, a reliable prediction of band-broadening (BB) functions in SEC of polymers, especially of non-symmetric BB functions near the exclusion limit, seems not possible at this point. The relation of the theory with the characteristic-function approach developed by Dondi is also discussed.

RF inductors with suspended and copper coated thick crystalline silicon spirals for monolithic MEMS LC circuits

This letter presents a novel radio frequency (RF) inductor in a monolithic inductor-capacitor circuit developed by using micro-electromechanical systems (MEMS) fabrication technology. The inductor consists of 40-μm-thick single crystalline silicon spiral with copper surface coating as the conductor, which is suspended on a glass substrate. The fabricated inductor exhibits a self-resonance frequency higher than 15GHz, with the quality factor more than 35 and inductance over 5nH at 11.3GHz. Simulations based on a compact equivalent circuit model with parameters extracted using a characteristic-function approach have also been carried out for the inductor, and good agreement with measurements is obtained.

Empirical characteristic function approach to goodness-of-fit tests for the Cauchy distribution with parameters estimated by MLE or EISE

We consider goodness-of-fit tests of the Cauchy distribution based on weighted integrals of the squared distance between the empirical characteristic function of the standardized data and the characteristic function of the standard Cauchy distribution. For standardization of data Gurtler and Henze (2000,Annals of the Institute of Statistical Mathematics,52, 267–286) used the median and the interquartile range. In this paper we use the maximum likelihood estimator (MLE) and an equivariant integrated squared error estimator (EISE), which minimizes the weighted integral. We derive an explicit form of the asymptotic covariance function of the characteristic function process with parameters estimated by the MLE or the EISE. The eigenvalues of the covariance function are numerically evaluated and the asymptotic distributions of the test statistics are obtained by the residue theorem. A simulation study shows that the proposed tests compare well to tests proposed by Gurtler and Henze and more traditional tests based on the empirical distribution function.

Performance evaluation for quaternary DS-SSMA communications with complex signature sequences over Rayleigh-fading channels

Performance of quaternary direct-sequence spread-spectrum multiple-access (DS-SSMA) systems with complex signature sequences and complex modulators and receivers in flat Rayleigh fading is investigated in this paper. Due to the availability of potentially large sets of complex sequences with good correlation characteristics, the interest of using complex spreading sequences in DS-SSMA has increased dramatically. The complex spreading sequences investigated in this paper include the recently introduced orthogonal unified complex Hadamard transform (UCHT) sequences. In this paper, complex processing in modulators and receivers is also employed in order to take advantage of the correlation properties of complex signature sequences. The average bit error rate (BER) for quaternary synchronous systems is obtained first, and then the BER for quaternary asynchronous systems is evaluated using characteristic function approach. Result based on Gaussian approximation method is also presented for asynchronous systems. The numerical examples illustrate that the systems based on UCHT spreading sequences perform generally better than the Gold sequences and the 4-phase family A-sequences.

On the Error Performance Analysis of Space–Time Trellis Codes

We present analytical performance results for space-time trellis codes over spatially correlated Rayleigh fading channels. Bit-error-probability estimates are obtained based on the derivation of an exact pairwise error probability expression using a residue technique combined with a characteristic function approach. We investigate both quasi-static and interleaved channels and demonstrate how the spatial fading correlation affects the performance of space-time codes over these two different channel models. Simulation results are also included to confirm the accuracy of analytical estimates.

Stochastic theory of size exclusion chromatography by the characteristic function approach

A general stochastic theory of size exclusion chromatography (SEC) able to account for size dependence on both pore ingress and egress processes, moving zone dispersion and pore size distribution, was developed. The relationship between stochastic-chromatographic and batch equilibrium conditions are discussed and the fundamental role of the ‘ergodic’ hypothesis in establishing a link between them is emphasized. SEC models are solved by means of the characteristic function method and chromatographic parameters like plate height, peak skewness and excess are derived. The peak shapes are obtained by numerical inversion of the characteristic function under the most general conditions of the exploited models. Separate size effects on pore ingress and pore egress processes are investigated and their effects on both retention selectivity and efficiency are clearly shown. The peak splitting phenomenon and peak tailing due to incomplete sample sorption near to the exclusion limit is discussed. An SEC model for columns with two types of pores is discussed and several effects on retention selectivity and efficiency coming from pore size differences and their relative abundance are singled out. The relevance of moving zone dispersion on separation is investigated. The present approach proves to be general and able to account for more complex SEC conditions such as continuous pore size distributions and mixed retention mechanism.

Characteristic functional approach to multiplicative fractal noise with application to environmental fluctuations in nonlinear chemical systems far from equilibrium:: A temporal analog of Anderson localization

We compute the characteristic functional of a nonlinear transformation of a set of random variables. The equation is applied to the study of fractal stochastic processes with multiplicative noise. For illustration we investigate two types of chemical systems subjected to environmental fluctuations. We analyze the interaction between the intrinsic chemical fluctuations and environmental noise and evaluate the stochastic properties of the system in the presence of multiplicative noise. For an autocatalytic reaction subjected to external noise and with Gaussian and generally non-Markovian multiplicative external noise the averaged joint probability density of concentrations displays long tails of the negative power law type, which are independent of the detailed properties of the noise. We also study the influence of multiplicative noise on a nonlinear chemical system with a stable limit cycle. In the presence of multiplicative noise the limit cycle collapses and the average trajectory in the composition space becomes a spiral. This effect is a temporal analog of Anderson localization. We evaluate the damping factors and localization times for different types of noise ranging from noise with infinite memory (static disorder) and noise with long memory (dynamic disorder with fractal noise) to Markovian noise (dynamic disorder with short memory).

Sum of gamma variates and performance of wireless communication systems over Nakagami-fading channels

Capitalizing on the Moschopoulos (1985) single gamma series representation of the probability density function (PDF) of the sum of gamma variates, we provide a PDF-based approach for the performance analysis of maximal-ratio combining and postdetection equal-gain combining diversity techniques as well as cochannel interference of cellular mobile radio systems over Nakagami (1960)-fading channels with arbitrary parameters. Aside from putting under the same umbrella many of the past results obtained via characteristic function (CF) or moment generating function (MGF)-based approaches, the proposed approach also allows the derivation of additional performance measures, which are harder to analyze via CF or MGF-based approaches.

Multivariate Distributional Tests in Risk Management: An Empirical Characteristic Function Approach

Multivariate Distributional Tests in Risk Management: An Empirical Characteristic Function Approach

Levy diffusion in a force field, huber relaxation kinetics, and nonequilibrium thermodynamics: H theorem for enhanced diffusion with Levy white noise

A characteristic functional approach is suggested for Levy diffusion in disordered systems with external force fields. We study the overdamped motion of an ensemble of independent particles and assume that the force acting upon one particle is made up of two additive components: a linear term generated by a harmonic potential and a second term generated by the interaction with the disordered system. The stochastic properties of the second term are evaluated by using Huber's approach to complex relaxation [Phys. Rev. B 31, 6070 (1985)]. We assume that the interaction between a moving particle and the environment can be expressed by the contribution of a large number of relaxation channels, each channel having a very small probability of being open and obeying Poisson statistics. Two types of processes are investigated: (a) Levy diffusion with static disorder for which the fluctuations of the random force are frozen and last forever and (b) diffusion with strong dynamic disorder and independent Levy fluctuations (Levy white noise). In both cases we show that the probability distribution of the position of a diffusing particle tends towards a stationary nonequilibrium form. The characteristic functional of concentration fluctuations is evaluated in both cases by using the theory of random point processes. For large times the fluctuations of the concentration field are stationary and the corresponding probability density functional can be evaluated analytically. In this limit the fluctuations depend on the distribution of the total number of particles but are independent of the initial positions of the particles. We show that the logarithm of the stationary probability functional plays the role of a nonequilibrium thermodynamic potential, which has a structure similar to the Helmholtz free energy in equilibrium thermodynamics: it is made up of the sum of an energetic component, depending on the external mechanical potential, and of an entropic component, depending on the concentration field. We show that the conditions for the existence and stability of the nonequilibrium steady state, which emerges for large times, can be expressed in terms of the stochastic potential. For Levy white noise the average concentration field can be expressed as the solution of a fractional Fokker-Planck equation. We show that the stochastic potential is a Lyapunov function of the fractional Fokker-Planck equation, which ensures that all transient solutions for the average concentration field tend towards a unique stationary form.