Saddle-point Integration Research Articles

Performance Evaluation for Forward-Link Cellular DS-CDMA overFrequency-Selective Nakagami Multipath Fading Channels

The bit-error rate (BER) for a forward-link cellular direct-sequence code-division multiple-access (DS-CDMA) system is evaluated. This analysis takes into account the effects of multi-cell interference resulting from Nakagami multipath fading, frequency selectivity, path loss and mobile user spatial distribution. Exponentially decaying multipath intensity profile (MIP) is adopted in the model to investigate its significance on the overall performance. In order to evaluate the BER performance efficiently, saddlepoint integration (SPI) is applied in the analysis. By fixing the number of resolvable multipaths and varying the number of fingers for the RAKE receiver, it can be shown that the capacity of a DS-CDMA cellular system increases almost linearly with the number of RAKE fingers.

Evaluating the error probability in lightwave systems with chromatic dispersion, arbitrary pulse shape and pre- and postdetection filtering

A novel approach to analytically evaluate the bit error probability in optically preamplified direct-detection systems is presented, which can take into account the effects of pulse shaping, chirping, filtering at the transmitter and the receiver, both pre- and postdetection, chromatic dispersion, and ASE noise. The method is computationally very fast in that the saddle point integration method for solving the resulting line integral of a particular moment generating function is adopted. A closed-form approximation for the bit error probability is also provided, which is within 0.01 dB from the exact numerical results.

Universal renormalization of saddle-point integrals for condensed Bose gases.

When treating the ground-state contribution exactly, a variant of the saddle-point method emerges that works even for condensed Bose gases. Results thus obtained, such as canonical partition functions, differ by universal renormalization factors from those provided by the conventional but incorrect scheme. The amended method yields the statistical properties of ideal and very weakly interacting Bose gases with a fixed number of particles with particular simplicity.

Distributions of singular values for some random matrices.

The singular value decomposition is a matrix decomposition technique widely used in the analysis of multivariate data, such as complex space-time images obtained in both physical and biological systems. In this paper, we examine the distribution of singular values of low-rank matrices corrupted by additive noise. Past studies have been limited to uniform uncorrelated noise. Using diagrammatic and saddle point integration techniques, we extend these results to heterogeneous and correlated noise sources. We also provide perturbative estimates of error bars on the reconstructed low-rank matrix obtained by truncating a singular value decomposition.

Forward-link analysis of cellular DS-CDMA over generalised fading channels using saddlepoint integration

The bit-error rate (BER) analysis of an asynchronous cellular direct-sequence code-division multiple-access (DS-CDMA) system with channel fading and random mobile locations often results in a multiintegral expression, even though the Gaussian approximation is applied. In the paper, saddlepoint integration (SPI) is proposed to simplify the analysis. The computational complexity grows only linearly with the number of interfering cells in the system, regardless of the number of mobiles in each cell. By making use of the derived expression, the BER performance for arbitrary Nakagami parameters and path-loss exponents is evaluated efficiently. Simulation results show that the SPI technique is accurate and reliable.

Error probabilities for QAM systems on partially coherent channels with intersymbol interference and crosstalk

We describe an efficient procedure to calculate the probability of error P/sub e/ for a quadrature amplitude modulation (QAM) communications system operating over a channel that introduces distortion, interference, and noise. The method is an extension of the saddlepoint integration technique introduced by Helstrom (1986) to efficiently evaluate P/sub e/ for one-dimensional pulse-amplitude modulation (PAM) systems with intersymbol interference (ISI) and crosstalk. We consider the effects of noise, random carrier phase offset, ISI, and crosstalk between the I and Q channels. The error probability is written as a double Laplace inversion integral and can be easily applied to any rectangular constellation. This integral is calculated by extending the saddlepoint integration technique to two complex dimensions. Results are presented for QAM systems with 16, 64, and 256 symbols. The technique can be directly extended to environments with cochannel interference consisting of other QAM signals.

Quantum Hall transition in an array of quantum dots

A two-dimensional array of quantum dots in a magnetic field is considered. The electrons in the quantum dots are described as unitary random matrix ensembles. The strength of the magnetic field is such that there is half a flux quantum per plaquette. This model exhibits the integer quantum Hall effect. For N electronic states per quantum dot the limit N\ensuremath{\rightarrow}\ensuremath{\infty} can be solved by a saddle-point integration of a supersymmetric field theory. The effect of level statistics on the density of states and the Hall conductivity is compared with the effect of temperature fluctuations.

Computing the distribution of sums of random sine waves and of Rayleigh-distributed random variables by saddle-point integration

The method of saddle-point integration is applied to calculating the cumulative distribution of sums of independent random variables that either have a Rayleigh distribution or are sine waves with independently random phases in the presence of Gaussian random noise.

Calculating the Distribution of the Serial Correlation Estimator by Saddlepoint Integration

The efficient method of numerical saddlepoint integration is described and applied to calculating the probability distribution of the maximum likelihood and Yule-Walker estimators of the correlation coefficient a of a first-order autoregressive normal time series with initial value either zero or nonzero when a finite number n of data are at hand. Stationary time series of the same type are also treated. Significance points are computed in a number of examples to show how, as n increases, the finite-sample distributions approach the asymptotic distributions that have appeared in the literature.

Nearside–farside analysis of differential cross sections: Diffraction and rainbow scattering in atom–atom and atom–molecule rotationally inelastic sudden collisions

Nearside–farside (NF) theory, as used to understand nuclear heavy-ion differential cross sections, is applied for the first time to the angular scattering of atom–atom and atom–diatom collisions. A NF decomposition of the partial wave series (PWS) for the scattering amplitude has the following advantages: (a) it is exact, (b) it uses PW scattering matrix elements (quantum or semiclassical) as calculated by standard computer programs, (c) it is easily incorporated into existing computer programs which calculate angular distributions, (d) semiclassical techniques, such as stationary phase or saddle point integration, are not invoked for the PWS, although the semiclassical picture is still evident. A disadvantage of a NF decomposition is that it is not unique. The Fuller and Hatchell NF decompositions are used to analyze the angular scattering of four collision systems whose PWS involve Legendre polynomials: (a) atom–atom He+Ne elastic diffraction scattering, (b) atom–atom H++Ar elastic rainbow scattering, (c) atom rigid-rotator Ne+D2(j=0) →Ne+D2(j) diffraction scattering under sudden conditions so that the infinite-order-sudden (IOS) approximation is valid, (d) atom rigid-rotator He+N2(j=0)→He+N2(j) rotational rainbow IOS scattering. The utility of these two NF decompositions is assessed by comparison with results from the semiclassical complex angular momentum (CAM) representation of the scattering amplitude. This is chosen because it allows an unambiguous separation of the scattering amplitude into nearside and farside subamplitudes under semiclassical conditions. The Fuller NF decomposition, unlike the Hatchell NF decomposition, provides a physically clear explanation of the angular scattering, which always agrees with the semiclassical CAM interpretation (except for scattering angles ≊180°). The Fuller NF decomposition is therefore recommended for applications to atomic and molecular collisions. The NF theory for the decomposition of Legendre polynomials is generalized to scattering amplitudes whose PWS involve associated Legendre functions or reduced rotation matrix elements.

Spatial diversity equalization for MIMO ocean acoustic communication channels

The available bandwidth of ocean acoustic channels is inherently narrow that impedes high-transmission rate and makes it difficult for multiple users to communicate simultaneously. To alleviate this problem, spatial diversity antennas are used to increase the date rate. In this paper, we employ the spatial diversity equalizers (SDE) to increase the effective channel bandwidth by minimizing the mean-square error (mse). Although joint equalizers have been used in digital telephone subscriber lines to suppress crosstalk, we apply the concept to ocean acoustic channels and show that multiuser communication is possible despite the narrow-channel bandwidth. In addition, we will show that the advantage of SDE is not because we use more taps, but because we collect the data carried through various ocean paths. By applying the saddle point integration method to multiinput multioutput (MIMO) channels, we compute the probability of error (PE) to show that a factor of 3-4 of channel reuse is possible.

Approximating the pairwise error probability forfading channels

The computation of the pairwise error probability (PEP) for coded digital communications in the presence of fading is often performed by using the Chernoff upper bound, which is simple to evaluate but rather loose. Exact evaluation of the PEP is possible in principle by computing the residues of a complex function, but this may be complicated in practice. In the Letter the authors derive two approximations to the PEP. The first one, based on saddle-point integration, is simple to compute and generally tighter than the Chernoff bound. The second one, based on Gauss–Chebyshev quadrature rules, is both easy to use and computationally stable, and approximates the true value of PEP within any degree of accuracy.

Macroscopic Lyapunov functions for separable stochastic neural networks with detailed balance

We derive macroscopic Lyapunov functions for large, long-range, Ising-spin neural networks with separable symmetric interactions, which evolve in time according to local field alignment. We generalize existing constructions, which correspond todeterministic (zero-temperature) evolution and to specific choices of the interaction structure, to the case ofstochastic evolution and arbitrary separable interaction matrices, for both parallel and sequential spin updating. We find a direct relation between the form of the Lyapunov functions (which describe dynamical processes) and the saddle-point integration that results from performing equilibrium statistical mechanical studies of the present type of model.

Performance evaluation of QAM and QPRS systems with interfering channels in frequency selective fading

An analytical technique is developed to evaluate the error rate performance of M-ary quadrature amplitude modulation (QAM) and quadrature partial response signalling (QPRS) in the presence of adjacent channel and cochannel interference with frequency selective fading. Error probability expressions are derived and evaluated numerically using the method of saddle-point integration. In addition to the above impairments, the system model also incorporates other disturbances, namely stationary white gaussian noise, quadrature channel interference, random timing misalignments and carrier phases of interfering signals. An extensive set of results and comparisons are presented.

On Ramanujan's Q-function

This study provides a detailed analysis of a function which Knuth discovered to play a central rôle in the analysis of hashing with linear probing. The function, named after Knuth Q( n), is related to several of Ramanujan's investigations. It surfaces in the analysis of a variety of algorithms and discrete probability problems including hashing, the birthday paradox, random mapping statistics, the “rho” method for integer factorization, union-find algorithms, optimum caching, and the study of memory conflicts. A process related to the complex asymptotic methods of singularity analysis and saddle point integrals permits to precisely quantify the behaviour of the Q( n) function. In this way, tight bounds are derived. They answer a question of Knuth ( The Art of Computer Programming, Vol. 1, 1968, [Ex. 1.2.11.3.13]), itself a rephrasing of earlier questions of Ramanujan in 1911–1913.

Calculating the K-distribution by saddlepoint integration

An important probability distribution for modelling non-Gaussian phenomenon is the K-distribution. Because of its widespread use in radar and underwater acoustics, a method for calculating the distribution is required which is both easy and efficient. The paper presents a method for calculating the K-distribution, in particular the K-Bessel function, using saddlepoint integration. The paper provides a detailed truncation-error analysis of the technique, and an implementation in MATLAB.

Binary nucleation kinetics: A matrix method

The kinetics of binary nucleation is investigated and an exact method to evaluate the nucleation rate is developed. The exact method accounts for all flows to the stable clusters, and thus the nucleation rate is obtained independently of determining the location of the saddle point in the Gibbs free energy surface. The exact method is based on determination of cluster concentrations, which can be obtained after solving a set of linear equations. The values of the nucleation rate and the direction of flow at the saddle point are calculated for water–sulphuric acid and water–ammonia systems. The results are compared to the results obtained by Stauffer’s saddle point integration method for binary nucleation kinetics. The saddle point integration method is found to give a reasonably accurate estimation of nucleation rate in the water–sulphuric acid system. However, in the water–ammonia system the difference is found to be as high as three orders of magnitude.

Dilute system of hard-core bosons: a soluble limit

A model for bosons with N degenerate energy levels per site and a hard-core interaction of bosons at the same level is introduced. The limit N → ∞ can be solved exactly in terms of a saddle point integration. The zero temperature limit is investigated where the quantum fluctuations drive a transition from an insulating to a superfluid phase. The results of the calculations agree with those of the mean-field theory for weakly interacting bosons only near the phase transition.

The dyadic Green's function for the cylindrical chirostrip antenna

Using the eigenfunction expansion of dyadic Green's function in unbounded chiral medium, and the method of scattering superposition, the expressions of dyadic Green's function for the one and two—layer cylindrical chirostrip antennas are derived, when the field sources are placed in any layer of cylindrical chirostrip antenna. Also, using the method of saddle point integration, the asymptotic expression of dyadic Green's function for the wraparound chirostrip antenna is obtained. The results can be directly used to analyse the radiation characteristics of cylindrical chirostrip dipolc antenna, et al.

The large-N magnetization in matrix models revisited

We analyse the large-N limit of a matrix model with quartic self-interaction and external scalar field, by the technique of orthogonal polynomials. The rise of a «spontaneous magnetization» is studied and compared with the previous known results, based on saddle point integration or Schwinger-Dyson equations.