Use Of Gaussian Approximation Research Articles

On the Use of Gaussian Approximation in Analyzing the Performance of Optical Receivers

The analytical calculation of the bit error rate (BER) of digital optical receivers that employ avalanche photodiodes (APDs) is challenging due to 1) the stochastic nature of the avalanche photodiode's impulse-response function and 2) the presence of intersymbol interference (ISI). At ultrafast transmission rates, ISI becomes a dominant component of the BER, and its effect on the BER should be carefully addressed. One solution to this problem, termed the bit-pattern-dependent (PD) approach, is to first calculate the conditional BER given a specific bit pattern and then average over all possible bit patterns. Alternatively, a simplifying method, termed the bit-pattern-independent (PI) approach, has been commonly used whereby the average bit stream is used to calculate the distribution of the receiver output, which, in turn, is used to calculate the BER. However, when ISI is dominant, the PI approximation is inaccurate. Here, the two approaches are analytically compared by analyzing their asymptotic behavior and their bounds. Conditions are found to determine when the PI method overestimates the BER. The BER found using the PD method exponentially decays with the received optical power, whereas for the PI approach, the BER converges to a constant, which is unrealistic. For an InP-based APD receiver with a 100-nm multiplication layer, the PI method is found to be inaccurate for transmission rates beyond 20 Gb/s.

Modeling Spectral Envelopes Using Restricted Boltzmann Machines and Deep Belief Networks for Statistical Parametric Speech Synthesis

This paper presents a new spectral modeling method for statistical parametric speech synthesis. In the conventional methods, high-level spectral parameters, such as mel-cepstra or line spectral pairs, are adopted as the features for hidden Markov model (HMM)-based parametric speech synthesis. Our proposed method described in this paper improves the conventional method in two ways. First, distributions of low-level, un-transformed spectral envelopes (extracted by the STRAIGHT vocoder) are used as the parameters for synthesis. Second, instead of using single Gaussian distribution, we adopt the graphical models with multiple hidden variables, including restricted Boltzmann machines (RBM) and deep belief networks (DBN), to represent the distribution of the low-level spectral envelopes at each HMM state. At the synthesis time, the spectral envelopes are predicted from the RBM-HMMs or the DBN-HMMs of the input sentence following the maximum output probability parameter generation criterion with the constraints of the dynamic features. A Gaussian approximation is applied to the marginal distribution of the visible stochastic variables in the RBM or DBN at each HMM state in order to achieve a closed-form solution to the parameter generation problem. Our experimental results show that both RBM-HMM and DBN-HMM are able to generate spectral envelope parameter sequences better than the conventional Gaussian-HMM with superior generalization capabilities and that DBN-HMM and RBM-HMM perform similarly due possibly to the use of Gaussian approximation. As a result, our proposed method can significantly alleviate the over-smoothing effect and improve the naturalness of the conventional HMM-based speech synthesis system using mel-cepstra.

Turbo equalization of doubly selective channels

ABSTRACTIn this paper, turbo equalization for transmission over doubly selective channels is proposed. The maximum a posteriori probability (MAP) algorithm is used for channel detection as well as for channel decoding. The detection/decoding constituents can exchange soft information in an iterative manner resulting in the so‐called turbo equalization. The time‐varying multi‐path fading channel is modeled using the basis expansion model (BEM). In this BEM, the time‐varying channel is viewed as a bank of time‐invariant finite impulse response filters, and the time variation is captured by means of time‐varying complex exponential basis functions. Therefore, the time‐varying transition tables that characterize the time‐varying channel can also follow a similar BEM. The complexity of the MAP channel detector is rather prohibitive for practical applications. This motivates the search for lower‐complexity soft‐output channel detectors. For this purpose, soft‐output linear minimum‐mean square error (LMMSE)‐based channel detectors are proposed for single carrier as well as for multi‐carrier systems. With the use of Gaussian approximation, expressions for the a posteriori and extrinsic log‐likelihood ratios have been derived. The performance of the proposed turbo equalization schemes are evaluated using numerical simulations. Copyright © 2012 John Wiley & Sons, Ltd.

A Quadratic Volatility Cheyette Model

In this paper we present an extension of the one factor blended Cheyette model for pricing single currency exotics, allowing for a more adequate fit to the swaption volatility smile. We first present a general framework based on the HJM model and then make a separability assumption on the instantaneous forward rate volatility, thus enabling a representation of the discount curve in a finite number of Markovian state variables. We show a practical application of this family of models by analyzing calibration and pricing in the case of a quadratic volatility function. By doing so, we provide a novel and parsimonious specification of the Cheyette model. Then for calibration purposes, we develop fast and accurate approximations for European swaptions, based on standard projection and averaging techniques. We also improve the usual naive mean state estimation by the use of Gaussian approximations. Last we present an efficient large step Monte-Carlo simulation for strongly path dependent exotics.

Explicit Volatility Specification for the Linear Cheyette Model

In this paper we present an extension of the classical Hull-White framework for pricing single currency exotics, which allows for a more adequate fit to the swaption volatility smile. We first present a general framework based on the HJM model and then make a separability assumption on the instantaneous forward rate volatility, thus enabling a representation of the discount curve in a finite number of Markovian state variables.We show a practical application of this family of models by analyzing calibration and pricing in the case where the forward rate volatility is a linear function of the short rate. By doing so, we provide a novel and parsimonious specification of the Cheyette model. Then for calibration purposes, we develop fast and accurate approximations for European swaptions, based on standard projection and averaging technics. We also improve the usual naive mean state approximation by the use of Gaussian approximations. We give numerical examples.

Energy-efficient multiuser SIMO: achieving probabilistic robustness with gaussian channel uncertainty

This paper addresses the joint optimization of power control and receive beamforming vectors for a multiuser singleinput multiple-output (SIMO) antenna system in the uplink in which mobile users are single-antenna transmitters and the base station receiver has multiple antennas. Channel state information at the receiver (CSIR) is exploited but the CSIR is imperfect with its uncertainty being modeled as a random Gaussian matrix. Our objective is to devise an energy-efficient solution to minimize the individual users' transmit power while meeting the users' signal-to-interference plus noise ratio (SINR) constraints, under the consideration of CSIR and its error characteristics. This is achieved by solving a sum-power minimization problem, subject to a collection of users' outage probability constraints on their target SINRs. Regarding the signal power minus the sum of inter-user interferences (SMI) power as Gaussian, an iterative and convergent algorithm which is proved to reach the global optimum for the joint power allocation and receive beamforming solution, is proposed, though the optimization problem is indeed non-convex. A systematic scheme to detect feasibility and find a feasible initial solution, if there exists any, is also devised. Simulation results verify the use of Gaussian approximation and robustness of the proposed algorithm in terms of users' probability constraints, and indicate a significant performance gain as compared to the zero-forcing (ZF) and minimum mean square- error (MMSE) beamforming systems.

The use of Gaussian approximation for the solution of suboptimal control of radio system state vector problem

The problem of optimal non-linear stochastic systems control on the basis of information criteria use is solved. A theorem is introduced proving the possibility of reducing partial derivatives equation solution to the integration of common differential equations. On the example of Shannon criterion the control of a dynamic object is synthesized. Numerical results are obtained illustrating the efficiency of the suggested approach practical use.

Long-Distance Decoy-State Quantum Key Distribution in Optical Fiber

The theoretical existence of photon-number-splitting attacks creates a security loophole for most quantum key distribution (QKD) demonstrations that use a highly attenuated laser source. Using ultralow-noise, high-efficiency transition-edge sensor photodetectors, we have implemented the first version of a decoy-state protocol that incorporates finite statistics without the use of Gaussian approximations in a one-way QKD system, enabling the creation of secure keys immune to photon-number-splitting attacks and highly resistant to Trojan horse attacks over 107 km of optical fiber.

P-Value Simulation for Affected Sib Pair Multiple Testing

A standard approach to calculation of critical values for affected sib pair multiple testing is based on: (a) fully informative markers, (b) Haldane map function assumptions leading to a Markov chain model for inheritance vectors, (c) central limit approximation to averages of sampled inheritance vectors leading to an Ornstein-Uhlenbeck process approximation, and (d) simple approximations to the maximum of such a process. Under these assumptions, assuming equispaced or close to equispaced markers, if the sample size is large, an approximation is available that is easy to calculate and performs well. However, for small sample sizes, a large number of markers, and for small p-values, there is good reason to be cautious about the use of the Gaussian approximation. We develop an algorithm for calculation of multiple testing p-values based on the standard Markov chain model, avoiding the use of Gaussian (large sample) approximation. We illustrate the use of this algorithm by demonstrating some inadequacies of the Gaussian approximation.

Bayesian Methods for Neural Networks and Related Models

Models such as feed-forward neural networks and certain other structures investigated in the computer science literature are not amenable to closed-form Bayesian analysis. The paper reviews the various approaches taken to overcome this difficulty, involving the use of Gaussian approximations, Markov chain Monte Carlo simulation routines and a class of non-Gaussian but “deterministic” approximations called variational approximations.

An optically preamplified intersatellite PPM receiver employing maximum likelihood detection

An original analysis, facilitated by the use of Gaussian approximation (GA) and Chernoff bound (CB) techniques, is presented for an optically preamplified intersatellite pulse position modulation (PPM) receiver employing maximum likelihood detection (MLD). The theoretical results, from calculations performed at a wavelength of 1.54 /spl mu/m and bit rate of 25 Mb/s demonstrate that the receiver is approximately 1.5 dB more sensitive than an earlier optically preamplified PPM receiver and also more sensitive than the fundamental limit of optically preamplified on-off keyed (OOK) NRZ signalling. Thus, it is demonstrated that the proposed receiver configuration is a strong contender for implementation in future laser intersatellite communication systems.

Some Aspects of Reliability Growth

Reliability growth is examined with the use of a variant of the Duane model. This variant is based on stepwise growth and avoids some of the mathematical and philosophic dificulties associated with the Duane model. Monte Carlo simulation, while confirming the noisy behavior of reliability growth data, indicates that central tendencies are sufficiently strong to justify the use of Gaussian approximations for analysis of some aspects of growth. Some exact methods and some approximations for parameter estimation are presented.

The Use of Gaussian Approximations in Nuclear Calculations

Gaussian approximations can be extremely useful in calculations involving deformed nuclei. In particular, Nilsson type wave functions of a deformed Saxon–Wood type single-particle potential can be evaluated extremely rapidly by using Gaussians to describe the potential. Furthermore, a single 6-Gaussian approximation to the Coulomb interaction can be used to calculate the Coulomb energy to an appropriate accuracy for both light and heavy nuclei.