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18
The FourierSeries Method For Inverting Transforms Of Probability Distributions
, 1991
"... This paper reviews the Fourierseries method for calculating cumulative distribution functions (cdf's) and probability mass functions (pmf's) by numerically inverting characteristic functions, Laplace transforms and generating functions. Some variants of the Fourierseries method are remarkably easy ..."
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Cited by 149 (51 self)
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This paper reviews the Fourierseries method for calculating cumulative distribution functions (cdf's) and probability mass functions (pmf's) by numerically inverting characteristic functions, Laplace transforms and generating functions. Some variants of the Fourierseries method are remarkably easy to use, requiring programs of less than fifty lines. The Fourierseries method can be interpreted as numerically integrating a standard inversion integral by means of the trapezoidal rule. The same formula is obtained by using the Fourier series of an associated periodic function constructed by aliasing; this explains the name of the method. This Fourier analysis applies to the inversion problem because the Fourier coefficients are just values of the transform. The mathematical centerpiece of the Fourierseries method is the Poisson summation formula, which identifies the discretization error associated with the trapezoidal rule and thus helps bound it. The greatest difficulty is approximately calculating the infinite series obtained from the inversion integral. Within this framework, lattice cdf's can be calculated from generating functions by finite sums without truncation. For other cdf's, an appropriate truncation of the infinite series can be determined from the transform based on estimates or bounds. For Laplace transforms, the numerical integration can be made to produce a nearly alternating series, so that the convergence can be accelerated by techniques such as Euler summation. Alternatively, the cdf can be perturbed slightly by convolution smoothing or windowing to produce a truncation error bound independent of the original cdf. Although error bounds can be determined, an effective approach is to use two different methods without elaborate error analysis. For this...
Fast And Exact Simulation Of Stationary Gaussian Processes Through Circulant Embedding Of The Covariance Matrix
, 1997
"... . Geostatistical simulations often require the generation of numerous realizations of a stationary Gaussian process over a regularly meshed sample grid## This paper shows that for many important correlation functions in geostatistics, realizations of the associated process over m +1 equispaced point ..."
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Cited by 40 (1 self)
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. Geostatistical simulations often require the generation of numerous realizations of a stationary Gaussian process over a regularly meshed sample grid## This paper shows that for many important correlation functions in geostatistics, realizations of the associated process over m +1 equispaced points on a line can be produced at the cost of an initial FFT of length 2m with each new realization requiring an additional FFT of the same length. In particular, the paper first notes that if an (m+1)×(m+1) Toeplitz correlation matrix R can be embedded in a nonnegative definite 2M×2M circulant matrix S, exact realizations of the normal multivariate y #N(0,R) can be generated via FFTs of length 2M . Theoretical results are then presented to demonstrate that for many commonly used correlation structures the minimal embedding in which M = m is nonnegative definite. Extensions to simulations of stationary fields in higher dimensions are also provided and illustrated. Key words. geostatistics, ...
Option Pricing by Transform Methods: Extensions, Unification, and Error Control
 Journal of Computational Finance
"... We extend and unify Fourieranalytic methods for pricing a wide class of options on any underlying state variable whose characteristic function is known. In this general setting, we bound the numerical pricing error of discretized transform computations, such as DFT/FFT. These bounds enable algorith ..."
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Cited by 38 (3 self)
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We extend and unify Fourieranalytic methods for pricing a wide class of options on any underlying state variable whose characteristic function is known. In this general setting, we bound the numerical pricing error of discretized transform computations, such as DFT/FFT. These bounds enable algorithms to select efficient quadrature parameters and to price with guaranteed numerical accuracy.
Efficient signal processing techniques for exploiting transmit antenna diversity on fading channels
 IEEE Trans. Signal Processing
, 1997
"... Abstract — A class of powerful and computationally efficient strategies for exploiting transmit antenna diversity on fading channels is developed. These strategies, which require simple linear processing at the transmitter and receiver, have attractive asymptotic characteristics. In particular, give ..."
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Cited by 20 (4 self)
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Abstract — A class of powerful and computationally efficient strategies for exploiting transmit antenna diversity on fading channels is developed. These strategies, which require simple linear processing at the transmitter and receiver, have attractive asymptotic characteristics. In particular, given a sufficient number of transmit antennas, these techniques effectively transform a nonselective Rayleigh fading channel into a nonfading, simple white marginally Gaussian noise channel with no intersymbol interference. These strategies, which we refer to as linear antenna precoding, can be efficiently combined with trellis coding and other popular errorcorrecting codes for bandwidthconstrained Gaussian channels. Linear antenna precoding requires no additional power or bandwidth and is attractive in terms of robustness and delay considerations. The resulting schemes have powerful and convenient interpretations in terms of transforming nonselective fading channels into frequency and timeselective ones. I.
Wavelets, Fractals, and Radial Basis Functions
 IEEE TRANS. SIGNAL PROCESSING
, 2002
"... Wavelets and radial basis functions (RBFs) lead to two distinct ways of representing signals in terms of shifted basis functions. RBFs, unlike wavelets, are nonlocal and do not involve any scaling, which makes them applicable to nonuniform grids. Despite these fundamental differences, we show that t ..."
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Cited by 14 (4 self)
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Wavelets and radial basis functions (RBFs) lead to two distinct ways of representing signals in terms of shifted basis functions. RBFs, unlike wavelets, are nonlocal and do not involve any scaling, which makes them applicable to nonuniform grids. Despite these fundamental differences, we show that the two types of representation are closely linked together ... through fractals. First, we identify and characterize the whole class of selfsimilar radial basis functions that can be localized to yield conventional multiresolution wavelet bases. Conversely, we prove that for any compactly supported scaling function 9(0c), there exists a onesided central basis function p+ (x) that spans the same multireso lution subspaces. The central property is that the multiresolution bases are generated by simple translation of p+ without any dilation. We also present an explicit timedomain representation of a scaling function as a sum of harmonic splines. The leading term in the decomposition corresponds to the fractional splines: a recent, continuousorder generalization of the polynomial splines.
A new approach for option pricing under stochastic volatility
 Review of Derivatives Research
, 2007
"... Abstract We develop a new approach for pricing Europeanstyle contingent claims written on the time T spot price of an underlying asset whose volatility is stochastic. Like most of the stochastic volatility literature, we assume continuous dynamics for the price of the underlying asset. In contrast ..."
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Cited by 4 (1 self)
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Abstract We develop a new approach for pricing Europeanstyle contingent claims written on the time T spot price of an underlying asset whose volatility is stochastic. Like most of the stochastic volatility literature, we assume continuous dynamics for the price of the underlying asset. In contrast to most of the stochastic volatility literature, we do not directly model the dynamics of the instantaneous volatility. Instead, taking advantage of the recent rise of the variance swap market, we directly assume continuous dynamics for the time T variance swap rate. The initial value of this variance swap rate can either be directly observed, or inferred from option prices. We make no assumption concerning the real world drift of this process. We assume that the ratio of the volatility of the variance swap rate to the instantaneous volatility of the underlying asset just depends on the variance swap rate and on the variance swap maturity. Since this ratio is assumed to be independent of calendar time, we term this key assumption the stationary volatility ratio hypothesis (SVRH). The instantaneous volatility of the futures follows an unspecified stochastic process, so both the underlying futures price and the variance swap rate have unspecified stochastic volatility. Despite this, we show that the payoff to a pathindependent contingent claim can be perfectly replicated by dynamic trading in futures contracts and variance swaps of the same maturity. As a result, the contingent claim is uniquely valued relative to its underlying’s futures price and the assumed observable variance swap rate. In contrast to standard models of stochastic volatility, our approach does not require specifying the market price of volatility risk or observing the initial level of instantaneous volatility.
A Novel Discontinuity Metric for Unit Selection TexttoSpeech Synthesis
"... The level of quality that can be achieved by modern concatenative texttospeech synthesis heavily depends on the optimization criteria used in the unit selection process. While effective cost functions arise naturally in the assessment of prosodic characteristics, the criteria typically selected to ..."
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Cited by 2 (0 self)
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The level of quality that can be achieved by modern concatenative texttospeech synthesis heavily depends on the optimization criteria used in the unit selection process. While effective cost functions arise naturally in the assessment of prosodic characteristics, the criteria typically selected to quantify discontinuities at the speech signal level do not tightly reflect users' perception of the resulting acoustic waveform. This paper introduces a novel discontinuity measure which jointly, albeit implicitly, accounts for both interframe incoherence and discrepancies in formant frequencies /bandwidths. This metric is derived from a distinct feature extraction paradigm, eschewing general purpose Fourier analysis in favor of a separately optimized modal decomposition for each boundary region. This alternative transform framework preserves, by construction, those properties of the waveform which are globally relevant to each concatenation considered. Experimental evaluations are conducted to characterize the behavior of the new measure, first on a contiguity prediction task, and then via a systematic listening comparison using a conventional metric as baseline. The results underscores the viability of the proposed approach in quantifying the perception of discontinuity between acoustic units.
Positivedefiniteness, integral equations and Fourier transforms
, 2000
"... In this paper we define classes of functions which we call positive definite kernel functions and positive definite kernels. The first class may be thought of as a generalization to two dimensions of the classical positive definite functions of BochnerKhinchin type. We study their properties in dep ..."
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Cited by 2 (1 self)
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In this paper we define classes of functions which we call positive definite kernel functions and positive definite kernels. The first class may be thought of as a generalization to two dimensions of the classical positive definite functions of BochnerKhinchin type. We study their properties in depth and show how the second class arises by considering the associated integral operators. We give necessary and su#cient conditions for the existence of a bilinear expansion of Mercer type and show the analog of Bochner's theorem in the L setting, namely that a function is a positive definite kernel if and only if its Fourier transform is a positive definite kernel. A simple and elegant su#cient condition for compactness of support of positive definite kernels is given, namely that they are compactly supported along the main diagonal. Several corollaries relating compactness of support of the Fourier transform and analyticity are derived. 1
Calculation of Spectral Peaks in a Chaotic DcDc Converter
"... A simple mapping is derived which describes the behaviour of a peak currentmode controlled boost converter operating chaotically. The invariant density of this mapping is calculated iteratively, and from this the power spectral density of the input current at the clock frequency and its harmonics i ..."
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A simple mapping is derived which describes the behaviour of a peak currentmode controlled boost converter operating chaotically. The invariant density of this mapping is calculated iteratively, and from this the power spectral density of the input current at the clock frequency and its harmonics is deduced. The calculation is presented, along with experimental verification, and the possibility of EMC improvement by chaos is discussed. 1. Introduction In a previous publication [1] we presented the idea that chaos, a naturally occurring phenomenon in switchmode power supplies, might be used to improve their electromagnetic compatibility. We illustrated this proposal with some measurements. In the present paper we show how to calculate the power spectral density (PSD) at the clock frequency and its harmonics, of the input current of a currentmode controlled boost converter operating chaotically. This work builds mainly on [2][4], in which we derived an exact twodimensional mapping...
MultiDimensional Transform Inversion
 Ann. Appl. Prob
, 1994
"... We develop an algorithm for numerically inverting multidimensional transforms. Our algorithm applies to any number of continuous variables (Laplace transforms) and discrete variables (generating functions). We use the Fourierseries method; i.e., the inversion formula is the Fourier series of a per ..."
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We develop an algorithm for numerically inverting multidimensional transforms. Our algorithm applies to any number of continuous variables (Laplace transforms) and discrete variables (generating functions). We use the Fourierseries method; i.e., the inversion formula is the Fourier series of a periodic function constructed by aliasing. This amounts to an application of the Poisson summation formula. By appropriately exponentially damping the given function, we control the aliasing error. We choose the periods of the multidimensional periodic function so that each infinite series is a finite sum of nearly alternating infinite series; then we apply the Euler transformation to compute the infinite series from finitely many terms. The multidimensional inversion algorithm enables us, evidently for the first time, to quickly and accurately calculate probability distributions from several classical transforms in queueing theory. For example, we apply our algorithm to invert the twodimensional transforms of the joint distribution of the duration of a busy period and the number served in that busy period, and the timedependent of the transient queuelength and workload distributions, in the M/G/1 queue. In other related work, we have applied the inversion algorithms here to calculate timedependent distributions in the transient BMAP/G/1 queue (with a batch Markovian arrival process) and the piecewisestationary M t /G t / 1 queue.