In this section, we consider approximative methods for the fast computation of multivariate discrete Fourier transforms for nonequispaced data (NDFT) in the time domain and in the frequency domain. In particular, we are interested in the approximation error as function of the arithmetic complexity of the algorithm. We discuss the robustness of NDFT-algorithms with respect to roundoff errors and apply NDFT-algorithms for the fast computation of Bessel transforms.

The FFT is used widely in signal processing for efficient computation of the Fourier transform (FT) of finitelength signals over a set of uniformly-spaced frequency locations. However, in many applications, one requires nonuniform sampling in the frequency domain, i.e.,a nonuniform FT . Several papers have described fast approximations for the nonuniform FT based on interpolating an oversampled FFT. This paper presents an interpolation method for the nonuniform FT that is optimal in the min-max sense of minimizing the worst-case approximation error over all signals of unit norm. The proposed method easily generalizes to multidimensional signals. Numerical results show that the min-max approach provides substantially lower approximation errors than conventional interpolation methods. The min-max criterion is also useful for optimizing the parameters of interpolation kernels such as the Kaiser-Bessel function.

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Active probes of network performance represent samples of the underlying performance of a system. Some effort has gone into considering appropriate sampling patterns for such probes: i.e. there has been significant discussion of the importance of sampling using a Poisson process to avoid biases introduced by synchronization of system and uniformly spaced probes. However, there are unanswered questions about whether Poisson probing has costs in terms of sampling efficiency, and there is some misinformation about what types of inferences are possible based on different probe patterns. This paper provides a quantitative comparison of different sampling methods, both in terms of efficiency. The paper also shows that the irregularity in probing patterns is useful not just in avoiding synchronization, but also in determining frequency domain properties of a system. The paper provides a firm basis for practitioners or researchers to use in making such a decision about the type of sampling they should use in a particular application, along with methods for the analysis of their outputs.

F E U U S A D by James Dennis Reimann Doctor of Philosophy in Statistics University of California at Berkeley Professor John Rice, Chair This thesis studies estimation of the frequency of a periodic function of time, when the function is observed with noise at a collection of unequally-spaced times. This research was motivated by the detection and classification of variable stars in astronomy. Most of the statistical literature on frequency estimation assumes equally-spaced times, but observation times in astronomy are often unequally-spaced with a sampling distribution that contains periodic effects due to being able to collect data only at certain times of day. In Chapter 1 we describe the database of variable stars collected by the MACHO collaboration and present examples which illustrate the common types of variable stars and the nature of the estimation problem. In Chapter 2 we provide background material and give models for the periodic function and sampling times. We derive the ...

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The Lomb-Scargle periodogram approach was applied to the search for periodically expressed genes in a Plasmodium falciparum dataset. The Lomb-Scargle algorithm has several computational advantages over more common approaches, such as Fourier analysis, including direct treatment of missing values and and a periodogram that has known statistical properties. Hierarchical clustering of periodograms shows explicit partitioning of multiple periodicities present in some gene expression patterns. The Lomb-Scargle algorithm performance was compared to earlier analysis of CAMDA 2004 challenge dataset by Bozdech et al. [1], based on Fast Fourier Transforms (FFT). We identified an additional 265 genes with 48-hr periodic expression, which were not considered by the FFT because they had too many missing values. We also automatically detected, in the same analysis, expression patterns with periodicity close to 24 hr, and other interesting patterns. Keywords Lomb-Scargle periodogram, periodicity, gene expression profile,

The spectral analysis of non uniform sampled data sequences using Fourier Periodogram method is the classical approach.In view of data fitting and computational standpoints why the Least squares periodogram (LSP) method is preferable than the “classical ” Fourier periodogram and as well as to the frequentlyused form of LSP due to Lomb and Scargle is explained. Then a new method of spectral analysis of nonuniform data sequences can be interpreted as an iteratively weighted LSP that makes use of a data-dependent weighting matrix built from the most recent spectral estimate. It is iterative and it makes use of an adaptive (i.e., data-dependent) weighting, we refer to it as the iterative adaptive approach (IAA).LSP and IAA are nonparametric methods that can be used for the spectral analysis of general data sequences with both continuous and discrete spectra. However, they are most suitable for data sequences with discrete spectra (i.e., sinusoidal data), which is the case we emphasize in this paper. Of the existing methods for nonuniform sinusoidal data, Welch, MUSIC and ESPRIT methods appear to be the closest in spirit to the IAA proposed here. Indeed, all these methods make use of the estimated covariance matrix that is computed in the first iteration of IAA from LSP. Comparative study of LSP with MUSIC and ESPRIT methods are discussed.

Intraday pattern in bid-ask spreads and its