: One use of spectral analysis of time series is to determine if two different time series are realizations from the same process This thesis develops the theory behind Krishnaiah and Schuurmann's theoretical work reported in their report Approximations to the Distributions of the Traces of Complex Multivariate Beta and F Matrices We take the trace of a test statistic calculated from the spectral density matrices of the time series and test it The thesis applies the theory to two small sample simulations (Author)

Spectral Analysis of Time Series.

Preface Prologue: The Exponential Function Chapter 1: Abstract Integration Set-theoretic notations and terminology The concept of measurability Simple functions Elementary properties of measures Arithmetic in [0, ] Integration of positive functions Integration of complex functions The role played by sets of measure zero Exercises Chapter 2: Positive Borel Measures Vector spaces Topological preliminaries The Riesz representation theorem Regularity properties of Borel measures Lebesgue measure Continuity properties of measurable functions Exercises Chapter 3: Lp-Spaces Convex functions and inequalities The Lp-spaces Approximation by continuous functions Exercises Chapter 4: Elementary Hilbert Space Theory Inner products and linear functionals Orthonormal sets Trigonometric series Exercises Chapter 5: Examples of Banach Space Techniques Banach spaces Consequences of Baire's theorem Fourier series of continuous functions Fourier coefficients of L1-functions The Hahn-Banach theorem An abstract approach to the Poisson integral Exercises Chapter 6: Complex Measures Total variation Absolute continuity Consequences of the Radon-Nikodym theorem Bounded linear functionals on Lp The Riesz representation theorem Exercises Chapter 7: Differentiation Derivatives of measures The fundamental theorem of Calculus Differentiable transformations Exercises Chapter 8: Integration on Product Spaces Measurability on cartesian products Product measures The Fubini theorem Completion of product measures Convolutions Distribution functions Exercises Chapter 9: Fourier Transforms Formal properties The inversion theorem The Plancherel theorem The Banach algebra L1 Exercises Chapter 10: Elementary Properties of Holomorphic Functions Complex differentiation Integration over paths The local Cauchy theorem The power series representation The open mapping theorem The global Cauchy theorem The calculus of residues Exercises Chapter 11: Harmonic Functions The Cauchy-Riemann equations The Poisson integral The mean value property Boundary behavior of Poisson integrals Representation theorems Exercises Chapter 12: The Maximum Modulus Principle Introduction The Schwarz lemma The Phragmen-Lindelof method An interpolation theorem A converse of the maximum modulus theorem Exercises Chapter 13: Approximation by Rational Functions Preparation Runge's theorem The Mittag-Leffler theorem Simply connected regions Exercises Chapter 14: Conformal Mapping Preservation of angles Linear fractional transformations Normal families The Riemann mapping theorem The class L Continuity at the boundary Conformal mapping of an annulus Exercises Chapter 15: Zeros of Holomorphic Functions Infinite Products The Weierstrass factorization theorem An interpolation problem Jensen's formula Blaschke products The Muntz-Szas theorem Exercises Chapter 16: Analytic Continuation Regular points and singular points Continuation along curves The monodromy theorem Construction of a modular function The Picard theorem Exercises Chapter 17: Hp-Spaces Subharmonic functions The spaces Hp and N The theorem of F. and M. Riesz Factorization theorems The shift operator Conjugate functions Exercises Chapter 18: Elementary Theory of Banach Algebras Introduction The invertible elements Ideals and homomorphisms Applications Exercises Chapter 19: Holomorphic Fourier Transforms Introduction Two theorems of Paley and Wiener Quasi-analytic classes The Denjoy-Carleman theorem Exercises Chapter 20: Uniform Approximation by Polynomials Introduction Some lemmas Mergelyan's theorem Exercises Appendix: Hausdorff's Maximality Theorem Notes and Comments Bibliography List of Special Symbols Index

Real and Complex Analysis. By W. Rudin. Pp. 412. 84s. 1966. (McGraw-Hill, New York.)

In this paper we survey some recent work directed towards generalizing the fast Fourier transform (FFT). We work primarily from the point of view of group representation theory. In this setting the classical FFT can be viewed as a family of efficient algorithms for computing the Fourier transform of either a function defined on a finite abelian group, or a bandlimited function on a compact abelian group. We discuss generalizations of the FFT to arbitrary finite groups and compact Lie groups.

Generalized FFTS - A Survey of Some Recent Results

We describe various basic facts about Gabor multipliers and their continuous analogue which we will call STFT-multipliers. These operators are obtained by going from the signal domain to some transform domain, and applying a pointwise multiplication operator before resynthesis. Although such operators have been in use implicitly for quite some time, this paper appears to be the first systematic mathematical treatment of Gabor multipliers. Indeed, typical time-frequency localization operators, or thresholding algorithms involve simple 0/1-multipliers. The main results of this chapter are of a qualitative nature and describe how the properties of the Gabor multiplier depend on the decay of the multiplier sequence, the time-frequency (TF) concentration properties of the Gabor atom in use, and the time-frequency-lattice. These properties will be described in terms of the mapping properties of the corresponding Gabor multiplier between modulation spaces, or membership in some operator ideal (such as trace-class or Hilbert—Schmidt operator). It is also possible to give relatively precise estimates on behaviour of the sequence of eigenvalues of such operators, especially for the case of tight Gabor frames. We shall also discuss the problem of injectivity of the linear mapping from the multiplier symbol to the operator, recovery of Gabor multipliers from lower symbols, and a related question concerning best approximation of operators (e.g., from the Hilbert—Schmidt class) by Gabor multipliers of a certain type.

A First Survey of Gabor Multipliers

One of the most important stages of floating-point to fixed-point conversion, is the evaluation of the fixed-point specification accuracy. This evaluation is required to optimize the data word-length according to accuracy constraints. Classical methods for accuracy evaluation are based on fixed-point simulations but they lead to very long optimization times.A new method based on an analytical approach is presented for the case of linear-time-invariant (LTI) systems. The use of this method in data word-length minimization processes reduces significantly the optimization time compared to simulation based methods. Our approach allows the automatic determination of the signal-to-quantization-noise-ratio (SQNR) expression at the system output according to the fixed-point data format. This method is valid for recursive and non-recursive LTI systems and takes account of the quantization modes (truncation or rounding). The theoretical concepts and the different methodology stages are explained. Then, the ability to efficiently evaluate the fixed-point specification accuracy is demonstrated through examples.

Analytical Fixed-Point Accuracy Evaluation in Linear Time-Invariant Systems

An operator-theoretic overview of discrete-time single-channel signal processing is given, covering such topics as the time and frequency domain structure of weakly stationary stochastic processes,...

Richard B. Holmes

Papers

Mathematical foundations of signal processing