Fourier Transforms : Principles and Applications.
- 1st ed.
- 1 online resource (774 pages)
- New York Academy of Sciences Series .
- New York Academy of Sciences Series .
Intro -- FOURIER TRANSFORMS -- Contents -- Preface -- Philosophy and Distinctives -- Flow of the Book -- Suggested Use -- Acknowledgments -- 1 Review of Prerequisite Mathematics -- 1.1 Common notation -- 1.2 Vectors in space -- 1.3 Complex numbers -- 1.4 Matrix algebra -- 1.5 Mappings and functions -- 1.6 Sinusoidal functions -- 1.7 Complex exponentials -- 1.8 Geometric series -- 1.9 Results from calculus -- 1.10 Top 10 ways to avoid errors in calculations -- Problems -- 2 Vector Spaces -- 2.1 Signals and vector spaces -- 2.2 Finite-dimensional vector spaces -- 2.2.1 Norms and Metrics -- 2.2.2 Inner Products -- 2.2.3 Orthogonal Expansion and Approximation -- 2.3 Infinite-dimensional vector spaces -- 2.3.1 Convergent Sequences -- 2.3.2 Infinite Sequences and the p Spaces -- 2.3.3 Functions and the Lp Spaces -- 2.3.4 Orthogonal Expansions in 2 and L2 -- 2.4 Operators -- 2.5 Creating orthonormal bases-the Gram-Schmidt process -- 2.6 Summary -- Problems -- 3 The Discrete Fourier Transform -- 3.1 Sinusoidal sequences -- 3.2 The Discrete Fourier transform -- 3.3 Interpreting the DFT -- 3.4 DFT properties and theorems -- 3.5 Fast Fourier transform -- 3.6 Discrete cosine transform -- 3.7 Summary -- Problems -- 4 The Fourier Series -- 4.1 Sinusoids and physical systems -- 4.2 Definitions and interpretation -- 4.3 Convergence of the Fourier series -- 4.4 Fourier series properties and theorems -- 4.5 The heat equation -- 4.6 The vibrating string -- 4.7 Antenna arrays -- 4.8 Computing the Fourier series -- 4.9 Discrete time Fourier transform -- 4.9.1 Convergence Properties -- 4.9.2 Theorems -- 4.9.3 Discrete-time Systems -- 4.9.4 Computing the DTFT -- 4.10 Summary -- Problems -- 5 The Fourier Transform -- 5.1 From Fourier series to Fourier transform -- 5.2 Basic properties and some examples -- 5.3 Fourier transform theorems. 5.4 Interpreting the Fourier transform -- 5.5 Convolution -- 5.5.1 Definition and basic properties -- 5.5.2 Convolution and Linear Systems -- 5.5.3 Correlation -- 5.6 More about the Fourier transform -- 5.6.1 Fourier inversion in L1 -- 5.6.2 Fourier Transform in L2 -- 5.6.3 More about convolution -- 5.7 Time-bandwidth relationships -- 5.8 Computing the Fourier transform -- 5.9 Time-frequency transforms -- 5.10 Summary -- Problems -- 6 Generalized Functions -- 6.1 Impulsive signals and spectra -- 6.2 The delta function in a nutshell -- 6.3 Generalized functions -- 6.3.1 Functions and Generalized Functions -- 6.3.2 Generalized Functions as Sequences of Functions -- 6.3.3 Calculus of Generalized Functions -- 6.4 Generalized Fourier transform -- 6.4.1 Definition -- 6.4.2 Fourier Theorems -- 6.5 Sampling theory and Fourier series -- 6.5.1 Fourier Series, Again -- 6.5.2 Periodic Generalized Functions -- 6.5.3 The Sampling Theorem -- 6.5.4 Discrete-time Fourier Transform -- 6.6 Unifying the Fourier family -- 6.6.1 Basis Functions and Orthogonality Relationships -- 6.6.2 Sampling and Replication -- 6.7 Summary -- Problems -- 7 Complex Function Theory -- 7.1 Complex functions and their visualization -- 7.2 Differentiation -- 7.3 Analytic functions -- 7.4 exp z and functions derived from it -- 7.5 log z and functions derived from it -- 7.5.1 The Logarithm Function -- 7.5.2 The Square Root, Revisited -- 7.5.3 Rational powers, zm∕n -- 7.5.4 Irrational and Complex Powers of z -- 7.5.5 The Square Root of a Polynomial -- 7.5.6 Inverse Trigonometric Functions -- 7.6 Summary -- Problems -- 8 Complex Integration -- 8.1 Line integrals in the plane -- 8.2 The basic complex integral: ∫zndz -- 8.3 Cauchy's integral theorem -- 8.4 Cauchy's integral formula -- 8.5 Laurent series and residues -- 8.5.1 Laurent series -- 8.5.2 Residues and Integration. 8.6 Using contour integration to calculate integrals of real functions -- 8.6.1 Trigonometric Integrals -- 8.6.2 Improper Integrals -- 8.6.3 Singular Integrals -- 8.6.4 Integrals with Multivalued Functions -- 8.7 Complex integration and the fourier transform -- 8.8 Summary -- Problems -- 9 Laplace, Z, and Hilbert Transforms -- 9.1 The Laplace transform -- 9.1.1 Definition, Basic Properties -- 9.1.2 Laplace Transforms of Generalized Functions -- 9.1.3 Laplace Transform Theorems -- 9.1.4 The Inverse Laplace Transform -- 9.1.5 Laplace Transform of Sampled Functions -- 9.2 The Z transform -- 9.2.1 Definition -- 9.2.2 Z transform Theorems -- 9.2.3 The Inverse Z Transform -- 9.2.4 Discrete-time Systems -- 9.3 The Hilbert transform -- 9.3.1 The Fourier Transform of One-sided Functions -- 9.3.2 Hilbert Transform Properties -- 9.3.3 The Analytic Signal -- 9.4 Summary -- Problems -- 10 Fourier Transforms in Two and Three Dimensions -- 10.1 Two-Dimensional Fourier Transform -- 10.1.1 Definition and Interpretation -- 10.1.2 Fourier Transform Theorems -- 10.2 Fourier Transforms in Polar Coordinates -- 10.2.1 Circular Symmetry: Hankel Transform -- 10.2.2 Spherical Symmetry -- 10.3 Wave Propagation -- 10.3.1 Plane Waves -- 10.3.2 Fraunhofer Diffraction -- 10.3.3 Antennas -- 10.3.4 Lenses -- 10.4 Image Formation and Processing -- 10.4.1 Fourier Analysis of Imaging Systems -- 10.4.2 Image Reconstruction from Projections -- 10.5 Fourier Transform of a Lattice -- 10.5.1 Nonorthogonal Lattices and the Reciprocal Lattice -- 10.5.2 Sampling Theory -- 10.5.3 X-ray Diffraction -- 10.6 Discrete Multidimensional Fourier Transforms -- 10.7 Summary -- Problems -- Bibliography -- Index -- EULA.
9781118901694
Fourier analysis. Image processing-Mathematical models. Signal processing-Mathematical models.