MULTIDIMENSIONAL FOURRIER AND RELATED TRANSFORMS
Ouvrage 0-13-042151-0 : MULTIDIMENSIONAL FOURRIER AND RELATED TRANSFORMS
SUMMARY
This book discusses new applications of the Fourier
transform which
is one of the most pervasive ideas in mathematics. It
contains a
useful collection of ideas, formulas, and programs.
FEATURES
This book presents a unified framework for understanding
continuous-time, discrete-time, and discrete
Fourier-related
transforms.
This book gives equal weight to both complex and
real-basis
functions.
Contains coverage of several important topics:
One-sided and two-sided Laplace and z-transforms.
Correspondence between Fourier-related transforms,
linear
vector spaces and the method of least squares.
Applications in continuous-time and discrete-time
linear
time-invariant system analysis and filtering.
Fast algorithms for discrete Fourier-related transforms,
with
accompanying C and Fortran programs for the number of
data points
equal to the power of 2, or any number of data points.
Discrete-time convolutions.
Discrete-time correlations. Pg___
Toeplitz forms and their fast applications.
This book presents many exercises and examples
throughout, and
also provides guidance on when to use each type of
Fourier
transforms.
TABLE OF CONTENTS
(NOTE: Each chapter contains introduction, summary,
references and
bibliography, and problems.)
Preface.
1. Continuous-Time Fourier-Related Transforms.
The Fourier Transforms. Derivation of the Fourier
Transform Equations. A Discussion of the Real
Fourier
Transform. Existence of the Fourier Transforms.
Importance
of Sinusoids as Basis Functions. Amplitude and Phase
Spectra. The Cosine and Sine Transforms. Properties
of the
Fourier Transforms. Fourier Transform Pairs. The
Generalized Fourier Transforms. Phasors, 2-Tuples,
and
Integro-Differential Equations. The Uncertainty
Principle.
The Hartley Transform. The Mellin Transforms.
Properties
of the Mellin Transform. The Hilbert Transform.
Analytic
Signal.
2. Fourier-Related Series and Discrete-Time Transforms.
The Fourier Series. Derivation of the Fourier Series
Equations. Amplitude and Phase Spectra. The
Relationship
Between the Fourier Series and the Fourier
Transform.
Properties of the Fourier Series. The Cosine and
Sine
Series. The Gibbs Phenomenon and the Convergence of
the
Fourier Series. Steady-State Solution of
Integro-Differential
Equations. The Hartley Series. The Hilbert Series.
Sampling
of Continuous-Time Signals. The Discrete-Time
Fourier
Transforms. The Relationship Between the
Discrete-Time
Fourier Transform and the Fourier Transform.
Properties of
the Discrete-Time Fourier Transform. Amplitude and
Phase
Spectra. Eigenfunctions of Discrete-Time LTI
Systems. The
Discrete-Time Cosine and Sine Transforms. The
Discrete-Time Hartley Transform. The Discrete-Time
Hilbert Transform.
3. Linear Vector Spaces and the Method of Least-Squares.
Vector Spaces. Properties of Vector Spaces.
Inner-Product
Vector Spaces. Representation of a
Finite-Dimensional
Vector Space by a Matrix. Hilbert Spaces.
Approximation
by Least-Squares. Normal Equations. Signal Modeling
by
Basis Signal Vectors. The Method of Least-Squares
and the
Pseudoinverse. Projection Operators. Orthogonal
Basis.
Approximation by Weighted Least-Squares.
W-Orthogonal
Vectors and Projection Matrices. Orthonormalization
of
Basis Vectors. The Generalized Fourier Series.
Approximation of Signals by Truncated Fourier.
Series.
4. The Laplace Transform and the z-Transform.
The Laplace Transforms. Eigenfunctions of a LTI
System.
The Region of Convergence for the Laplace Transform.
The
One-Sided Laplace Transform. Laplace Transform
Pairs.
Laplace Transforms of Causal Periodic Signals.
Properties
of the Laplace Transforms. The Inverse Laplace
Transform.
Solution of Integro-Differential Equations. The
Initial-Value
and Final-Value Theorems. The z-Transform. The
Region of
Convergence for the z-Transform. Common z-Transform
Pairs. Properties of the z-Transform. Eigenfunct
ions of
Discrete-Time LTI Systems. Rational z-Transforms.
The
Inverse z-Transform. The One-Sided z-Transform.
Solution
of Linear Constant-Coefficient Difference Equations.
5. Continuous-Time Linear Time-Invariant
Systems
and Filtering. . Linear Time-Invariant Systems
and
Convolution. Transfer Functions of
Continuous-Time
LTI Systems. Nonzero Initial Conditions. Zeroes
and
Poles. Stability of Continuous-Time LTI
Systems.
Types of Responses of Continuous-Time Systems.
Sinusoidal Steady-State Response.
Continuous-Time
Filtering Ideal Frequency Selective Filters.
Nonideal
Frequency Selective Filters. Analog Filters
Described
by Differential Equations. Butterworth Filters.
Equiripple Filters. Systems Characterized by
Linear
Constant-Coefficient Differential Equations.
6. Discrete-Time Linear-Time Invariant Systems and
Filtering.
Discrete-Time LTI Systems and Discrete-Time
Convolution. Transfer Function of a LTI
Discrete-Time System. Types of Responses of
Discrete-Time LTI Systems. Stability of
Discrete-Time LTI Systems. Discrete-Time
Filters.
Conversion of Continuous-Time Systems into
Discrete-Time Systems Conversion by Impulse
Invariance. The Matched-Z Transformation. The
Bilinear Transformation. Finite Impulse
Response
Filters. Design of Lowpass FIR Filters. Design
of
Highpass, Bandpass, and Bandstop FIR Filters.
Windowing Techniques. Design of FIR Filters by
Frequency Sampling. Frequency Transformations.
Implementation of Digital IIR Filters.
Implementation
of Digital FIR Filters. Second-Order
Structures.
7. The Discrete Fourier-Related Transforms.
The Discrete Fourier Transforms. Discussion of
the
Real Discrete Fourier Transform. Aplitude and
Phase
Spectra. Properties of the Discrete Fourier
Transform. The Discrete Sine and Symmetric
Cosine
Transform. The Discrete Hartley Transform. The
Relationship between the DFT and the
Continuous-Time Fourie Transforms and Series.
The
Periodogram Numerical Computation of the
Fourier
Transform. Synthesis of a Continuous-Time
Signal
from its Samples.
8. The Generalized Discrete Fourier-Related
Transforms.
The Generalized Discrete Fourier Transforms.
The
Orthonormal Generalized RDFT. The Generalized
Discrete Cosine and Sine Transforms. Further
Generalization of Discrete Fourier Transforms.
Unconventional Discrete Transforms. Important
Matrices Diagonalized by the Generalized
Discrete
Fourier-Related Transforms. The Discrete Cosine
Transform. The Scrambled Discrete Fourier
Transforms. The Discrete Cosine-III Transform.
A
Performance Measure for Transform Signal
Compression.
9. Fast Algorithms for the Discrete Fourier-Related
Transforms.
Fundamental Building Blocks and Methodology of
FFT Algorithms. The Radix-2 Decimation-In-Time
(DIT) CFFT Algorithm. Generation of
Bit-Reversed
Indices. Fast Computation of the RDFT of a Real
Sequence of Length 2N by CFFT of Length N. The
Radix-2 Decimation-in-Time RFFT Algorithm. The
Radix-4 Decimation-In-Time RFFT Algorithm. The
Split-Radix Decimation-In-Time RFFT Algorithm.
Kronecker Product of Matrices. Fast Computation
of
the Real Discrete Fourier Transform for Any
Number
of Data Points. The Two-Factor DIT RFFT
Algorithm. Analysis of the Two-Factor DIT RFFT
Algorithm. Generalization of DIT RFFT to an
Arbitrary Number of Factors. Single Radix DIT
RFFT Algorithm. The Prime-Factor RFFT
Algorithm.
The Rader Prime RFFT Algorithm. The Winograd
RFFT Algorithm. The Winograd Small RFFT
Algorithms. Fast Computation of the Inverse
RDFT.
Fast Algorithms for the CDFT and the Inverse
CDFT. RFFT and CFFT Routines in Fortran and C.
The Chirp-Z Transform Algorithm. The Goertze l
Algorithm. Fast Computation of R2. Fast
Computation of C2 and C3. Fast Computation of
C1. Fast Computation of S2 and S3. Fast
Computation of S1. Fast Computation of R3. Fast
Computation of C4. Fast Computation of S4. Fast
Computation of R4. DSCT and DST Routines in
Fortran and C.
10. Discrete-Time Convolutions, Correlations,
Toeplitz
Forms, and Their Fast Computations.
The Relationship between the Continuous-Time
and
Discrete-Time Convolutions and Correlations.
Circular Convolution and Its Fast Computation.
Skew-Circular Convolution and Its Fast
Computation. Other Methods for the Fast
Computation of Skew-Circular Convolution and
Correlation. Linear Convolution by Circular
Convolution. Sectioned Linear Convolution by
Circular and Skew- Circular Convolutions.
Overlap-Add and Overlap-Save Methods of
Sectioned Convolutions. Linear Correlation.
Certain
Toeplitz Matrices.
Appendix A: Fundamentals of Systems and Signals.
System Modeling. Classification of Systems.
Discrete-Time Systems. Initial Conditions.
Types and
Classification of Signals.
Appendix B: Elementary Signals.
Elementary Continuous-Time Functions.
Elementary
Discrete-Time Signals.
Appendix C: Generalized Functions and Distributions.
The Impulse Function. Generalized Functions.
Distributions. The Fourier Transform of a
Distribution.
Appendix D: The Karhunen-Loeve Transform.
Index.
Auteur : ERSOY
Editeur : PRENTICE HALL
Nombre de pages : 336
Date de publication : 01 1995
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