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Fast Fourier Transform - Algorithms and Applications presents an introduction to the principles of the fast Fourier transform (FFT). It covers FFTs, frequency domain filtering, and applications to video and audio signal processing.



As fields like communications, speech and image processing, and related areas are rapidly developing, the FFT as one of the essential parts in digital signal processing has been widely used. Thus there is a pressing need from instructors and students for a book dealing with the latest FFT topics.



Fast Fourier Transform - Algorithms and Applications provides a thorough and detailed explanation of important or up-to-date FFTs. It also has adopted modern approaches like MATLAB examples and projects for better understanding of diverse FFTs.



Fast Fourier Transform - Algorithms and Applications is designed for senior undergraduate and graduate students, faculty, engineers, and scientists in the field, and self-learners to understand FFTs and directly apply them to their fields, efficiently. It is designed to be both a text and a reference. Thus examples, projects and problems all tied with MATLAB, are provided for grasping the concepts concretely. It also includes references to books and review papers and lists of applications, hardware/software, and useful websites. By including many figures, tables, bock diagrams and graphs, this book helps the reader understand the concepts of fast algorithms readily and intuitively. It provides new MATLAB functions and MATLAB source codes. The material in Fast Fourier Transform - Algorithms and Applications is presented without assuming any prior knowledge of FFT. This book is for any professional who wants to have a basic understanding of the latest developments in and applications of FFT. It provides a good reference for any engineer planning to work in this field, either in basic implementation or in research and development.

Recenzijos

From the reviews:

The new book Fast Fourier Transform - Algorithms and Applications by Dr. K.R. Rao, Dr. D.N. Kim, and Dr. J.J. Hwang is an engaging look in the world of FFT algorithms and applications. This book not only provides detailed description of a wide-variety of FFT algorithms, gives the mathematical derivations of these algorithms, plentiful helpful flow diagrams illustrating the algorithms, and MATLAB programsthe book also presents novel topics in depth (for example, integer FFTs, the non-uniform DFT, phase-only correlation, image and audio watermarking, the curvelet transform, and many more) with insightful motivation, explanation, and numerous examples and programs. The authors also provide insight into issues related to implementation of FFTs on different hardware platforms. Excellent aspects of the book are the plentiful numerical examples illustrating the properties of the DFT and a most impressive number of applications of the FFT- including applications in communications, image restoration, and descriptions of audio coding standards that rely on the FFT (MPEG, AC-2) to name only a few. The book also supplies a generous number of exercises and computer projects. In addition, throughout the book the authors present and discuss the research literature at length and include many healthy doses of pointers to the literature which provides a pathway for specialized further study.

Dr. Ivan W. Selesnick, Polytechnic University, New York, NY

The fast Fourier transform (FFT) is an essential tool in applied mathematics and digital signal processing. This monograph on the FFT is mainly written for graduate students and researchers in engineering and science. It consists of 8 chapters, 8 appendices and a comprehensive bibliography. (Manfred Tasche, Zentralblatt MATH, Vol. 1203, 2011)

This volume offers an account of the Discrete Fourier Transform (DFT) and its implementation, including the Fast Fourier Transform(FFT). The target audience is clearly instructors and students in engineering . book gives an excellent opportunity to applied mathematicians interested in refreshing their teaching to enrich their presentation of the DFT/FFT with modern applications. The exercises and projects are one of the most important feature of this volume. This is in itself a clear motivation to strongly recommend this volume . (Jean-Pierre Croisille Mathematical Reviews, Issue 2012 k)

1 Introduction 1(4)
1.1 Applications of Discrete Fourier Transform
2(3)
2 Discrete Fourier Transform 5(36)
2.1 Definitions
5(2)
2.1.1 DFT
5(1)
2.1.2 IDFT
5(1)
2.1.3 Unitary DFT (Normalized)
6(1)
2.2 The Z-Transform
7(6)
2.3 Properties of the DFT
13(5)
2.4 Convolution Theorem
18(6)
2.4.1 Multiplication Theorem
24(1)
2.5 Correlation Theorem
24(3)
2.6 Overlap-Add and Overlap-Save Methods
27(4)
2.6.1 The Overlap-Add Method
27(4)
2.7 Zero Padding in the Data Domain
31(1)
2.8 Computation of DFTs of Two Real Sequences Using One Complex FFT
32(2)
2.9 A Circulant Matrix Is Diagonalized by the DFT Matrix
34(3)
2.9.1 Toeplitz Matrix
34(1)
2.9.2 Circulant Matrix
34(1)
2.9.3 A Circulant Matrix Is Diagonalized by the DFT Matrix
35(2)
2.10 Summary
37(1)
2.11 Problems
37(3)
2.12 Projects
40(1)
3 Fast Algorithms 41(70)
3.1 Radix-2 DIT-FFT Algorithm
42(5)
3.1.1 Sparse Matrix Factors for the IFFT N = 8
46(1)
3.2 Fast Algorithms by Sparse Matrix Factorization
47(9)
3.3 Radix-2 DIF-FFT
56(5)
3.3.1 DIF-FFT N = 8
57(4)
3.3.2 In-Place Computations
61(1)
3.4 Radix-3 DIT FFT
61(2)
3.5 Radix-3 DIF-FFT
63(3)
3.6 FFT for N a Composite Number
66(1)
3.7 Radix-4 DIT-FFT
67(6)
3.8 Radix-4 DIF-FFT
73(2)
3.9 Split-Radix FFT Algorithm
75(3)
3.10 Fast Fourier and BIFORE Transforms by Matrix Partitioning
78(5)
3.10.1 Matrix Partitioning
78(2)
3.10.2 DFT Algorithm
80(2)
3.10.3 BT (BIFORE Transform)
82(1)
3.10.4 CBT (Complex BIFORE Transform)
82(1)
3.10.5 DFT (Sparse Matrix Factorization)
82(1)
3.11 The Winograd Fourier Transform Algorithm
83(9)
3.11.1 Five-Point DFT
83(1)
3.11.2 Seven-Point DFT
84(1)
3.11.3 Nine-Point DFT
85(1)
3.11.4 DFT Algorithms for Real-Valued Input Data
85(2)
3.11.5 Winograd Short-N DFT Modules
87(1)
3.11.6 Prime Factor Map Indexing
88(2)
3.11.7 Winograd Fourier Transform Algorithm (WFTA)
90(2)
3.12 Sparse Factorization of the DFT Matrix
92(5)
3.12.1 Sparse Factorization of the DFT Matrix Using Complex Rotations
92(2)
3.12.2 Sparse Factorization of the DFT Matrix Using Unitary Matrices
94(3)
3.13 Unified Discrete Fourier–Hartley Transform
97(7)
3.13.1 Fast Structure for UDFHT
101(3)
3.14 Bluestein's FFT Algorithm
104(2)
3.15 Rader Prime Algorithm
106(1)
3.16 Summary
107(1)
3.17 Problems
108(2)
3.18 Projects
110(1)
4 Integer Fast Fourier Transform 111(16)
4.1 Introduction
111(1)
4.2 The Lifting Scheme
112(1)
4.3 Algorithms
112(7)
4.3.1 Fixed-Point Arithmetic Implementation
117(2)
4.4 Integer Discrete Fourier Transform
119(6)
4.4.1 Near-Complete Integer DFT
119(2)
4.4.2 Complete Integer DFT
121(2)
4.4.3 Energy Conservation
123(1)
4.4.4 Circular Shift
123(2)
4.5 Summary
125(1)
4.6 Problems
126(1)
4.7 Projects
126(1)
5 Two-Dimensional Discrete Fourier Transform 127(58)
5.1 Definitions
127(4)
5.2 Properties
131(9)
5.2.1 Periodicity
131(1)
5.2.2 Conjugate Symmetry
131(2)
5.2.3 Circular Shift in Time/Spatial Domain (Periodic Shift)
133(1)
5.2.4 Circular Shift in Frequency Domain (Periodic Shift)
133(2)
5.2.5 Skew Property
135(1)
5.2.6 Rotation Property
135(1)
5.2.7 Parseval's Theorem
135(1)
5.2.8 Convolution Theorem
136(1)
5.2.9 Correlation Theorem
137(2)
5.2.10 Spatial Domain Differentiation
139(1)
5.2.11 Frequency Domain Differentiation
139(1)
5.2.12 Laplacian
139(1)
5.2.13 Rectangle
139(1)
5.3 Two-Dimensional Filtering
140(10)
5.3.1 Inverse Gaussian Filter (IGF)
142(1)
5.3.2 Root Filter
142(1)
5.3.3 Homomorphic Filtering
143(3)
5.3.4 Range Compression
146(2)
5.3.5 Gaussian Lowpass Filter
148(2)
5.4 Inverse and Wiener Filtering
150(6)
5.4.1 The Wiener Filter
151(3)
5.4.2 Geometric Mean Filter (GMF)
154(2)
5.5 Three-Dimensional DFT
156(2)
5.5.1 3-D DFT
156(1)
5.5.2 3-D IDFT
157(1)
5.5.3 3D Coordinates
157(1)
5.5.4 3-D DFT
157(1)
5.5.5 3-D IDFT
157(1)
5.6 Variance Distribution in the 1-D DFT Domain
158(2)
5.7 Sum of Variances Under Unitary Transformation Is Invariant
160(1)
5.8 Variance Distribution in the 2-D DFT Domain
160(2)
5.9 Quantization of Transform Coefficients can be Based on Their Variances
162(4)
5.10 Maximum Variance Zonal Sampling (MVZS)
166(2)
5.11 Geometrical Zonal Sampling (GZS)
168(1)
5.12 Summary
168(1)
5.13 Problems
169(1)
5.14 Projects
170(15)
6 Vector-Radix 2D-FFT Algorithms 185(10)
6.1 Vector Radix DIT-FFT
185(4)
6.2 Vector Radix DIF-FFT
189(4)
6.3 Summary
193(2)
7 Nonuniform DFT 195(40)
7.1 Introduction
195(1)
7.2 One-Dimensional NDFT
196(12)
7.2.1 DFT of Uniformly Sampled Sequences
196(1)
7.2.2 Definition of the NDFT
197(3)
7.2.3 Properties of the NDFT
200(3)
7.2.4 Examples of the NDFT-2
203(5)
7.3 Fast Computation of NDFT
208(9)
7.3.1 Forward NDFT
208(5)
7.3.2 Inverse NDFT
213(4)
7.4 Two-Dimensional NDFT
217(5)
7.4.1 2D Sampling Structure
218(3)
7.4.2 Example of Two-Dimensional Nonuniform Rectangular Sampling
221(1)
7.5 Filter Design Using NDFT
222(11)
7.5.1 Low-Pass Filter Design
222(8)
7.5.2 Example of Nonuniform Low-Pass Filter
230(3)
7.6 Summary
233(1)
7.7 Problems
233(2)
8 Applications 235(82)
8.1 Frequency Domain Downsampling
235(5)
8.1.1 Frequency Domain Upsampling (Zero Insertion)
238(2)
8.2 Fractal Image Compression
240(4)
8.3 Phase Only Correlation
244(3)
8.4 Image Rotation and Translation Using DFT/FFT
247(2)
8.5 Intraframe Error Concealment
249(1)
8.6 Surface Texture Analysis
250(1)
8.7 FFT-Based Ear Model
251(1)
8.8 Image Watermarking
251(2)
8.9 Audio Watermarking
253(3)
8.9.1 Audio Watermarking Using Perceptual Masking
255(1)
8.10 OFDM
256(2)
8.10.1 Signal Representation of OFDM Using IFFT/FFT
257(1)
8.11 FFT Processors for OFDM
258(2)
8.12 DF DFT-Based Channel Estimation Method
260(2)
8.12.1 DF DFT-Based Channel Estimation Method
260(2)
8.13 The Conjugate-Gradient Fast Fourier Transform (CG-FFT)
262(1)
8.14 Modified Discrete Cosine Transform (MDCT)
262(4)
8.15 Oddly Stacked TDAC
266(7)
8.16 Perceptual Transform Audio Coder
273(1)
8.17 OCF Coder
274(1)
8.18 NMR Measurement System
274(1)
8.19 Audio Coder for Mobile Reception
275(2)
8.20 ASPEC (Adaptive Spectral Perceptual Entropy Coding of High Quality Music Signals)
277(1)
8.21 RELP Vocoder (RELP: Residual Excited Linear Prediction)
278(1)
8.22 Homomorphic Vocoders
278(2)
8.23 MUSICAM (Masking-Pattern Universal Sub-Band Integrated Coding and Multiplexing)
280(1)
8.24 AC-2 Audio Coder
280(3)
8.25 IMDCT/IMDST Implementation via IFFT
283(4)
8.26 MDCT/MDST Implementation via IFFT
287(1)
8.27 Autocorrelation Function and Power Density Spectrum
287(2)
8.27.1 Filtered White Noise
288(1)
8.28 Three-Dimensional Face Recognition
289(2)
8.29 Two-Dimesional Multirate Processing
291(9)
8.29.1 Upsampling and Interpolation
292(2)
8.29.2 Downsampling and Decimation
294(6)
8.30 Fast Uniform Discrete Curvelet Transform (FUDCuT)
300(11)
8.30.1 The Radon Transform
300(1)
8.30.2 The Ridgelet Transform
301(2)
8.30.3 The Curvelet Transform
303(8)
8.31 Problems
311(3)
8.32 Projects
314(3)
8.32.1 Directional Bandpass Filters
315(2)
Appendix A: Performance Comparison of Various Discrete Transforms 317(8)
A.1 Transform Coding Gain
318(1)
A.2 Variance Distribution in the Transform Domain
319(1)
A.3 Normalized MSE
319(1)
A.4 Rate Versus Distortion (Rate-Distortion)
319(1)
A.5 Residual Correlation
320(2)
A.6 Scalar Wiener Filtering
322(1)
A.7 Geometrical Zonal Sampling (GZS)
323(1)
A.8 Maximum Variance Zonal Sampling (MVZS)
324(1)
Appendix B: Spectral Distance Measures of Image Quality 325(8)
Project B
328(5)
Appendix C: Integer Discrete Cosine Transform (INTDCT) 333(16)
C.1 Integer DCT Via Lifting
333(4)
C.1.1 Decomposition of DCT Via Walsh–Hadamard Transform
334(3)
C.1.2 Implementation of Integer DCT
337(1)
C.2 Integer DCT by the Principle of Dyadic Symmetry
337(9)
C.2.1 Generation of the Order-Eight Integer DCT
338(2)
C.2.2 Integer DCTs in Video Coding Standards
340(5)
C.2.3 Performance of Eight-Point Integer DCTs
345(1)
Problems
346(1)
Projects
347(2)
Appendix D: DCT and DST 349(14)
D.1 Kernels for DCT and DST
349(2)
D.2 Derivation of Unitary DCTs and DSTs
351(8)
D.3 Circular Convolution Using DCTs and DSTs Instead of FFTs
359(1)
D.4 Circular Shifting Property of the DCT
360(1)
Problems
361(1)
Projects
361(2)
Appendix E: Kronecker Products and Separability 363(4)
E.1 Kronecker Products
363(1)
E.2 Generalized Kronecker Product
364(1)
E.3 Separable Transformation
365(2)
Appendix F: Mathematical Relations 367(2)
Problem
368(1)
Appendix G: Basics of MATLAB 369(10)
G.1 List of MATLAB Related Websites
375(1)
G.1.1 MATLAB Tutorial
375(1)
G.1.2 MATLAB Commands and Functions
375(1)
G.1.3 MATLAB Summary and Tutorial
375(1)
G.1.4 A MATLAB Primer
375(1)
G.1.5 MATLAB FAQ
375(1)
G.2 List of MATLAB Related Books
375(4)
Appendix H: 379(4)
H.1 MATLAB Code for 15-Point WFTA
379(3)
H.2 MATLAB Code for Phase Only Correlation (POC)
382(1)
Bibliography 383(32)
Index 415
Prof. K. R. Rao received the Ph. D. degree in electrical engineering from The University of New Mexico, Albuquerque in 1966. Since 1966, he has been with the University of Texas at Arlington where he is currently a professor of electrical engineering. He, along with two other researchers, introduced the Discrete Cosine Transform in 1975 which has since become very popular in digital signal processing. Some of his books have been translated into Japanese, Chinese, Korean and Russian. He has conducted workshops/tutorials on video/audio coding/standards worldwide. He has published extensively in refereed journals and has been a consultant to industry, research institutes and academia. He is a Fellow of the IEEE.
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