Fast Fourier Transform
Description
This book presents an introduction to the principles of the fast Fourier transform. This book 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 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.
This book provides 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.
Reviews / Votes
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 programs the 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)
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Persons
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.
Content
Preface. Acknowledgements. Acronyms.
1 Introduction. 1.1 Applications of Discrete Fourier Transform.
2 Discrete Fourier Transform. 2.1 Definitions. 2.2 The Z-transform. 2.3 Properties of the DFT. 2.4 Convolution Theorem. 2.5 Correlation Theorem. 2.6 Overlap-Add and Overlap-Save Methods. 2.7 Zero Padding in the Data Domain. 2.8 Computation of DFTs of Two Real Sequences Using One Complex FFT. 2.9 A circulant matrix is diagonalized by the DFT matrix. 2.10 Summary. Projects. Problems.
3 Fast Algorithms. 3.1 Radix-2 DIT-FFT. 3.2 Fast Algorithms by Sparse Matrix Factorization. 3.3 Radix-2 DIF-FFT. 3.4 Radix-3 DIT-FFT. 3.5 Radix-3 DIF-FFT. 3.6 FFT for N a Composite Number. 3.7 Radix-4 DIT-FFT. 3.8 Radix-4 DIF-FFT. 3.9 Split-Radix FFT Algorithm. 3.10 Fast Fourier and BIFORE Transforms by Matrix Partitioning. 3.11 The Winograd Fourier Transform Algorithm. 3.12 Sparse Factorization of the DFT Matrix. 3.13 Unified Discrete Fourier-Hartley Transform. 3.14 Bluestein's FFT Algorithm. 3.15 Rader Prime Algorithm. 3.16 Summary. Projects. Problems.
4 Integer Fast Fourier Transform. 4.1 Introduction. 4.2 Lifting Scheme. 4.3 Integer FFT. 4.4 Integer Discrete Fourier Transform. 4.5 Summary. Projects. Problems.
5 Two-Dimensional Discrete Fourier Transform. 5.1 Definitions. 5.2 Properties. 5.3 Two-Dimensional Filtering. 5.4 Inverse and Wiener Filtering. 5.5 Three-Dimensional DFT. 5.6 Variance Distribution in the 1-D DFT Domain. 5.7 Sum of variances under orthogonal transformation is invariant. 5.8 Variance Distribution in the 2-D DFT Domain. 5.9 Quantization of transform coefficients can be based on their variances. 5.10 Maximum Variance Zonal Sampling (MVZS). 5.11 Geometrical Zonal Sampling (GZS). 5.12 Summary. Projects. Problems.
6 Vector-Radix 2-D FFT Algorithm. 6.1 Vector Radix DIT-FFT. 6.2 Vector Radix DIF-FFT. 6.3 Summary. Projects. Problems.
7 Nonuniform DFT. 7.1 Introduction. 7.2 One-Dimensional NDFT. 7.3 Fast Computation of NDFT. 7.4 Two-Dimensional NDFT. 7.5 Filter Design Using NDFT. 7.6 Summary. Problems.
8 Applications. 8.1 Frequency Domain Downsampling. 8.2 Fractal Image Compression. 8.3 Phase Only Correlation. 8.4 Image Rotation and Translation Using DFT/FFT. 8.5 Intraframe Error Concealment. 8.6 Surface Texture Analysis. 8.7 FFT-Based Ear Model. 8.8 Image Watermarking. 8.9 Audio Watermarking. 8.10 OFDM. 8.11 FFT Processors for OFDM. 8.12 DF DFT-Based Channel Estimation Method. 8.13 The Conjugate-Gradient Fast Fourier Transform (CG-FFT). 8.14 Modified Discrete Cosine Transform (MDCT). 8.15 Oddly Stacked TDAC. 8.16 Preceptual Transform Audio Coder. 8.17 OCF Coder. 8.18 NMR Measurement System. 8.19 Audio Coder for Mobile Reception. 8.20 ASPEC. 8.21 RELP Vocoder. 8.22 Homomorphic Vocoders. 8.23 MUSICAM. 8.24 AC-2 Audio Coder. 8.25 IMDCT/IMDST Implementation via IFFT. 8.26 MDCT/MDST Implementation via IFFT. 8.27 Autocorrelation Function and Power Density Spectrum. 8.28 Three-Dimensional Face Recognition. 8.29 Two-Dimensional Multirate Processing. 8.30 Fast Uniform Discrete Curvelet Transform. 8.31 Problems. 8.32 Projects.
Appendix A: Performance Comparison of Various Discrete Transforms. A.1 Transform Coding Gain. A.2 Variance Distribution in the Transform Domain. A.3 Normalized MSE. A.4 Rate Versus Distortion. A.5 Residual Correlation. A.6 Scalar Wiener Filtering. A.7 Geometrical Zonal Sampling. A.8 Maximum Variance Zonal Sampling.
Appendix B: Spectral Distance Measures of Image Quality. Project B.
Appendix C: Integer Discrete Cosine Transform. C.1 Integer DCT Via Lifting. C.2 Integer DCT by the Principle of Dyadic Symmetry. Problems. Projects.
Appendix D: DCT and DST. D.1 Kernels for DCT and DST. D.2 Derivation of Unitary DCTs and DSTs. D.3 Circular Convolution Using DCTs and DSTs Instead of FFTs. D.4 Circular Shifting Property of the DCT. Problems. Projects.
Appendix E: Kronecker Products and Separability. E.1 Kronecker Products. E.2 Generalized Kronecker Product. E.3 Separable Transformation.
Appendix F: Mathematical Relations. Problem.
Appendix G: Basics of MATLAB. G.1 List of MATLAB Related Websites. G.2 List of MATLAB Related Books.
Appendix H. H.1 MATLAB Code for 15-Point WFTA. H.2 MATLAB Code for Phase only Correlation.
Bibliography. Index.