The FFT in the 21st Century: Eigenspace Processing / Edition 1by James Beard
This short reference provides four in-depth reference tutorials on the FFT: Fourier transforms, including Fourier series and discrete Fourier transforms, the Cooley-Tukey algorithm, the bit-reverse index reordering problem and two of its solutions, and spectral windows. Each area has problems of various scopes and difficulties designed to provoke thought and… See more details below
This short reference provides four in-depth reference tutorials on the FFT: Fourier transforms, including Fourier series and discrete Fourier transforms, the Cooley-Tukey algorithm, the bit-reverse index reordering problem and two of its solutions, and spectral windows. Each area has problems of various scopes and difficulties designed to provoke thought and curiosity, and to illuminate the topic in ways not always presented in the text. The scope of the material is specifically intended to apply to applications for the foreseeable future as well as provide a reference for applications that have not yet emerged, such as 3-D processing of multiple parallel data streams for true time-delay broadband beamforming and ambiguity resolution using sparse arrays. This scope is supported through a unified treatment of one, two, and three-dimensional FFTs with seamless extension to higher dimensionality. The section on spectral window weighting is alone worth the book, as it provides a unified in-depth tutorial, including many plots, of all commonly used one-dimensional and two-dimensional spectral weightings, with extensions to higher dimensionality. Several new high-performance spectral weights are introduced, including a monopulse weighting for planar arrays with good antenna efficiency and excellent sidelobe control that is based on Chebychev polynomials of the second kind, and a five-cosine extension to the Blackman-Harris window with 118 dB performance. This unified treatment will leave you in complete command of the theory, algorithm, and signal processing trade space in areas where the FFT is used for the present and for many years into the future.
- Springer US
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- 9.21(w) x 6.14(h) x 0.56(d)
Table of Contents
Dedication. Author. Preface. Foreword.
The Fourier Transform. 1: Overview. 2: Conventions and Notations. 2.1. Complex Variables and the Complex Conjugate. 2.2. Vectors and Matrices. 2.3. Matrix Transpose and Hermitian. 2.4. Common Functions. 3: The Fourier Transform. 3.1. The Classical Fourier Transform. 3.2. Our First Encounter with the Dirac Delta Function. 3.3. Meaning, Usefulness, and Limitations of Formal Identities. 3.4. The Inverse Fourier Transform. 3.5. Parseval's Theorem. 3.6. Multivariate Fourier Transforms. 3.7. Fourier Transform Pairs. 3.8. The Hilbert Transform. 4: The Classical Fourier Series. 4.1. The Fourier Series. 4.2. Parseval's Theorem. 5: The Discrete Fourier Transform. 5.1. The Transform - A Trigonometric Identity. 5.2. Parseval's Theorem. 5.3. The Dirichlet Kernel. 5.4. DFT Pairs. 6: Gibb's Phenomenon. 7: Spatial and Matrix Representations and Interpretations. 7.1. The Continuous Fourier Transform. 7.2. The Classical Fourier Series. 7.3. The Discrete Fourier Transform. 8: Problems. 8.1. General. 8.2. Classical Fourier Transform. 8.3. Classical Fourier Series. 8.4. Discrete Fourier Transform. 8.5. Greater Time and Difficulty. 8.6. Project.
Introduction to the Radix 2 FFT. 1: Historical Note. 2: Notations and Conventions. 3: Ordering the Bits in the Addresses. 4: Examples. 4.1. Simple DIF and DIT. 4.2. Multivariate FFT. 5: Problems. 5.1. General. 5.2. Greater Difficulty. 5.3. Project.
The Reordering Problem and its Solutions. 1: Introduction. 2: Different types of Cooley-Tukey FFTs. 3: In-place, self reordering FFTs. 4: Conclusions. 4.1. Summary. 4.2. Execution Speeds. 4.3. Variable Radix Algorithms. 4.4. Multivariate FFTs. 5: Examples. 6: Problems. 6.1. General. 6.2. Greater Difficulty. 6.3. Project.
Spectral Window Weightings. 1: Overview. 1.1. Base Concepts - The DFT Trade Space. 1.2. Continuous and Discrete Spectral Windows. 1.3. Sampled Continuous Spectral Windows. 1.4. Noise Bandwidth. 1.5. Array Efficiency. 1.6. Spectral Window Frequency Response. 2: Discrete Spectral Windows. 2.1. The Dolph-Chebychev Window. 2.2. Chebychev 2 Window. 2.3. Split Chebychev 2 Window for Monopulse. 2.4. Finite Impulse Response Filters. 3: Continuous Spectral Windows and Sampled Continuous Windows. 3.1. Sampled Continuous Window Functions as Discrete Windows. 3.2. Cosine Windows. 3.3. Continuous Extensions of the Dolph-Chebychev Window. 4: Two-Dimensional Window Weightings. 4.1. Planar Radar Antennas and Two-Dimensional DFTs. 4.2. The Two-Dimensional Dolph-Chebychev Weighting. 4.3. Two-Dimensional Chebychev 2 Window. 4.4. Monopulse with Split Two-Dimensional Chebychev Window. 4.5. Limiting Form of the Two-Dimensional Chebychev Windows for Large N. 4.6. Two-Dimensional Taylor Weighting. 4.7. Lambda Functions and a Unified Theory. 4.8. Monopulse with the Bayliss Window Weights. 5: Three and More Dimensions. 5.1. Chebychev. 6: Linear Programming Window Function Design. 7: Conclusions. 7.1. One-Dimensional Weightings. 7.2. Two-Dimensional Weightings. 7.3. Three and Higher Dimensions. 8: Problems. 8.1. General. 8.2. Greater Difficulty. 8.3. Chebychev Windows. 8.4. Cosine Windows. 8.5. Bessel Windows. 8.6. Two-Dimensional Windows. 8.7. Greater Time and Difficulty. 8.8. Project.
Acknowledgments. References. Index.
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