The FFT in the 21st Century

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.