| Summary: | 博士 === 國立交通大學 === 電機與控制工程系 === 88 === The main purpose of this dissertation is to develop efficient fast-Fourier-transform (FFT) algorithms for different applications. These algorithms are designed to best enhance the performance of FFT computation, in different conditions, mainly by reducing the arithmetic operations and reducing the storage requirement. Better FFT algorithms can be designed by using some particular properties of the computational architecture. For example, the input/output pruning strategy can be employed to reduce unnecessary computations; and the symmetrical property for real input data can be explored to save computational time. In addition, we discuss an efficient approach for computing the running spectra based on short-time Fourier transform (STFT). The approach is competitive in computing the moving-windowed STFT’s.
Two constructive subjects in this research are: 1) the concept and implementation of tunnel FFT, and 2) realization of 2D svr-FFT (split-vector-radix FFT). In tunnel FFT, the main idea is to efficiently utilize the memory space to accomplish heavy FFT computations. The idea is important especially for multidimensional signal. In 2D svr-FFT, we, for the first time, explore its structure, derive a theorem to identify the DFT blocks at a given stage, and discover the phenomenon that the distribution of DFT blocks (DFT image) exhibits fractal structure── the well-known Sierpinski triangle. The result enables us to develop an efficient algorithm that actually implements the concept of 2D svr-FFT. Most importantly, it can be easily extended to higher-dimensional svr-FFT algorithms.
This research work also proposes an unconventional method, instead of the widely-used recursive approach, for analyzing the number of arithmetic operations required by FFT algorithms. The proposed method is straightforward and more comprehensible. In summary, the dissertation is mainly focused on how to accomplish a given FFT computation by utilizing the minimum computer resources.
1.1 Beginning
1.2 Organization of this Dissertation
2 Background
2.1 Defintion and View of DFT
2.2 Beginning of FFT
2.3 Basic Architecture of FFT
2.4 Basic Programming Skill
3 Split-Radix FFT
3.1 Background
3.2 The Relation between DFT Blocks
3.3 Programming Aspects for 1D and 2D DIT sr-FFT
3.4 Computational Complexity
3.4.1 Theoretical Analysis
3.4.2 Computer Simulation
4 Real-Time Shift-Pruning FFT
4.1 Background
4.2 Input/Output Pruning FFT
4.3 Moving FFT
4.4 Real-Time FFT
4.5 Real-Time Shift-Pruning FFT
4.6 Computational Complexity
4.6.1 Theoretical Analysis
4.6.2 Computer Simulation
5 2D Vector-Radix FFT for Real and Conjugate-Symmetric Data
5.1 Background
5.2 Data Layout of the 2D DFT Block
5.3 Computational Complexity
5.3.1 Theoretical Analysis
5.3.2 Computer Simulation
6 Tunnel FFT
6.1 Background
6.2 Tunnel FFT and Tunnel IFFT
6.3 Computational Complexity
6.3.1 Theoretical Analysis
6.3.2 Computer Simulation
7 Conclusion
Appendix A 1D sr-FFT Code
Appendix B 2D svr-FFT Pseudo Code
Appendix C Computational Scheme for 2D Real Input
Appendix D Computational Scheme for 2D Conjugate-Symmetric Input
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