Fractional Fourier Transform and Its Applications

• Main Authors: Chi-chin Lien, 連錡晉
• Other Authors: Soo-Chang Pei
• Format: Others
• Language: zh-TW
• Published: 1999
• Online Access: http://ndltd.ncl.edu.tw/handle/44974497829606381679

碩士 === 國立臺灣大學 === 電信工程學研究所 === 87 === The properties and applications of fractional transform are widely discussed in recent years. Among these transforms, the fractional Fourier Transform (FrFT) attracts most attention. FrFT has a parameter more than Fourier transform. When the parameter equals one, FrFT is the same as FT. When the parameter equals zero, FrFT is identity transform. Additionally, the parameter satisfies additive property. We treat FrFT as an extension of Fourier transform. No matter in the fields of signal processing or optics, we all deeply expect its potential. In this dissertation, we give complete introduction and research of fractional Fourier transform. This dissertation consists of five chapters. In chapter 1, we simply introduce FrFT and explain the organization of this thesis. In chapter 2, we introduce continuous FrFT. Besides of its definition, we also derive some properties, and explain its relationship to Fourier optics, affine Fourier transform, and time-frequency distribution. In chapter 3, we introduce discrete FrFT. There are four methods introduced. The main method is to eigen-decompose the transform matrix of DFT, and it is the only method that preserves the additive property of fractional transform now. The other three are approximate methods of FrFT. In chapter4, we apply DFrFT to pattern recognition, beam shaping, and random phase encoding. We do some computer experiments and analysis. The results show that we can control the shift-variant extent of object recognition by fractional order and use DFrFT to detect objects with different scales. DFrFT also has better performance than DFT in beam shaping and random phase encoding. In chapter 5, we conclude this dissertation and suggest our future works.