| Summary: | 博士 === 國立臺灣大學 === 電信工程學研究所 === 89 === Fourier transform (FT) is a very popular mathematical tool. It has been widely applied in engineering, signal processing, etc.
In this thesis, we will introduce the generalization of FT, i.e., fractional Fourier transform (FRFT) and linear canonical transform (LCT).
FRFT has one parameter alpha. The FRFT with parameter alpha= R*pi/2 just means we do FT for R times. So when alpha= pi/2, FRFT becomes the conventional FT. When alpha= pi, FRFT be-comes the time-reverse operation. When alpha= 3pi/2, FRFT becomes the inverse Fourier transform (IFT). And when alpha= 0, FRFT becomes the identity operation. Now we can ask an interesting question: What does the FRFT becomes when alpha= R*pi/2 and R isn’t an integer number? In Chap. 1, we will see in this case FRFT will correspond to the input function multiplied by a chirp function, then transformed by a scaled Fourier transform, then multiplied by a chirp function.
LCT, however, is the further generalization of FRFT. LCT has 4 parameters {a, b, c, d}, and FRFT is the special case of LCT that {a, b, c, d} = {cos(alpha), sin(alpha), -sin(alpha), cos(alpha)}.
FRFT is more flexible than FT, and LCT is more flexible than FRFT. Since FRFT and LCT are more flexible than FT, so their utilities are stronger than FT. They can solve some problems that can’t be solved well by FT.
Recently, there are many research works about FRFT. FRFT has been used for many applications (We use Fig. 7-1 to list the applications of FRFT and LCT systematically). Until now, the research works about LCT are not as many as those of FTFT. But since LCT is very flexible, so it has a lot of potentiality in the future.
In this thesis, I will introduce the research works about FRFT and LCT systematically, including the research works of my professor and I.
In Chap. 1, I will introduce the definition and basic ideas of FRFT and LCT.
In Chap. 25, I will introduce the properties of FRFT and LCT, especially their relation with Wigner discuss function (WDF) and other time-frequency analysis tool (Chap. 3), and their eigenfunctions (Chaps. 4, 5).
In Ch. 6, I will illustrate how to implement FRFT and LCT. In Chap. 7, I will introduce all the applications of FRFT and LCT.
In Chaps. 8, 9, 10, I will introduce discrete FRFT and LCT. In Chap. 8 I will make an overview introduction. In Chap. 9 we will introduce one type of discrete FRFT, i.e., the closed form discrete FRFT in detail. In Chap. 10, discuss the eigenfunctions of the discrete FRFT that has additivity property.
In Chap. 11, I will discuss the simplified FRFT. In Chap. 12, I will discuss 2-D FRFT / LCT. In Chap. 13, I will discuss the fractional, canonical, and simplified fractional sine, cosine, and Hartley transforms. In Chap. 14, I will discuss the fractional Hilbert transform. In Chap. 15, I will discuss other transforms and operations related to FRFT and LCT.
In Chap. 16, I make a conclusion. In the Reference, I will list the papers and books related to FRFT and LCT, and classify them.
May this thesis be helpful for you.
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