This thesis is a supplement to chapter 13 in book Fractional Fourier Transform and Its Applications whose authors are Ran Tao, Bing Deng and Yue Wang[1] . LCT was first proposed by Moshinsky and Collins in the 1970s[2][3] . And the special case was first proposed by Bargmann[4] . LCT is introduced systematically in [5] in In 1997, Bargmann et al. had discussed how to design filter by applying LCT[6] . LCT plays an important role with very broad applicability in many fields of science and engineering. Although it is not known widely, its special cases are frequently used in various fields, especially in signal processing field. Therefore, understanding the LCT may help to gain more insights on its special cases and we can use LCT to solve problems more proficiently. Linear canonical transform (LCT) is a generalization of the Fourier transform and fractional Fourier transform (FRFT). At the earliest LCT was used for differential equation and optical system analysis. With the development of FRFT from 1990s, LCT becomes emphasis in signal processing field. LCT is a four-parameter (a, b, c, d) class of linear integral transform. This thesis have applied some concerned theories in [7]. T is called a tempered distribution when T is continuous linear functional from S := S (R) to H, where S (R) is the Schwarz class of rapidly decreasing functions and H is the Hamilton. If T ∈ S (R), we write T [ϕ] := Z R T (x) ϕ (x)dx. Fourier transform ϕb(u) = Z R ϕ (x)e jxudx is performed by the formula Tb [ϕ] = T [ϕb] ,, where ϕ ∈ S (R). And the following result will be used in this thesis b1 (u) = 2πδ (u). (∗) The outline of this thesis is as follows. In chapter 1 we summarize briefly the definition of LCT, some important properties, eigenfunctions of LCT[10] and convolution theorem[11] . In chapter 2 we show frame theory in LCT domain[12] . In chapter 3 Hilbert transform[13] in LCT domain is presented and in chapter 4 the discrete LCT[14] is derived. In chapter 5 applications of LCT are stated.