Linear Canonical Transform

Linear canonical transform (LCT) is a generalization of the Fourier transform (FT) and the fractional Fourier transform (FRFT). Analyzing a signal by the LCT can be viewed as analyzing a signal in a domain between time and frequency. From the view of the Wigner distribution function, the LCT has the effect of twisting the distributions of a function in the time-frequency plane. In this chapter, we introduce the definitions, theories, physical meanings, and applications of the LCT and its generalized versions. Since the LCT is more general than the FT, in digital signal processing applications, using the LCT is more flexible than using the FT. Therefore, most applications of the FT are also the applications of the LCT, and one can apply the LCT to achieve even better performances. The LCT has been applied to filter design, signal sampling, modulation, multiplexing, image processing, optics, phase retrieval, radar system analysis, and communication. It will play a very important role in signal processing in the future.