The Discrete Linear Chirp Transform and its Applications

In many applications in signal processing, the discrete Fourier transform (DFT) plays a significant role in analyzing characteristics of stationary signals in the frequency domain. The DFT can be implemented in a very efficient way using the fast Fourier transform (FFT) algorithm. However, many actual signals by their nature are non--stationary signals which make the choice of the DFT to deal with such signals not appropriate. Alternative tools for analyzing non--stationary signals come with the development of time--frequency distributions (TFD). The Wigner--Ville distribution is a time--frequency distribution that represents linear chirps in an ideal way, but it has the problem of cross--terms which makes the analysis of such tools unacceptable for multi--component signals. In this dissertation, we develop three definitions of linear chirp transforms which are: the continuous linear chirp transform (CLCT), the discrete linear chirp transform (DLCT), and the discrete cosine chirp transform (DCCT). Most of this work focuses on the discrete linear chirp transform (DLCT) which can be considered a generalization of the DFT to analyze non--stationary signals. The DLCT is a joint frequency chirp--rate transformation, capable of locally representing signals in terms of linear chirps. Important properties of this transform are discussed and explored. The efficient implementation of the DLCT is given by taking advantage of the FFT algorithm. Since this novel transform can be implemented in a fast and efficient way, this would make the proposed transform a candidate to be used for many applications, including chirp rate estimation, signal compression, filtering, signal separation, elimination of the cross--terms in the Wigner--Ville distribution, and in communication systems. In this dissertation, we will explore some of these applications