Integral Signal Represent at ions The integral transform is one of the most important tools in signal theory. The best known example is the Fourier transform,buttherearemany other transforms of interest. In the following, W will first discuss the basic concepts of integral transforms. Then we will study the Fourier, Hartley, and Hilbert transforms. Finally, we will focus on real bandpass processes and their representation by means of their complex envelope. | Signal Analysis Wavelets Filter Banks Time-Frequency Transforms and Applications. Alfred Mertins Copyright 1999 John Wiley Sons Ltd Print ISBN 0-471-98626-7 Electronic ISBN 0-470-84183-4 Chapter 2 Integral Signal Representations The integral transform is one of the most important tools in signal theory. The best known example is the Fourier transform but there are many other transforms of interest. In the following we will first discuss the basic concepts of integral transforms. Then we will study the Fourier Hartley and Hilbert transforms. Finally we will focus on real bandpass processes and their representation by means of their complex envelope. Integral Transforms The basic idea of an integral representation is to describe a signal x t via its density xis with respect to an arbitrary kernel p t s x t I xts pit s ds Js t G T. 2-1 Analogous to the reciprocal basis in discrete signal representations see Section a reciproal kernel 0 s f may be found such that the density xis can be calculated in the form xis J x t is t dt sES. 2-2 22 . Integral Transforms 23 Contrary to discrete representations we do not demand that the kernels p t s and ớ s t be integrable with respect to t. From and we obtain x t a T dr ds s T 2-3 x t 0 s t ip t s ds dr. JT J s In order to state the condition for the validity of in a relatively simple form the so-called Dirac impulse ỗ i is required. By this we mean a generalized function with the property x t í ố í t x t dr X G L1 IR . oo The Dirac impulse can be viewed as the limit of a family of functions ga t that has the following property for all signals x t continuous at the origin ga t dt 1 lim ga t x t dt x 0 . J-oo - 0 J-oo An example is the Gaussian function ổaơ J e fc a 0. V2 ra Considering the Fourier transform of the Gaussian function that is 00 _ i e jojt dt _ụ e 2a we find that it approximates the constant one for a 0 that is Ga w re 1 cj G IR. For the Dirac impulse the correspondence d t t 1
