The Application of the Single Sided Laplace Transform and Fourier Transform Embodied Within Double Sided Laplace Transform by Using Generalized Functions, Eliminating Regions of Convergence Restraints
Abstract
Roland Khera
The Double-Sided Laplace Transform (DSLT) and the Fourier Transform (FT) are the same at s=jω, but the Unit Step function (Heaviside Function), U(t) does not have the same DSLT and FT at s=jω. This is now solved. It will be shown that the DSLT of f is the Single-Sided Laplace transform (SSLT) of f(t). With the use of generalized function (in particular the complex delta function and its derivatives), the DSLT can be used where ever SSLT and FT are used in engineering applications. The SSLT of an even rational function is shown to be odd, and visa-versa. The problem of the region of convergence in the “s” complex plane is eliminated by including the complex delta function for solving divergent DSLT integrals. The solution for solving divergent integrals are already well established for solving divergent integrals for FT by using the real delta function. An example is provided for solving the Phase Retrieval problem exactly by measuring the signal's even and odd components autocorrelation functions. This has not been possible with the use of the SSLT because one is then only considers cases for time greater than zero, whereas an even and odd function in time needs to be both positive and negative.