Generalisation of the Dirac-delta impulse extending Laplace and z transform domains

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A generalisation of the Dirac-delta impulse and its derivatives as two generalised distributions, namely, the xi and zeta impulses, and their derivatives, defined on the complex s-plane and z-plane of continuous-time and discrete-time functions, respectively, is proposed. The generalised impulses extend the existence of Laplace and z transforms to a large class of infinite duration two-sided functions, which hitherto had no transform or had only a Fourier transform in the form of distributions. The proposed generalised impulses are shown to bridge the gap between the theory of generalised functions and both the unilateral and bilateral Laplace and z transforms. The generalised impulses extend the existence of Laplace and z transforms to include both functions that have a Fourier transform as a distribution as well as exponentially rising infinite duration two-sided functions that have no Fourier transform. It is shown that a modulation theory can now be added to the properties of bilateral transforms. No such theorem has hitherto existed for these transforms. The proposed generalised impulses and the resulting extended Laplace and z transforms are shown to lead to new complex-plane operations, such as spatial convolution, and to simplify operations such as ordinary convolution, sampling and the solution of differential and difference equations. Bilateral Laplace and z transforms may receive greater attention now that these transforms can be applied to a new, large and basic class of functions, such as two-sided infinite duration exponentials and rising trigonometric and hyperbolic functions.