Laplace transform and Z-transform: unification and extension

Laplace Transform and Z-transform: Unification and Extension

We introduce the Laplace transform for an arbitrary time scale. Two particular choices of time scales, namely the reals and the integers, yield the concepts of the classical Laplace transform and of the classical Z-transform. Other choices of time scales yield new concepts of our Laplace transform, which can be applied to find solutions of higher order linear dynamic equations with constant coe...

Laplace Transform

We have seen before that Fourier analysis is very useful in the study of signals and linear and time invariant (LTI) systems. The main reason is that a lot of signals can be expressed as a linear combination of complex exponentials of the form e with s = jw. There are many properties that still apply when s is not restricted to be pure imaginary. That is why we introduce a generalization of the...

the Laplace transform.

The spherical phylon group and invariants of the Laplace transform. Abstract We introduce the spherical phylon group, a subgroup of the group of all formal diffeomorphisms of R d that fix the origin. The invariant theory of the spherical phylon group is used to understand the invariants of the Laplace transform.

Laplace Transform and Continuous-Time Frequency Response 1 Definition of Laplace Transform

Laplace Transform, Dynamics and Spectral Geometry

We consider vector fields X on a closed manifold M with rest points of Morse type. For such vector fields we define the property of exponential growth. A cohomology class ξ ∈ H(M ;R) which is Lyapunov for X defines counting functions for isolated instantons and closed trajectories. If X has exponential growth property we show, under a mild hypothesis generically satisfied, that these counting f...