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The extend to the Fourier Transform for non-repetitive finite "transient" waveforms. Then you hit the difficulty that this does not always converge, a requirement of the Dirichlet conditions. Some Laplace Transforms do not converge, but all the ones which you will encounter during your training period will converge. Then you come into the digital world where all hell breaks loose and you are asked to undertake some flights of fancy which contradict all the maths that you will have studied up until now.

The train of sampled digital pulses is related to the Unit Impulse, and you now need a transform to model Delayed Unit Impulses. Note that T is a fixed constant and not the continuous variable t. Like laplace transform the z-transform of a sampled insight into the transient behavior, the steady state behaviour and the stability of discrete time systems. A working knowledge of the z-transform is essential to the study of digital filters and systems. Derivation of the z-Transform The z-transform is the discrete-time counterpart of the laplace transform.

The Relationship between the Laplace, the Fourier and the z-Transform The laplace transform, the fourier transform and the z-transform are closely related in that they all employ complex exponential as their basis function. A similar relationship exists between the laplace transform and the fourier transform of a continuous time signal.

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