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In case of DTFT of a system function, it can do the transform only if the h[n] is a stable sequence Σh[n] should be less than infinite. So we can take DTFT only for stable systems. But by using z-transform we can transform even the unstable systems and can analysis the stability criterion.
In casse there are super exponential signals, one cannot find the Fourier Transform. In order to obtain a transform, not necessarily the spectrum, the signal is first "killed" by a super-super exponential and the transform taken.The advantage it offers is that a larger family of signals can be transformed.In the digital domain it is the DTFT and z-tranform, while in the analog domain it is the Fourier transform and the Laplace transform