If the ROC contains the unit circle (i.e., z 1) then the system is stable. This extends to cases with multiple poles: the ROC will never contain poles. This is intentional to demonstrate that the transform result alone is insufficient.Ĭreating the polezero plot for the causal and anticausal case show that the ROC for either case does not include the pole that is at 0.5. The discrete-time Fourier transform (DTFT)not to be confused with the discrete Fourier transform (DFT)is a special case of such a Z-transform obtained by restricting z to lie on the unit circle. The inverse Z-transform simplifies to the inverse discrete-time Fourier transform. In the case where the ROC is causal (see Example 2 ), this means the path C must encircle all of the poles of X(z). Thus, care is required to note which definition is being used by a particular author. This convention is used, for example, by Robinson and Treitel 7 and by Kanasewich.įor example, the location of zeros and poles move from inside the unit circle using one definition, to outside the unit circle using the other definition. Z Transform Pairs Proof Series Where Oneįrom a mathematical view the Z-transform can also be viewed as a Laurent series where one views the sequence of numbers under consideration as the (Laurent) expansion of an analytic function. It was later dubbed the z-transform by Ragazzini and Zadeh in the sampled-data control group at Columbia University in 1952. Z Transform Pairs Proof Series Where One.
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