Ch7. The Z-Transform
4
In general, the z-transform of a sequence has associated with it a
range of values of z for which X(z) converges, and this range of values
is referred to as the region of convergence (ROC).
The z-transform reduces to the
Fourier transform for values of z
on the unit circle.
For convergence of the z-transform, we require that the Fourier
transform of converge.
n
rnx
][
z-plane
ω
1
Im
Re
Unit circle
j
ez
If the ROC includes the unit circle, then the Fourier transform also
converges.
Different from the continuous-time case, the z-transform reduces to
the Fourier transform on the contour in the complex z-plane
corresponding to a circle with a radius of unity .
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