文档介绍：INTRODUCTION
z-transform is the discrete-time counterpart of the Laplace transform, however, they have some important distinctions that arise from the fundamental differences between continuous-time and discrete-time signals and systems.
z-transform expand the application in which Fourier analysis can be used.
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1 . The Z-Transform
The z-transform of a general discrete-time signal x[n] is defined as
where z is plex variable.
Expressing plex variable z in polar form as
is the Fourier transform of x[n] multiplied by a real exponential
For r = 1, or equivalently, |z| = 1, z-transform equation reduces to the Fourier transform.
We will denote the transform relationship between x[n] and X(z) as
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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.
z-plane
ω
1
Im
Re
Unit circle
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 plex z-plane corresponding to a circle with a radius of unity .
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For convergence of X(z), we require that
Consequently, the region of convergence is the range of values of z for which
Then
Unit circle
z-plane
Im
a 1 Re
Pole-zero plot and region of convergence for Example for 0 < α< 1
Example Consider the signal
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Example Determine the z-transform of
If , this sum converges and
Unit circle
z-plane
Im
a 1 Re
Pole-zero plot and region of convergence for Example for 0 < α< 1
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Example Consider a signal that is the sum of two real exponentials:
The z-transform is then
Im
1/3 1/2 1 3/2 Re
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Example Consider the signal
The z-transform of this sig