Example 4.4: Find the Fourier transform of the time function
yt e
Àajtj
, a > 0:
This equation can be written as
yt xÀt xt,
where
xt e
Àat
ut, a > 0:
From Table 4.1,we have XV 1=a jV. From Table 4.2,we have
YV XÀV XV. This results in
YV
1
a À jV
1
a jV
2a
a
2
V
2
:
4.2 The z-Transform
Continuous-time signals and systems are commonly analyzed using the Fourier trans-
form and the Laplace transform (will be introduced in Chapter 6). For discrete-time
systems,the transform corresponding to the Laplace transform is the z-transform. The
z-transform yields a frequency-domain description of discrete-time signals and systems,
and provides a powerful tool in the design and implementation of digital filters. In this
section,we will introduce the z-transform,discuss some important properties,and show
its importance in the analysis of linear time-invariant (LTI) systems.
4.2.1 Definitions and Basic Properties
The z-transform (ZT) of a digital signal, xn, À I < n < I,is defined as the power
series
Xz
I
nÀI
xnz
Àn
,
4:2:1
where Xz represents the z-transform of xn. The variable z is a complex variable,and
can be expressed in polar form as
z re
jy
,
4:2:2
where r is the magnitude (radius) of z,and y is the angle of z. When r 1, jzj 1 is
called the unit circle on the z-plane. Since the z-transform involves an infinite power
series,it exists only for those values of z where the power series defined in (4.2.1)
THE Z-TRANSFORM
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