Real-time digital signal processing: implementations, ... changes in the input signal is limited by its internal clock rate, so that it may be slow to
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xnT xt
I
nÀI
dt À nT
I
n0
xnTdt À nT:
6:1:13
To obtain the frequency characteristics of the sampled signal, take the Laplace
transform of x(nT ) given in (6.1.13). Integrating term-by-term and using the property
of the impulse function
I
ÀI
xtdt À tdt xt, we obtain
Xs
I
ÀI
I
n0
xnTdt À nT
4
5
e
st
dt
I
n0
xnTe
ÀnsT
:
6:1:14
When defining a complex variable
z e
sT
,
6:1:15
Equation (6.1.14) can be expressed as
Xz Xsj
ze
sT
I
n0
xnTz
Àn
,
6:1:16
where X(z) is the z-transform of the discrete-time signal x(nT). Thus the z-transform can
be viewed as the Laplace transform of the sampled function x(t) with the change of
variable z e
sT
.
As discussed in Chapter 4, the Fourier transform of a sequence x(nT) can be obtained
from the z-transform by replacing z with e
j!
. That is, by evaluating the z-transform on
the unit circle of jzj 1. The whole procedure can be summarized in Figure 6.1.
6.1.3 Mapping Properties
The relationship z e
sT
defined in (6.1.15) represents the mapping of a region in the s-
plane to the z-plane since both s and z are complex variables. Since s s jV, we have
z e
sT
e
sT
e
jVT
jzje
j!
,
6:1:17
Sampling
Laplace
transform
z-transform
Fourier
transform
z
= e
sT
z
= e
jw
x(t)
x(nT)
X(s)
X(z)
X(w)
Figure 6.1 Relationships between the Laplace, Fourier, and z-transforms
246
DESIGN AND IMPLEMENTATION OF IIR FILTERS
Summary :
Integrating term-by-term and using the property of the impulse function I ÀI xtdt À tdt xt, we obtain Xs I ÀI I n0 xnTdt À nT 4 5 e st dt I n0 xnTe ÀnsT : 6:1:14 When defining a complex variable z e sT , 6:1:15 Equation (6.1.14) can be expressed as Xz Xsj ze sT I n0 xnTz Àn , 6:1:16 where X(z) is the z-transform of the discrete-time signal x(nT).
Tags :
xnt,transform,ztransform,laplace,fourier,figure,since,xntdt,6115,function,signal,sampled,mapping