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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transform. Given a positive-time function, xt 0, for t < 0, a simple way to find the
Fourier transform is to multiply x(t) by a convergence factor e
Àst
, where s is a positive
number such that
I
0
xte
Àst
dt < I:
6:1:1
Taking the Fourier transform defined in (4.1.10) on the composite function xte
Àst
, we
have
Xs
I
0
xte
Àst
e
ÀjVt
dt
I
0
xte
ÀsjVt
dt
I
0
xte
Àst
dt,
6:1:2
where
s s jV
6:1:3
is a complex variable. This is called the one-sided Laplace transform of x(t) and is
denoted by Xs LTxt. Table 6.1 lists the Laplace transforms of some simple time
functions.
Example 6.1: Find the Laplace transform of signal
xt a be
Àct
, t ! 0:
From Table 6.1, we have the transform pairs
a 6
a
s
and e
Àct
6
1
s c
:
Using the linear property, we have
Xs
a
s
b
s c
:
The inverse Laplace transform can be expressed as
xt
1
2pj
sjI
sÀjI
Xse
st
ds:
6:1:4
The integral is evaluated along the straight line s jV in the complex plane from
V ÀI to V I, which is parallel to the imaginary axis jV at a distance s
from it.
242
DESIGN AND IMPLEMENTATION OF IIR FILTERS
Summary :
I: 6:1:1 Taking the Fourier transform defined in (4.1.10) on the composite function xte Àst , we have Xs I 0 xte Àst e ÀjVt dt I 0 xte ÀsjVt dt I 0 xte Àst dt, 6:1:2 where s s jV 6:1:3 is a complex variable.
Tags :
transform,àst,xte,laplace,hae,function,àct,complex,find,table,fourier,simple,multiply