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Chapter 4 The Laplace Transformation
f f f f
^
L f t *f t ` = ³ f W ³ f O e – s O W + dO dW = ³ f W e – sW dW ³ f O e – sO dO
1
1
2
1
2
2
0 0 0 0
= F s F s
2
1
13. Convolution in the Complex Frequency Domain
Convolution in the complex frequency domain divided by 12Sj , corresponds to multiplication in the
e
time domain. That is,
1
f t f t -------- F s *F s (4.36)
1
2
1
2
2Sj
Proof:
f
L f t f t ^ 1 2 ` = ³ f t f t e – st dt (4.37)
1
2
0
and recalling that the Inverse Laplace transform from (4.2) is
1 V + jZ Pt
f t = -------- ³ V jZ F P e dP
1
1
2Sj
–
by substitution into (4.37), we get
f 1 V + jZ
Pt
L f t f t ^ 1 2 ` = ³ -------- ³ F P e dP f t e – st dt
2
1
0 2Sj V jZ
–
1
–
= -------- ³ V + jZ F P ³ f f t e s – P t dt dP
2Sj V – jZ 1 0 2
–
We observe that the bracketed integral is F s P ; therefore,
2
1 V + jZ 1
Lf t f t ^ 1 2 ` = -------- ³ F P F s P– 2 dP = --------F s *F s
1
2
1
2Sj
2Sj
V – jZ
For easy reference, we have summarized the Laplace transform pairs and theorems in Table 4.1.
4.3 The Laplace Transform of Common Functions of Time
In this section, we will present several examples for finding the Laplace transform of common func-
tions of time.
Example 4.1
^
Find L u t `
0
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