348
Practical MATLAB
® Applications for Engineers
e. Differentiation in freq(s)
Ϫt f t
dF s
ds
⋅
↔
( )
( )
or a more general relation is given by
( )
( )
[ ( )]
Ϫ1
n n
n
n
t f t
d
ds
F s
↔
f. Differentiation in t
df t
dt
sF s
f
( )
( )
( )
↔
Ϫ
Ϫ
0
The second derivative is given by
d
dt
df t
dt
d
dt
f t
s sF s
f
f
s F s
s
( )
[ ( )]
[ ( )
( )]
( )
( )

 

 
↔
ϭ
2
2
2
0
0
Ϫ
Ϫ Ј
Ϫ
ϩ
ϩ
f f
f
( )
( )
0
0
ϩ
ϩ
Ϫ Ј
The preceding process can be repeated for higher-order derivatives.
g. Integration in s
f t
t
F s ds
( )
( )
↔
∞
∫
0
h. Periodic functions f(t), with period T > 0
f t
e f t dt
e
st
T
sT
( )
( )
(
)
↔
∫
Ϫ
Ϫ
Ϫ
0
1
i. Integration in t
f d
s F s
f
s
( )
( )
( )
0
1
0
∫
↔
ϩ
j. Convolution in t or product in s
f t
f t
f
f t
d
F s F s
1
2
1
2
1
2
( )
( )
( ) (
)
[ ( ) ( )]
⊗
↔
∞
∞
∫
ϭ
Ϫ
Ϫ
k. Product in t or convolution in s
f t f t
j
F s
F s
1
2
1
2
1
2
( ) ( )
[ ( )
( )]
↔
⊗
CRC_47760_CH004.indd 348
CRC_47760_CH004.indd 348
7/28/2008 12:25:57 PM
7/28/2008 12:25:57 PM