348 Practical MATLAB® Applications for Engineers
e. Differentiation in freq(s)
tft
dF s
ds
⋅↔()
()
or a more general relation is given by
() () 1 nn [()]
n
tft n
d
ds
↔ Fs
f. Differentiation in t
df t
dt
sF s f
()
↔ () ( ) 0
The second derivative is given by
d
dt
df t
dt
d
dt
ft ssFs f f
sFs s
()
[ ( )] [ ( ) ( )] ( )
()
↔
2
2
2
00
fff() ()00
The preceding process can be repeated for higher-order derivatives.
g. Integration in s
ft
t
Fsds
()
↔ ()
∞
∫
(^0)
h. Periodic functions f(t), with period T > 0
ft
e ftdt
e
T st
() sT
()
()
↔
∫
0
(^1)
i. Integration in t
fd sFs
f
() () s
()
0
1 0
∫ ↔
j. Convolution in t or product in s
ft 12 ()⊗↔ft() f 12 ( ) (ft )d [ () ()]FsFs 12
∞
∞
∫
k. Product in t or convolution in s
ftft
j
12 Fs 1 Fs 2