Biomedical Engineering Reference
In-Depth Information
Rewriting (10.33), as moment generating functions, is shown in (10.34) and (10.35).
Equation 10.36 shows that the moment about zero of the convolution function
f
3
is
equal to the product of the moments about zero of the two functions
f
1
and
f
2
.
∞
M
0
(
f
3
)
=
f
1
(
τ
)[area under
f
2
(
t
)]
d
τ
(10.34)
−∞
M
0
(
f
3
)
=
A
f
3
(
t
)
=
[area under
f
1
(
t
)]
•
A
f
2
(
t
)
(10.35)
M
0
(
f
3
)
=
M
0
(
f
1
)
•
M
0
(
f
2
)
(10.36)
Another useful relationship deals with the center of gravity (centroid) of the con-
volution in terms of the center of gravity of the factors (10.37). If you recall, the center
of gravity of a waveform was defined in terms of the energy of the signal as
t
0
.
∞
tf
(
t
)
dt
−∞
n
=
(10.37)
∞
f
(
t
)
dt
−∞
The
n
th moment of a function or waveform is defined by the general (10.38), and
∞
t
n
M
n
(
f
)
=
f
(
t
)
dt
(10.38)
−∞
then the centroid is defined as given by (10.39).
M
1
(
f
)
M
0
(
f
)
=
M
1
[area under the waveform]
n
=
(10.39)
The first moment of the convolution with infinite limits is given by (10.40).
t
dt
M
1
(
f
3
)
=
tf
3
(
t
)
dt
=
f
1
(
τ
)
f
2
(
t
−
τ
)
d
τ
(10.40)
The net result in moments is given by expression (10.41):
M
1
(
f
3
)
=
M
1
(
f
1
)
M
0
(
f
2
)
+
M
1
(
f
2
)
M
0
(
f
1
)
(10.41)
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