Biomedical Engineering Reference
In-Depth Information
In the example, for the next translations the limits are reset to the interval between
2
<
t
<
∞
as shown in (10.19).
t
e
−2
τ
∂τ
f
3
(
t
)
=
6
(10.19)
−
2
t
10.1.3 Convolution as a Summation
In evaluation of the convolution integral by digital computer, it is necessary to replace
the integral (10.20) with a finite summation.
t
v
2
(
t
)
=
v
1
(
τ
)
h
(
t
−
τ
)
∂τ
(10.20)
0
To accomplish this conversion, the infinitesimal
∂τ
is replaced with a time interval of
finite width
T
and
τ
is replaced with
nT
.
The convolution integral is written as the
summation for
KT
≤
t
<
(
K
+
1)
T
as shown in (10.21).
K
v
2
(
t
)
=
T
v
1
(
nT
)
h
(
t
−
nT
)
(10.21)
n
=
0
where
K
is one of the values of the integer
n
.
Equation (10.22) gives the convolution of (10.21) in expanded form.
v
2
(
t
)
=
T
[
v
1
(0)
h
(
t
)
+
v
1
(
T
)
h
(
t
−
T
)
+
v
1
(2
T
)
h
(
t
−
2
T
)
+
v
1
3
T
)
h
(
t
−
3
T
)
+···
]
(10.22)
Each term in (10.22) is the impulse response of magnitude of
v
1
(
nT
) shifted along the
t
-axis and later is multiplied by
T
. Summation of the terms in the equation approximates
the function. The accuracy of the approximation will depend on the magnitude of
T
.
The smaller the value of
T
is, the better the approximation.
10.2 NUMERICAL CONVOLUTION
Numerical evaluation of the convolution integral is straightforward. When either func-
tion is of finite duration, there will be a finite number of terms in each summation.
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