Biomedical Engineering Reference
In-Depth Information
of two functions in the time domain corresponds to convolution in the frequency domain
as shown in (10.3).
∞
1
2
1
2
=
ζ
ω
−
ζ
∂ζ
=
ω
×
ω
(10.3)
f
1
(
t
)
f
2
(
t
)
F
1
(
)
F
2
(
)
F
1
(
)
F
2
(
)
π
π
−∞
Whereas convolution in the time domain corresponds to product in the frequency domain
as shown by (10.4.)
∞
f
1
(
t
)
×
f
2
(
t
)
=
f
1
(
λ
)
f
2
(
t
−
λ
)
∂λ
⇔
F
1
(
ω
)
F
2
(
ω
)
(10.4)
−∞
Convolution is commutative for time-invariant systems are given by (10.5) and
(10.6).
∞
y
(
t
)
=
h
(
λ
)
x
(
t
−
λ
∂λ
(10.5)
)
−∞
∞
y
(
t
)
=
x
(
λ
)
h
(
t
−
λ
∂λ
(10.6)
)
−∞
Often, the convolution operation is denoted by an asterisk as shown in (10.7).
∞
f
1
(
t
)
×
f
2
(
t
)
=
f
1
(
λ
)
f
2
(
t
−
λ
)
∂λ
(10.7)
−∞
The integration limits vary with the particular characteristics of the functions being
convolved. Next, let us examine and discuss the manner of “Evaluation and Interpretation
of Convolution Integrals and Numerical Convolution.”
10.1 CONVOLUTION EVALUATION
Formally the convolution operation is given as in (10.8).
∞
f
3
=
f
1
×
f
2
=
λ
−
λ
∂λ
(10.8)
f
1
(
)
f
2
(
t
)
−∞
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