Biomedical Engineering Reference
In-Depth Information
the order of convolution is not important as indicated by (10.23).
f
1
(
t
)
[
f
2
(
t
)
f
3
(
t
)]
=
(
f
1
f
2
)
f
3
=
f
1
f
2
f
3
(10.23)
∞
where
f
2
f
3
=
f
2
(
λ
1
)
f
3
(
t
−
λ
1
)
∂λ
1
, and
−∞
⎡
⎤
∞
∞
⎣
⎦
f
1
[
f
2
f
3
]
=
f
2
(
τ
)
∂τ
f
2
(
τ
1
)
f
3
(
t
−
τ
−
τ
2
)
∂λ
2
1
1
−∞
−∞
2
=
∂τ
f
1
(
τ
2
)
f
2
(
τ
1
)
f
3
(
t
−
τ
−
τ
2
)
∂τ
1
1
Second, “Convolution is Commutative,” which means that the order of the func-
tions being convolved is immaterial as shown in (10.24). The limits of integration will
be different for different orders of the convolved functions.
f
1
f
2
=
f
2
f
1
(10.24)
The final property is that “Convolution is Distributive,” which means that con-
volution of a function with the sum of two functions can be performed as shown in
(10.25).
f
1
(
f
2
+
f
3
)
=
f
1
f
2
+
f
1
f
3
(10.25)
An interesting property of convolution pertains convolving a function with theo-
retical delta functions, which results in the original function (see (10.26)). This property
is the same as sampling.
∞
δ
=
λ
δ
−
λ
∂λ
=
(10.26)
z
(
t
)
(
t
)
z
(
)
(
t
)
z
(
t
)
−∞
The convolution of a function with a unit step function is given by (10.27).
t
×
μ
=
λ
∂λ
(10.27)
z
(
t
)
(
t
)
z
(
)
−∞
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