Geology Reference
In-Depth Information
Now consider a linear, continuous system with impulse response
h
(
t
). By defin-
ition of the impulse response, if the input is a Dirac delta function, δ(
t
), then the
output of the system is
h
(
t
). The Fourier transform of the Dirac delta function, δ(
t
),
is unity (2.260). Thus, for this input, the Fourier transform of the output from the
linear system is
H
(
f
)
·
1, where
H
(
f
) is called the
system function
. Because the
system is linear, at a given frequency its output is proportional to the input at that
frequency. If the input is g(
t
) with Fourier transform
G
(
f
), the Fourier transform of
the output is
H
(
f
)
·
G
(
f
),
(2.307)
the product of the two Fourier transforms. Therefore, by (2.269), the output
o
(
t
)is
the convolution
∞
o
(
t
)
=
h
(λ)g(
t
−
λ)
d
λ.
(2.308)
−∞
The autocorrelation of the output at lag τ is then
T
/2
∞
1
T
φ
oo
(τ)
=
lim
T
→∞
h
(λ)g(
t
−
λ)
d
λ
−
T
/2
−∞
∞
h
∗
(λ
)g
∗
(
t
−
τ
−
λ
)
d
λ
dt
·
−∞
∞
∞
h
(λ)
h
∗
(λ
)φ
gg
(τ
+
λ
−
λ)
d
λ
d
λ
.
=
(2.309)
−∞
−∞
If the Fourier transform of the autocorrelation, φ
gg
(τ), considered as a lag domain
function, is
S
gg
(
f
), its Fourier integral representation is
∞
S
gg
(
f
)
e
i
2π
f
τ
df
.
φ
gg
(τ)
=
(2.310)
−∞
Thus,
∞
∞
∞
h
(λ)
h
∗
(λ
)
S
gg
(
f
)
e
i
2π
f
(τ
+
λ
−
λ)
dfd
λ
d
λ
.
φ
oo
(τ)
=
(2.311)
−∞
−∞
−∞
The Fourier transform of the impulse response is
∞
h
(
t
)
e
−
i
2π
ft
dt
,
H
(
f
)
=
(2.312)
−∞
and its conjugate is
∞
H
∗
(
f
)
h
∗
(
t
)
e
i
2π
ft
dt
.
=
(2.313)
−∞
Search WWH ::
Custom Search