Geology Reference
In-Depth Information
G
(
f
+2
f
N
)
G
(
f
)
G
(
f
-2
f
N
)
∞
∞
-3
f
N
-2
f
N
-
f
N
0
f
N
2
f
N
3
f
N
f
Figure 2.14 If the function
G
(
f
) is band limited, and vanishes outside the Nyquist
band
−
f
N
<
f
<
f
N
, then discrete sampling only produces a periodic repetition of
the true transform up and down the frequency axis, without any other distortion.
frequency encountered in practice can be made band limited. For a band limited
function of frequency, the Fourier transform calculated from the sampled record is
simply
∞
H
(
f
)
=
G
(
f
−
2
nf
N
).
(2.301)
n
=−∞
This represents a periodic repetition of the true transform,
G
(
f
), along the fre-
quency axis, as shown in Figure 2.14.
If
G
(
f
) is not band limited, and extends beyond the range of frequencies
−
f
N
<
f
<
f
N
,then
H
(
f
)
=···+
G
(
f
+
2
f
N
)
+
G
(
f
)
+
G
(
f
−
2
f
N
)
+···
.
(2.302)
The transform in the interval
f
N
≤
f
≤
3
f
N
is
folded back
onto the interval
−
f
N
≤
f
≤
f
N
, and the transform in the interval
−
3
f
N
≤
f
≤−
f
N
is folded back onto
the interval
f
N
, and so on. The folding of the frequency axis is illus-
trated in Figure 2.15. The contents of portions of the frequency axis, multiples of
2
f
N
away, are
aliased
back onto the principal Nyquist band
−
f
N
≤
f
≤
f
N
. Thus,
if significant portions of
G
(
f
) lie outside the principal Nyquist band,
H
(
f
) will be
seriously distorted by
aliasing
. To prevent distortion by aliasing, it is necessary
to choose
−
f
N
≤
f
≤
t
is larger than the highest signi-
ficant frequency, in both the content of the desired signal and the content of the
ever present noise. If
T
N
Δ
t
su
ciently small, so that
f
N
=
1/2
Δ
=
1/
f
N
is the period of the highest frequency component
present, then
t
must be chosen to be no greater than
T
N
/2, or we must sample at
least
twice per cycle of the highest frequency
present, in either signal or noise, to
prevent distortion by aliasing.
Δ
Search WWH ::
Custom Search