Geology Reference
In-Depth Information
The digitiser operates by integrating the signal over a small but finite length of
time. The sampling process can be modelled by taking the product of the modified
boxcar, shown on the left of Figure 2.7, with the continuous signal g(
t
). We take
the mean value of g(
t
) over the interval [
−
T
/2,
T
/2] about the point to be sampled,
→
andthenlet
T
0. The modified boxcar, in this limit, becomes the Dirac delta
function. Taking 2
N
+
1 samples is equivalent to forming the product of g(
t
) with
the
finite Dirac comb
, to obtain the sequence of samples,
h
(
t
). Thus,
j
=−
N
δ(
t
N
h
(
t
)
=
g(
t
)
·
−
j
Δ
t
),
(2.286)
where
t
is the uniform sample interval.
To see the e
Δ
ect of discrete sampling in the frequency domain, we take the Four-
ier transform of the finite Dirac comb. It is
∞
ff
j
=−
N
δ(
t
N
N
t
)
e
−
i
2π
ft
dt
e
−
i
2π
fj
Δ
t
−
j
Δ
=
.
(2.287)
−∞
j
=−
N
The right-hand side is the sum of a finite geometric progression, with first term
e
i
2π
fN
Δ
t
a
=
,
(2.288)
and common ratio
e
−
i
2π
f
Δ
t
r
=
.
(2.289)
The sum is then
a
1
r
2
N
+
1
N
−
e
−
i
2π
fj
Δ
t
=
1
−
r
j
=−
N
e
−
i
2π
f
(2
N
+
1)
Δ
t
−
e
i
2π
fN
Δ
t
1
=
e
−
i
2π
f
Δ
t
1
−
e
i
π
f
(2
N
+
1)
Δ
t
e
−
i
π
f
(2
N
+
1)
Δ
t
−
=
e
i
π
f
Δ
t
e
−
i
π
f
Δ
t
−
sin[π
f
(2
N
+
1)
Δ
t
]
=
sin(π
f
Δ
t
)
sin(π
fT
)
sin(π
f
Δ
t
)
,
=
(2.290)
Search WWH ::
Custom Search