Global Positioning System Reference
In-Depth Information
(
)
X
f
f
−
B
0
B
x
δ
(
f
)
...
...
−
B
B
s
−
B
f
s
+
B
f
−
−
f
s
f
s
2
f
s
0
2
f
s
FIGURE 1.8. Sampling operation shown in the frequency domain. Top: Signal
X
(
f
)
with
bandwidth
B
. Bottom:
x
δ
(
f
)
when
f
s
>
2
B
.
Causal and noncausal systems
A system is said to be causal if its response does
not begin before the input is applied, or in other words, the value of the out-
put at
t
=
t
0
depends only on the values of
x
(
t
)
for
t
≤
t
0
. In mathematical
terms, we have
y
f
x
)
for
t
0
<
∞
.
Noncausal systems do not satisfy the condition given above. Moreover, they
do not exist in a real world but can be approximated by the use of time delay.
The classification of continuous-time systems easily carries over to discrete-
time systems. Here the input and output signals are sequences, and the system
maps the input sequence
x
(
t
0
)
=
(
t
t
≤
t
0
and
−∞
<
t
,
.
A simple example of a discrete-time linear system is a system that is a linear
combination of the present and two past inputs. Such a system can in general be
described by
(
n
)
into the output sequence
y
(
n
)
y
(
n
)
=
x
(
n
)
+
a
1
x
(
n
−
1
)
+
a
2
x
(
n
−
2
)
(1.18)
and is illustrated in Figure 1.10.
Input
System
Output
f
x
(
t
)
y
(
t
)
FIGURE 1.9. Block diagram representation of a system.
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