Global Positioning System Reference
In-Depth Information
(
)
f
t
(
−
T
/
2
,
1
)
(
T
/
2
,
1
)
time
t
−
/
/
T
2
T
2
FIGURE 1.1. Rectangular pulse.
In discrete time the unit sample, also called a unit impulse sequence, is defined as
1
,
n
=
0,
δ(
n
)
=
0
,
n
=
0.
It follows that a continuous-time signal
x
(
t
)
may be represented as
∞
x
(
t
)
=
x
(τ )δ(
t
−
τ)
d
τ
for all
t
.
−∞
Similarly, a sequence
x
(
n
)
may be represented as
∞
x
(
n
)
=
x
(
k
)δ(
n
−
k
)
for all
n
.
(1.6)
k
=−∞
The Fourier transform of the unit impulse
δ(
t
)
is given by
∞
e
−
j
2
π
ft
dt
δ(
t
)
=
1
,
−∞
which gives us the following Fourier transform pair:
δ(
t
)
↔
1
.
The spectrum of the unit sample is obtained by
∞
e
−
j
2
π
fn
δ(
n
)
=
1
,
n
=−∞
which gives us the following Fourier transform pair:
δ(
n
)
↔
1
.
1.1.4 Rectangular Pulse
Let us now consider a single rectangular pulse
f
with amplitude 1 and pulse
width equal to
T
. In Figure 1.1 we have shifted the pulse
(
t
)
−
T
/
2toplaceitsym-
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