Global Positioning System Reference
In-Depth Information
1.1.2 Discrete-Time Deterministic Signals
Let us suppose that
x
is a real- or complex-valued deterministic sequence,
where
n
takes integer values, and which is obtained by uniformly sampling the
continuous-time signal
x
(
n
)
(
)
(
)
E
=
t
; read Section 1.2. If
x
n
has finite energy
n
=−∞
|
2
(
)
|
<
∞
, then it has the frequency domain representation (discrete-
time Fourier transform)
x
n
∞
e
−
j
ω
n
X
(ω)
=
x
(
n
)
,
n
=−∞
or equivalently
∞
e
−
j
2
π
fn
X
(
f
)
=
x
(
n
)
.
n
=−∞
It should be noted that
X
(
f
)
is periodic with a period of one and
X
(ω)
is periodic
with a period of 2
.
The inverse discrete-time Fourier transform that yields the deterministic se-
quence
x
π
(
n
)
from
X
(ω)
or
X
(
f
)
is given by
1
/
2
π
1
2
e
j
2
π
fn
df
e
j
ω
n
d
x
(
n
)
=
X
(
f
)
=
X
(ω)
ω.
π
−
1
/
2
−
π
Notice that the integration limits are related to the periodicity of the spectra.
We refer to
2
as the energy density spectrum of
x
|
X
(
f
)
|
(
n
)
and denote it as
)
=
X
)
2
E
x
(
f
(
f
.
The energy density spectrum
E
x
(
f
)
of a deterministic discrete-time signal
x
(
n
)
can also be found by means of the
autocorrelation sequence
∞
x
∗
(
r
x
(
k
)
=
n
)
x
(
n
+
k
)
n
=−∞
via the discrete-time Fourier transform
∞
e
−
j
2
π
fk
E
x
(
f
)
=
r
x
(
k
)
.
k
=−∞
That is, for a discrete-time signal, the Fourier transform pair is
r
x
(
k
)
↔
E
x
(
f
).
1.1.3 Unit Impulse
In signal analysis a frequently used deterministic signal is the unit impulse. In
continuous time the unit impulse
δ(
t
)
, also called the delta function, may be de-
fined by the following relation:
∞
δ(
t
)
x
(
t
)
dt
=
x
(
0
),
−∞
0. Its area is
−∞
δ(
where
x
(
t
)
is an arbitrary signal continuous at
t
=
t
)
dt
=
1.
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