Global Positioning System Reference
In-Depth Information
1.1.1 Continuous-Time Deterministic Signals
Let us consider a deterministic
continuous-time signal x
(
t
)
,real-orcomplex-
E
=
−∞
|
valued with finite energy defined as
denotes the
absolute value, or magnitude, of the complex quantity. In the frequency domain
this signal is represented by its
Fourier transform
:
x
(
t
)
|
dt
. The symbol
|·|
∞
e
−
j
ω
t
dt
X
(ω)
=
x
(
t
)
,
(1.1)
−∞
√
−
where
j
=
1 and the variable
ω
denotes angular frequency. By definition
ω
=
and
f
are radian and cycle, respectively. In general,
the Fourier transform is complex:
2
π
f
and the units for
ω
X
e
j
arg
(
X
(ω))
.
(ω)
=
X
(ω)
+
X
(ω)
=
X
j
(ω)
(1.2)
The quantity
X
(ω)
is often referred to as the
spectrum
of the signal
x
(
t
)
because
the Fourier transform measures the frequency content, or spectrum, of
x
(
t
)
.Sim-
,andtoarg
X
(ω)
=
ilarly, we refer to
|
X
(ω)
|
as the
magnitude spectrum
of
x
(
t
)
arctan
(
(ω))
as the
phase spectrum
of
x
X
(ω))/
(
X
(
t
)
. Moreover, we refer to
2
as the
energy density spectrum
of
x
|
(ω)
|
(
)
because it represents the distribu-
tion of signal energy as a function of frequency. It is denoted
X
t
2
.
E
x
(ω)
=|
(ω)
|
X
The
inverse Fourier transform x
(
t
)
of
X
(ω)
is
∞
1
2
e
j
ω
t
dt
x
(
t
)
=
X
(ω)
.
(1.3)
π
−∞
We s a y t h a t
x
(
t
)
and
X
(ω)
constitute a
Fourier transform pair
:
x
(
t
)
↔
X
(ω).
The energy density spectrum
E
x
(ω)
of a deterministic continuous-time signal
x
can also be found by means of the (time-average)
autocorrelation function
(ACF) of the finite energy signal
x
(
t
)
.Let
∗
denote complex conjugation, and then
(
t
)
the ACF of
x
(
t
)
is defined as
∞
x
∗
(
r
x
(τ )
=
t
)
x
(
t
+
τ)
dt
,
(1.4)
−∞
E
x
(ω)
(
)
and the energy density spectrum
of
x
t
is defined as
∞
e
−
j
ωτ
d
E
x
(ω)
=
r
x
(τ )
τ.
(1.5)
−∞
Again, we say that
r
x
(τ )
and
E
x
(ω)
constitute a Fourier transform pair:
r
x
(τ )
↔
E
x
(ω).
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