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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