Global Positioning System Reference
In-Depth Information
through the Hilbert transform.
(
10
)
A detailed discussion on the Hilbert transform
will not be included. Only the procedure will be presented here.
First the Hilbert transform from Matlab will be presented. The approach is
through discrete Fourier transform (DFT) or fast Fourier transform (FFT). The
following steps are taken:
1. The DFT result can be written as:
N
−
1
x(n)e
−
j
2
πnk
X(k)
=
(
6
.
8
)
N
n
=
0
where
x
(
n
) is the input data,
X
(
k
) is the output frequency components,
k
−
1. Since the input data
are real, the frequency components have the following properties:
=
0
,
1
,
2
,...
,
N
−
1, and
n
=
0
,
1
,
2
,...
,
N
N
2
−
1
k)
∗
X(k)
=
X(N
−
for
k
=
1
∼
(
6
.
9
)
where
∗
represents complex conjugate. If the input data are complex the
relationship in this equation does not exist.
2. Find a new set of frequency components
X
1
(k)
. They have the follow-
ing values:
1
2
X(
0
)
X
1
(
0
)
=
N
2
−
X
1
(k)
=
X(k)
for
k
=
1
∼
1
X
1
N
2
2
X
N
1
=
2
N
2
+
1
∼
N
−
1
X
1
(k)
=
0f r
k
=
(6.10)
1.
3. The new data
x
1
(n)
in time domain can be obtained from the inverse DFT
of the
X
1
(k)
as:
These new frequency components also have
N
values from
k
=
0to
N
−
N
−
1
1
N
j
2
πnk
N
x
1
(n)
=
X
1
(k)e
(
6
.
11
)
k
=
0
From this approach, if there are
N
points of real input data, the result will be
N
points of complex data. Obviously, additional information is generated
through this operation. This is caused by padding the
X
1
(k)
values with
zeros as shown in Equation (6.10). Padding with zeros in the frequency
domain is equivalent to interpolating in the time domain.
(
10
)
Search WWH ::
Custom Search