Global Positioning System Reference
In-Depth Information
FIGURE 8.5
Phase angle from two consecutive data sets.
where
θ
presents the initial phase of the sine wave with respect to the Kernel func-
tion. If
k
is an integer, the initial phase of the Kernel function is zero. In general,
if the frequency of the input signal is an unknown quantity, all the components
of
k(k
=
0
∼
N
−
1
)
must be calculated. However, only half of the
k
values
(
k
N/
2
−
1) provide useful information as discussed in Section 6.13. The
highest component
X(k
i
)
can be found by comparing all the
X
(
k
) values. For this
operation, the fast Fourier transform (FFT) is often used to save calculation time.
If the frequency of the input signal can be found within a frequency resolution
cell, which is equal to 1
/Nt
s
(where
t
s
is the sampling time), the desired
X
(
k
)
can be found from one component of the DFT. It should be noted that to calculate
one component of
X
(
k
), the
k
value need not be an integer as in the case of FFT.
Since the input frequency is estimated from the acquisition method, the
X
(
k
) can
be found from one
k
value of Equation (8.40). The purpose of this operation is
to find the fine frequency of the input signal.
The phase angle
θ
can be used to find the fine frequency of the input signal as
discussed in Section 7.13. Figure 8.5 shows that the data points are divided into
two different time domains. In each time domain, the same
X(k
i
)
are calculated.
The corresponding phase angles are
θ
n
and
θ
n
+
m
and they are separated by time
m
. The fine frequency can be obtained as
=
0
∼
δθ
m
≡
θ
n
+
m
−
θ
n
f
=
(
8
.
42
)
m
This relation can provide much finer frequency resolution than the DFT result.
The frequency resolution depends on the angle resolution measured.
(
9
)
The dif-
ference angle
δθ
must be less than 2
π/
5 as discussed in Section 7.14 and this
requirement limits the time between the two consecutive DFT calculations. Since
the frequency
k
is very close to the input frequency, which changes slowly with
time, the unambiguous frequency range is not a problem. This approach is used
to find the correct frequency and update it accordingly.
8.9 DISCONTINUITY IN KERNEL FUNCTION
In conventional DFT operation the
k
value in Equation 8.40 is an integer. How-
ever, in applying Equation (8.40) to the tracking program the
k
value is usually
a noninteger, because in using an integer value of
k
, the frequency generated
from the kernel function
e
−
j
2
πnk/N
can be too far from the input frequency. If
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