Tracking GPS Signals - Fundamentals of Global Positioning System Receivers: A Software Approach - page 172

the k value is far from the input signal, the amplitude of X ( k ) obtained from
Equation (8.40) will be small, which implies that the sensitivity of the process-
ing is low. In order to avoid this problem, the k value should be kept as close to
the input frequency as possible. A k value close to the input frequency can also
reduce the frequency ambiguity. Under this condition, the k value is usually no
longer an integer.
When the k value is an integer, the initial phase of the Kernel function
e − j 2 πnk/N is zero and the values obtained from two consecutive sets are con-
tinuous. The beginning point of the first set is n = 0 and the beginning point of

the second set is n = N . It is easily shown that

e − j 2 πnk/N | n = 0 = e − j 2 πnk/N | n = N if k = integer ( 8 . 43 )

If k is not an integer this relation no longer holds. The following example is used
to illustrate this point. Assume that N = 0 ∼ 255. For any integer

value of k , 256 data points can be generated from e − j 2 πnk/N

= 256, and n

= 0 ∼ 255.
Two sets of the same 256 data points are placed in cascade to generate a total
of 512 data points. There is no discontinuity from data point 256, the last data
point of the first set, to data point 257, the beginning of the second set. Since
the values generated from e − j 2 πnk/N are complex, the continuity can be shown
graphically only in real and imaginary parts of e − j 2 πnk/N . Figure 8.6a shows the
real and imaginary results of k

for n

= 20. In this figure only the points from 240 to
270 are plotted and there is no discontinuity. Figure 8.6b shows the results of
k

= 20 . 5 and there is a discontinuity between point 256 and 257 in both the real
and imaginary portions of e − j 2 πnk/N . The discontinuity will affect the application
of Equation (8.42).
Figure 8.7 shows a cw input signal and two sections of the real part of
e − j 2 πnk/N .If e − j 2 πnk/N is continuous, the two sets of DFT can be considered
as the correlation of the input signal with one complex cw signal. Under this
condition, Equations (8.41) and (8.42) can be used to find the fine frequency.
If the kernel function has a discontinuity, the two sets of DFT are the input
signal correlated with two sets of kernel functions. Under this condition there is
a phase discontinuity in the phase relation. In order to use Equation (8.42), the
phase discontinuity must be taken into consideration.
This discontinuity can be found by calculating the phase angle at n

=

N .In

= 0 ∼ 255 are used to generate the values
of the kernel function. In order to generate a continuous kernel function, the
value of e − j 2 πnk/N

= 256 and n

the previous example, N

256) must equal e − 0 (or zero degree). If k is
an integer, this relation is true. If k is not an integer, this relation does not hold
and the phase difference between e − j 2 πk and e − 0 is the phase discontinuity. This
phase must be subtracted from the phase angle before Equation (8.42) can be
properly used.
If the difference phase from the Kernel function is subtracted at the end of
each millisecond, two situations can occur between the two adjacent milliseconds.
One is that there is no phase change and the other one is that there is a π phase

=

e − j 2 πk

=

( n