Acquisition of GPS C/A Code Signals - Fundamentals of Global Positioning System Receivers: A Software Approach - page 146

6. There are overall 1 . 25 × 10 6 (250 × 5 , 000) outputs in the frequency
domain. The highest amplitude that crosses a certain threshold will be
the desired value. From this value the beginning of the C/A code and
the Doppler frequency can be obtained. The frequency resolution obtained
is 100 Hz.

Although the straightforward approach is presented above, circular correlation
can be used to achieve the same purpose with fewer operations.

7.13 BASIC CONCEPT OF FINE FREQUENCY ESTIMATION ( 7 )

The frequency resolution obtained from the 1 ms of data is about 1 kHz, which
is too coarse for the tracking loop. The desired frequency resolution should be
within a few tens of Hertz. Usually, the tracking loop has a width of only a
few Hertz. Using the DFT (or FFT) to find fine frequency is not an appropriate
approach, because in order to find 10 Hz resolution, a data record of 100 ms is
required. If there are 5,000 data points/ms, 100 ms contains 500,000 data points,
which is very time consuming for FFT operation. Besides, the probability of
having phase shift in 100 ms of data is relatively high.
The approach to find the fine frequency resolution is through phase relation.
Once the C/A code is stripped from the input signal, the input becomes a cw
signal. If the highest frequency component in 1 ms of data at time m is X m ( k ), k
represents the frequency component of the input signal. The initial phase θ m ( k )
of the input can be found from the DFT outputs as

tan − 1 Im (X m (k))
Re (X m (k))

θ m (k) = ( 7 . 20 )

where Im and Re represent the imaginary and real parts, respectively. Let us
assume that at time n , a short time after m , the DFT component X n (k) of 1 ms
of data is also the strongest component, because the input frequency will not
change that rapidly during a short time. The initial phase angle of the input
signal at time n and frequency component k is
θ n (k) = tan − 1 Im (X n (k))
Re (X n (k))

( 7 . 21 )

These two phase angles can be used to find the fine frequency as

θ n (k) − θ m (k)

=

f ( 7 . 22 )

2 π(n − m)

This equation provides a much finer frequency resolution than the result obtained
from DFT. In order to keep the frequency unambiguous, the phase difference
θ n − θ m must be less than 2 π . If the phase difference is at the maximum value