Signal Processing - September 2017 - 56

is squared and added to form the decision statistic D s given
below, followed by M (optional) noncoherent combinations.
D s = ; Pk /2 R ^ ∆xh cos ^ 2r∆{h
#

2

sin ^r∆ fD T h sin ^rfD, k Tch
$
+ n IG
r∆ f D T
rfD, k Tc

+ ; Pk /2 R ^ ∆xh sin ^ 2r∆{h
#

2

sin ^r∆ fD T h sin ^rfD, k Tch
$
+ n QG .
r∆ f D T
rfD, k Tc

(6)

The decision statistic is compared with a predetermined
threshold v to declare the absence (H 0 hypothesis) or presence (H 1 hypothesis) of the L2C signal.

Satellite signal experiments
The low-rate Doppler search approach was tested with real GPS
L2C signals from a (PRN-6) block IIR-F satellite recorded with
a radio telescope, providing an IF of 32.42 MHz and sampling

Correlator Output

600
400
200
0
-200
-400
-600
0

5

10
Time (ms)
I Channel

15

20

Q Channel

FIGURE 4. A 100-Hz Doppler with frequency and phase information preserved by Tc = 0.1 ms averaging in a real GPS L2C signal.

rate of 128 Msps. After the IF carrier mixing, the L2C signal was
multiplied with RZ CM local replica code and then averaged by
Tc = 0.1 ms accumulations. Figure 4 presents the averaged output showing a 100-Hz Doppler carried over from the previous
processing stage with phase information preserved by the I
and Q components. The sampling rate of this Doppler signal
is dropped to 1 ksps (1/0.1 ms) to conduct the Doppler search
at 1 ksps processing rate.
Real L2C signals with a range of carrier Doppler ^ fD, k h
were then applied to the low-rate Doppler search system. The
signals were correlated with synchronized local PRN code
and averaged with 1 ms, 0.1 ms, and 0.01 ms intervals while
keeping the relative Doppler offset ^∆fDh fixed as 2/12T
(average expected offset for the typical Doppler search resolution of 2/3T Hz [23]).
Figure 5 shows the outcome of this test, measured as loss in
the received signal power relative to what is detected through
standard full-rate coherent processing of 20 ms of signal, for
the same ∆fD . The results show the impact of averaging interval Tc on power loss along the carrier Doppler fD, k .
These results are in agreement with the theoretical expression for the decision statistics presented in (6) with nulls of
the sinc-shaped envelope occurring at integer multiples of
1/Tc Hz in each case. A close-up of these results over the
standard 5-kHz GPS Doppler search range together with the
corresponding theoretical curves is shown in Figure 6. The
figure confirms that the relative power loss in the GPS Doppler search is inversely proportional to the averaging interval,
with the 0.01 ms interval detecting the end-of-range Doppler
without loss. A shorter averaging interval, however, implies a
relatively higher computational cost.

Probability of detection
For the H 1 hypothesis, i.e., L2C signal present with white Gaussian noise, the ratio of decision statistic to noise power, i.e.,
D s /v n2 , is a noncentral | 2 (chi-square) distribution with 2M
degrees of freedom. The expected value of the noncentrality
parameter with RZ CM replica code is

Relative Power Loss (dB)

Relative Power Loss (dB)

0
-10
-20
-30
-40
-50
20

40
60
80
100
Carrier Doppler (kHz)

Tc = 1 ms

Tc = 0.1 ms

120

0
-5
-10
-15
-20
-25
-30
0

140

1,000

2,000
3,000
4,000
Carrier Doppler (Hz)

Theory
Tc = 0.1 ms

Tc = 0.01 ms

5,000

Tc = 1 ms
Tc = 0.01 ms

FIGURE 5. Measured relative power loss as a function of carrier Doppler in

FIGURE 6. Measured and theoretical relative signal power loss over the

real GPS L2C signals for different averaging periods.

standard GPS Doppler search range for different averaging periods.

56

IEEE SIGNAL PROCESSING MAGAZINE

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September 2017

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Table of Contents for the Digital Edition of Signal Processing - September 2017
