The purpose of this section is to evaluate the performance of the proposed detector.
The number of samples, N, and the size Q of the vector 
have been
fixed respectively at 10000 and 30. Under the 
hypothesis, the noise is Gaussian and its spectral shape is given in
Fig. 4 (click here). All the numerical
processing, including the quantized autocorrelation, has been simulated numerically.
The quantization parameters are those given in Table 1 (click here). The
tests were made, first on synthetic data and, then, on actual data
acquired by the Nançay Decimetric Radio Telescope (NRT).
Figure 4: Spectrum of the noise under the hypothesis.
This typical shape has been measured from the NRT receiver
Figure 5: Spectra of the spread spectrum RFI. They are extracted from spectra similar
to those generated by Global Positioning System (GPS) or Global Navigation
Satellite System (GLONASS): a) spectrum of a spread
spectrum signal, b) spectrum of the tested narrow band filtered
spread spectrum RFI (NSP),
c) spectrum of the tested large band filtered spread spectrum RFI
(LSP)
Figure 6: 
against 
for the three types of RFI
retained: a sine wave (SW), a narrow band filtered spread spectrum RFI
(NSP), a large band filtered spread spectrum RFI (LSP).
N = 10000 samples are used for each one of the 800 trials. Q is the size of the
test vector
Figure 7: Experimental spectra resulting from the NRT
receiver. They are measured over 10 s on a 100 kHz band without (thin lines) and
with time-blanking (thick lines) processing. Detection time = 50 ms (equivalent to
N=10000 samples), 
, Q=30, 
: a)
12 sine waves RFI (SW), b) 2 large band spread spectrum RFI
(LSP)
Three typical RFI were chosen for the tests : a sine wave (SW), and two filtered
spread spectrum signals with two kinds of band limitation (see Fig. 5 (click here)).
The performance analysis is based on hypothesis testing. Two cases are possible,
whether the RFI is present (
hypothesis)
or not ( hypothesis). In order to obtain the needed
probabilities of detection and probabilities of false alarm
, it is necessary to generate many sample paths of the involved
data. For the results presented here, 800 sample paths for each
hypothesis were generated. Figure 6 (click here) shows the plots of
against (ROC curves
) for the three types of RFI retained.
The aim was to find the smallest INR (
) which yields at
least a superior to 
with a inferior to 
.
This objective is outperformed in the sine wave case, since
an INR of 
is reached. For the spread spectrum cases, the
performance decreases. The large band spread spectrum (LSP) case is still
detected well with a level of 
but the narrow band spread
spectrum (NSP) case does not reach the limit of 
. These
performance differences are related to the spectral appearance of the RFI
when they are observed through a resolution of 1/Q (see Sect.
5.2 (click here)). With Q=30, the NSP case appears spectrally as a white
noise, thereby diminishing the detection capabilities. By increasing the
resolution, the detection is improved (see Fig. 6 (click here) with Q=70).
To validate the results obtained with simulated data and verify the likelihood of the
hypothesis, the noise delivered by the receiver
was sampled at a rate of 200 kHz during 10 s. A dedicated RFI generator (see images available on the electronic version of the paper)
was used to emit the retained three types of RFI.
Firstly, the measured ROC curves were similar to those obtained with synthetic
data. This comparison has validated the hypothesis made on the noise
under the hypothesis. Secondly, a practical
test was made. The RFI generator was turned on and off manually at
random times and for random durations. The power of RFI was adjusted
to deliver an INR of .
The same algorithm (defined in Sect. 5 (click here)) was applied to the
stored data, the detection threshold being chosen to guarantee a
of . The detection window was fixed at 50 ms (N=10000
samples). When an RFI was detected, the corresponding data were discarded
from the final integration. The resulting spectra are shown in
Fig. 7 (click here). In both cases, the final spectrum shows significant
improvement.
In this section, the asymptotic performance is evaluated as a function of
INR and N. As shown in Sect. 5.2 (click here), the dependance on Q is
strongly linked with the RFI spectral appearance and is not included in this analysis.
Equation (6 (click here)) shows that 
is proportional to N.
When INR is low () the quantized correlator can be considered
as linear. Consequently, the term 
of Eq. (3 (click here))
is proportional to INR. Thus, the criterion 
is
proportional to 
. For example, if the detection time is
increased by a factor 100, a sine wave with an INR of 
can be
detected with a of and a of .
In the application presented here, only the second order statistics of
s(t) are tested. Nevertheless, it is also possible to exploit higher order
statistics of s(t) through the correlator. In this case, some channels of
the correlator must be devoted to the computation of these higher order
tests. For example, tests on the 
or 
order
statistics can be performed by feeding the correlator with s(t) and its
square version s2(t). Then, the non-linear functions are:
where 
and 
are parameters used to center and to normalize s2(t)
in relation to the quantization levels. Unfortunately, simulations for low
INR () have shown that performance is not improved
compared with the second order case given by Eq. (4 (click here)). In fact,
such modifications of increase its variance without increasing
the difference between itself and the
reference vector 
.
