GNSS Interference Detection
Using Statistical Analysis in the
Time-Frequency Domain
on the block-wise STFT using nonoverlapped samples. The canonical
STFT-based method shows better detection capability at the expense
of degraded false alarm performance caused by the PDF distortion
in the canonical STFT samples. The block-wise STFT-based method
alleviates the false alarm issue but slightly weakens the detection capability. Simulations show that the proposed canonical and block-wise
STFT-based methods improve the detection performance for both
narrow- and wideband interference in low jammer-to-noise ratio environments when compared with the existing GoF test applied to the
time-domain samples.
I. INTRODUCTION
PAI WANG
Beijing Institute of Technology, Beijing, China
EDIZ CETIN , Member, IEEE
Macquarie University, Sydney, Australia
ANDREW G. DEMPSTER , Senior Member, IEEE
University of New South Wales, Sydney Australia
YONGQING WANG , Member, IEEE
Beijing Institute of Technology, Beijing, China
SILIANG WU
Beijing Institute of Technology, Beijing, China
This paper presents a precorrelation interference detection
method based on statistical analysis in the time-frequency (TF) domain for global navigation satellite system signals. In particular, the
short-time Fourier transform (STFT) is considered as the TF tool due
to its linear property and low computational complexity. A goodnessof-fit (GoF) test is applied to each frequency slice in the spectrogram of
the received signal, which approximately follows a chi-square distribution in the absence of interference. The expected probability density
function (PDF) of the observed TF-domain samples can be computed
based on an interference-free signal or the noise power estimate. Two
versions of the proposed technique are presented: one based on the
canonical STFT with the maximum overlap size, and the other based
Manuscript received September 22, 2016; revised April 5, 2017; released
for publication September 12, 2017. Date of publication October 6, 2017;
date of current version February 7, 2018.
DOI. No. 10.1109/TAES.2017.2760658
Refereeing of this contribution was handled by J. T. Curran.
The work of P. Wang was supported by the China Scholarship Council.
This work was supported in part by the Australian Research Council
Linkage Grant LP140100252.
Authors’ addresses: P. Wang, Y. Wang, and S. Wu is with the School
of Information and Electronics, Beijing Institute of Technology, Beijing
100081, China, E-mail: (wendywang.bit@gmail.com; wangyongqing@
bit.edu.cn; siliangw@bit.edu.cn); E. Cetin was with the Australian Centre for Space Engineering Research, School of Electrical Engineering
and Telecommunications, University of New South Wales, Sydney, NSW
2052, Australia. He is now with the School of Engineering, Macquarie
University, North Ryde, NSW 2109, Australia, E-mail: (ediz.cetin@
mq.edu.au); A. G. Dempster is with the Australian Centre for Space Engineering Research, School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney, NSW 2052, Australia,
E-mail (a.dempster@unsw.edu.au). (Corresponding author: Ediz Cetin.)
C 2017 IEEE
0018-9251 416
Reliable navigation and timing have become important
requirements in a growing number of global navigation
satellite system (GNSS) based applications [1]. Direct sequence spread spectrum modulation inherently provides
GNSS signals with some degree of interference resistance.
However, the low received power level makes the GNSS signals vulnerable to either unintentional or intentional (jamming) radio frequency interference (RFI) [2], [3]. The RFI
may cause different degrees of impact on GNSS receivers,
such as degraded signal quality, severe jitter or large bias
in the navigation and timing results, or even complete loss
of lock in the tracking unit [4]. Thus, there is an everincreasing need for methods dealing with the RFI issue in
GNSS receivers.
Based on where the method is applied in the receiver
processing chain, there are several categories of interference
detection methods. Antenna-level techniques using antenna
arrays can provide promising detection performance for
various interference types; however, they suffer from the
major disadvantage of increased hardware complexity [5],
[6]. Automatic gain control (AGC) based methods monitor
the AGC level for interference detection. The performance
of these approaches degrades significantly in weak interference environments [7], [8]. Detection of weak interference,
before it can affect the operation of GNSS receivers, is
very important when trying to geo-locate it [9]–[12]. Postcorrelation techniques generally evaluate the observables
provided during signal acquisition, tracking, and navigation
processing, and thus, could reflect the specific effects of the
interference on receiver performance [13]–[15]. However,
these methods have a long processing delay to detect the
presence of interference.
Precorrelation methods based on the raw received signals have also been commonly used for interference detection. The specific properties of various interference
types were highlighted in different domains, such as time[16], [17], frequency- [18], transform- [19], or statistical domain [20], [21] to distinguish the interfered and
interference-free cases. Due to its spectral properties, the
frequency-domain transform was generally applied for
the detection of narrow-band carrier-wave (CW) interference [22], [23]. To deal with various interference types,
the Karhunen–Loève transform was considered to provide
the transform-domain representation of the received signal for interference detection [24]. Emphasis could also
be placed on the statistical characteristics of the received
signals. A number of Gaussianity tests concerning the
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distribution curve and moments of the received signals were
developed to show good detection performance for a large
class of interferences [20], [21], [25].
In real-life situations, nonstationary interference with
varying characteristics in both the time and frequency domains is often encountered. Since nonstationary interference is usually concentrated in a small region within the
time-frequency (TF) plane, TF-domain analysis has been
widely applied in GNSS interference detection [26]–[28].
The general procedure is to compare the peak magnitude
in the TF distribution (TFD) of the received signal with a
predefined threshold. The setting of the detection threshold
significantly impacts detection performance, including detection and false alarm probabilities. For linear TFDs, such
as the short-time Fourier transform (STFT), the threshold
for peak detection can be easily obtained using probability density function (PDF) analysis of the TF points [26].
However, the PDF of the TF points provided by quadratic
TFDs becomes much more complex. Thus, an experimental
threshold is usually used, which is sensitive to variations
in the channel noise characteristics and kernel parameters
used in the TFDs [27], [29]. Also, the TF peak detection
method cannot provide the same good performance for both
narrow- and wideband interference.
To safeguard the integrity of systems that rely on GNSS
to function, interference sources must be detected before
they disrupt the operation of the GNSS equipment and this
information conveyed to the end-users or trigger further
downstream processing. In the context of interference geolocation systems [12], the detection unit can activate the
geo-localization processing before the interference can affect the GNSS receivers resulting in increased coverage
area. The major motivation for our contribution is to apply
statistical analysis in the TF domain with a view to improve
the detection performance of weak interference. Since the
desired signal power is often much weaker than the receiver
noise power in a normal GNSS environment, the spectrogram (modulus square of STFT) points of an interferencefree signal approximately follow a chi-square distribution,
which is also used to determine the peak detection threshold. The proposed method applies the goodness-of-fit (GoF)
test used in [21] to each frequency slice in the spectrogram
of the received signal. The STFT implies the accumulation of several adjacent received samples, and thus, leads
to a jammer-to-noise power ratio (JNR) improvement for
the frequency slice within the interference bandwidth. The
quantitative JNR improvement depends on the interference
characteristics and window parameters used in the STFT.
The false alarm performance of the canonical STFT-based
method is degraded due to the PDF distortion. To eliminate
this undesired property, a solution based on the block-wise
STFT is also proposed at the expense of decreased detection capability. Simulations concerning both narrow- and
wideband interference have been conducted to compare the
detection performance of the proposed approach against
the aforementioned GoF test applied to the time-domain
samples and the TF peak detection method in low-JNR
environments.
The rest of this paper is organized as follows.
Section II describes the signal model and interference detection method in the statistical domain. In Section III, the
detection method based on statistical analysis in the TF
domain is proposed. The methods for computing the expected PDF are also illustrated in this section. In Section IV,
the effect of the PDF distortion in the canonical STFT on
the false alarm performance is analyzed, and a block-wise
STFT-based method is introduced to eliminate this undesired effect. Simulation results are presented in Section V to
compare the detection performance of our proposed method
and the GoF test applied to the time-domain samples, while
concluding remarks are given in Section VI.
II. SIGNAL MODEL AND INTERFERENCE DETECTION
IN THE STATISTICAL DOMAIN
A. Signal Model
For a single-antenna receiver, the received RF GNSS
signal contaminated by Gaussian noise and interference
can be modeled as
r(t) =
P
sp (t) + w(t) + u(t)
(1)
p=1
where P is the number of in-view satellites. The pth useful
GNSS signal has the following form:
sp (t) = 2Cp cp (t − τp )dp (t − τp )
· cos[2π(fc + fd,p )t + φp ]
(2)
where Cp is the pth GNSS signal power; cp (t) and dp (t) are
the spreading code and navigation data of the pth satellite,
respectively; τp is the code delay introduced by the propagation channel; fc is the carrier frequency; fd,p and φp are
the carrier Doppler frequency and phase, respectively.
In a receiver front-end, the RF signal is generally downconverted to in-phase and quadrature components by mixing the incoming signal with two local oscillators, which
are 90◦ offset in phase. The received signal is then sampled
and digitized by the analog-to-digital converter and can be
expressed as
r[i] =
P
2Cp cp,i dp,i ej (2πfd,p iTs +φp )
p=1
+ w[i] + u[i]
(3)
where Ts is the sampling interval, and w[i] is a zero-mean
complex Gaussian noise whose real and imaginary parts are
independent variables with same variance σ 2 /2.
Based on the bandwidth of the interference with respect to the GNSS signal, interference from various
undesired sources can be classified into either narrow- or
wideband. Narrow-band interference generally originates
from the harmonics transmitted by other radio or telecommunication systems [2], [30]. Wideband interference causes
the most disruption to the GNSS receivers [31]. A unified
model for the interference u[i], which covers various interference sources such as CW and chirp-type signals, can be
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417
hypothesis, the observed and expected cell numbers are
sufficiently close to each other. The alternative hypothesis
indicates that the observed cell numbers deviate substantially from the expected ones. A test statistic concerning
GoF is formed as a function of the squares of the deviations of the observed numbers from their expected values,
weighted by the reciprocals of their expected values, which
can be mathematically given by
T =
V
(nv − nv,e )2
v=1
Fig. 1. Instantaneous frequency law of a typical chirp-type jammer.
expressed as
u[i] = Au [i]ej 2π Fu [i]i
(4)
where the instantaneous amplitude Au [i] is generally a
slowly varying function of time and is treated as a constant Au in the subsequent analysis. Thus, the JNR of the
received samples is A2u /σ 2 . Fu [i] denotes the frequency
characteristics of the interference. Fig. 1 shows the unidirectional sweep instantaneous frequency law of a typical
chirp-type jammer, where Bw and TSW denote the sweep
bandwidth and period, respectively [20], [31].
B. Interference Detection in the Statistical Domain
The interference detection problem can be cast as hypothesis testing with the null and alternative hypotheses
formulated as
H0 :
Ha :
r[i] = w[i]
r[i] = w[i] + u[i]
(5)
where the GNSS signals are omitted as they are entirely
buried below the noise floor in the precorrelation stage.
Statistical properties of the received signal have been
explored to detect the presence of interference [20], [21].
The interference detection method in the statistical domain is generally applied to a baseband sample vector
R = {r[i], r[i + 1], . . . , r[i + N − 1]}, where N is the total number of samples. From the perspective of probability
distribution, each sample of R under the null hypothesis
is considered as a complex zero-mean Gaussian variable
with independent real and imaginary parts. However, the
Gaussian distribution would be distorted if interference is
present. Thus, the PDF analysis of the received signal handled by Person’s chi-squared GoF test was proposed for the
interference detection problem in [21].
The GoF test is generally used to indicate whether a specific model for a population distribution fits the observed
data. Supposing that the values of the observed data R
have V distinct cells, nv is the number of samples with
values falling into the vth cell. We hypothesize that the
expected cell number of the vth cell is nv,e . For the null
418
nv,e
.
(6)
For a discrete-valued population, the expected number in
the vth cell is nv,e = Npd (rv ), where rv is the value of
the vth cell with an expected probability pd (rv ). If the involved population is continuous-valued with an expected
cumulative density function Fc (r), the expected number
within the vth interval Iv = [rv , rv+1 ] can be computed
as nv,e = N[Fc (rv+1 ) − Fc (rv )]. As a rule of thumb, the
minimum expected cell number is equal to or greater than
five [32].
For large sample sizes, N, the test statistic T approximately follows a chi-square distribution with V − 1 degrees
of freedom under the null hypothesis [32]. Thus, the p-value
of the test is computed as
αT (R) = P χV2 −1 > T (R) .
(7)
In general, a fixed significance level that indicates the
false alarm probability of the hypothesis test is predefined
and compared with the p-value. The exclusive null and alternative hypotheses concerning the p-value can be rewritten
as
H0 :
Ha :
αT (R) ≥ α
αT (R) < α.
(8)
If the p-value provided by the test is equal to or larger than
the predefined significance level, H0 is accepted, which indicates the absence of interference. Otherwise, it should be
declared that the received signal is affected by interference.
It should be noted that the number of samples N involved
in each test highly affects the detection performance, which
is evaluated in the following sections.
III. PROPOSED DETECTION METHOD
In this section, an interference detection method analyzing the statistical properties of the STFT of the received
GNSS signal is proposed.
A. Detection Based on Statistical Analysis in the
TF Domain
Various linear and quadratic TFDs have been developed
in recent years [33]. As a linear transform with low computational complexity, the STFT has been widely used and
its canonical discrete form is defined as
S(n, k) =
r[n + m]h[m]e−j 2π mk/N
(9)
m
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where h[m] is a window function with length L and unit
power. The index m in the summation varies between
−(L − 1)/2 and (L − 1)/2. N is the total number of observed samples. The STFT does not generate cross-terms
for representing multicomponent signals; however, it suffers from an inevitable tradeoff between time and frequency
resolution.
In the absence of interference, each sample S(n, k; w)
in the frequency slice k is a zero-mean complex Gaussian
random variable due to the linear summation as shown
in (9). Since E[S(n, k; w)] = 0, the variance of the STFT
results can be computed as
2
Var[S(n, k; w)] = E w2 [n + m]
h [m] = σ 2 (10)
m
where w denotes the noise component in the received signal,
and σ 2 is the noise power.
The modulus square of the STFT, also known as the
spectrogram, is defined as
SP(n, k) = |S(n, k)|2 .
(11)
Thus, the spectrogram samples approximately follow a chisquare distribution [27]. The GoF test is then applied to each
frequency slice of the spectrogram results. All the samples
in each frequency slice are categorized into the predefined
intervals based on their values to obtain the observed PDF.
The method for generating the expected PDF is presented
in the next subsection. The test statistic is then computed
using these observed and expected PDFs based on (6). By
comparing the p-value of the GoF test to a predefined significance level, the interference detection problem is solved.
In the rest of the paper, the GoF tests applied to the TFand time-domain samples are referred to as the TF- and
time-domain methods, respectively.
If interference is present, due to the summation operation in the STFT, the interference component in each
frequency slice is equivalently accumulated. As shown in
(10), the noise power in the STFT samples is the same as
that of the received signal. Thus, the JNR of the evaluated TF-domain samples compared to that of the received
time-domain ones is improved. Substituting (4) into (9), the
frequency slice k in the STFT of the interference component
is rewritten as
j 2π Fu [n]n
S(n, k; u) ≈ Au e
(12)
h[m]ej 2π mδf (n,k)
m
where the frequency Fu [n + m] within a short window is
approximated as a constant Fu [n]. The frequency residual
is δf (n, k) = Fu [n] − k/N.
The JNR improvement is defined as the ratio between
the JNRs of the TF-domain samples in the frequency slice
k as shown in (12) and the received time-domain samples,
and is computed as
JNRIMP (n, k) =
m
2
h[m]e
j 2π mδf (n,k)
.
(13)
TABLE I
Theoretical JNR Improvement Introduced by the
TF-Domain Method For CW Interference
L
Rectangular
Hamming
3
4.77 dB
1.23 dB
5
6.98 dB
4.97 dB
Equation (13) shows that the JNRIMP depends on the
interference frequency characteristics, observed frequency
slice, and window parameters used in the STFT. The highest
JNRIMP occurs in the frequency slice equal to the interference frequency Fu for CW interference or within the sweep
bandwidth Bw for a chirp-type jammer.
For CW interference, the JNRIMP remains constant for
all the samples in the frequency slice k, which is shown in
Table I considering two window types and lengths. It can
be observed that a longer window yields a larger JNRIMP .
A rectangular window provides a larger JNRIMP than a
Hamming window of the same length, since the coefficients of a rectangular window are all equal to 1 while the
coefficients of a Hamming window decrease away from
the center. However, since the frequency residual δf for a
chirp-type jammer varies with time instant due to the sweep
frequency, the JNRIMP varies from sample to sample with
value depending on the observed frequency slice and sweep
frequency of the jammer. With the same window type and
length, the specific JNRIMP value for a chirp-type jammer
is slightly decreased compared to that for CW interference.
Moreover, it can be observed from (13) that the JNRIMP
decreases for a chirp-type jammer with increasing sweep
bandwidth. The specific effects of the jammer frequency
characteristics including the sweep bandwidth and period
are investigated in Section V.
B. Determination of the Expected PDF
In a GoF test, the PDF of observed samples is compared
to a preset statistical model to realize the hypothesis testing. In our case, the theoretical chi-square distribution for
the spectrogram samples is a continuous distribution with
unknown degrees of freedom.
To obtain the expected numbers in each cell, there are
two methods that can be used to convert the continuous
distribution to discrete probabilities. The first method is
to directly use the STFT of an interference-free signal.
Through an average operation of a large number of independent tests, the expected number and boundaries of each
cell are computed for the hypothesis testing. The second
method is to use the mean and variance of the expected
chi-square distribution, which can be computed based on
the noise power estimate as shown in (10). The noise
power estimate is a common observable in GNSS receivers
and is generally used to determine the thresholds in various interference detection approaches. An alternative approach is to normalize the noise powers of the in-phase and
quadrature baseband signals. In the absence of interference,
the observed TF-domain samples in each frequency slice
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419
Fig. 2. ACF and CCF of the real and imaginary parts of the canonical STFT computed using 8192 pure noise samples. (a) ACF for rectangular
window. (b) CCF for rectangular window. (c) ACF for Hamming window. (d) CCF for Hamming window.
approximately follow a standard chi-square distribution
with two degrees of freedom. The expected PDF can then
be obtained based on the cumulative probability function
of the standard chi-square distribution.
Although taking into account the presence of multiple
GNSS signals, it should be noted that the expected PDF
obtained from an interference-free signal needs to be updated if any parameter used in the test including the total
sample size or window length is changed. However, the
noise power estimate can be used for any parameter settings in the test. The method for computing the expected
PDF would not affect the performance comparison of the
time- and TF-domain methods. Thus, for simple implementation, the noise power estimate based method will be used
in the subsequent simulations. In practical applications, the
choice depends on the receiver’s characteristics and user
requirements.
IV. STATISTICAL PROPERTIES OF CANONICAL STFT
AND BLOCK-WISE STFT-BASED DETECTION
METHOD
As shown in (9), the samples in each frequency slice
of the canonical STFT results are computed using a sliding window along the time axis. A sample differs from the
previous one by just right sliding the window by one sample. This means that the adjacent STFT samples with time
distance shorter than the window length are actually not
independent.
In order to analyze the statistical properties of the canonical STFT results, the received baseband signal is rewritten
as
r[i] = rR [i] + j rI [i].
(14)
In the absence of interference, r[i] is treated as a complex noise component with independent real and imaginary
parts. The auto- and cross-correlation functions (ACF and
420
CCF) of rR [i] and rI [i] can be expressed as
E {rR [i]rR [i +
E {rR [i]rI [i +
]} = E {rI [i]rI [i +
]} = 0.
]} = (σ 2 /2)δ( )
(15)
Substituting (14) into (9), the ACF of the real part of
the canonical STFT result SR (n, k) is computed as
E {SR (n, k)SR (n + , k)}
=
h[m]h[j ] rR [n+m] cos γm +rI [n+m] sin γm
m
j
· rR [n +
+ j ] cos γj + rI [n +
+ j ] sin γj
(16)
where γm = 2πmk/N and is the time distance.
After some simplifications using (15), the ACF of the
SR (n, k) can be expressed as
E{SR (n, k)SR (n +
where
HL ( ) =
, k)} = (σ 2 /2) cos(2πk /N)HL ( )
(17)
m
0
j =m−
h[m]h[j ] | | < L
| | ≥ L.
(18)
Similarly, the ACF of the SI (n, k) and the CCF between
the SR (n, k) and SI (n, k) are denoted by
E{SI (n, k)SI (n +
, k)} = (σ 2 /2) cos(2πk /N)HL ( )
E{SR (n, k)SI (n +
, k)} = (σ 2 /2) sin(2πk /N)HL ( ).
(19)
The experimental ACF and CCF when using rectangular and Hamming windows of length 5 are plotted in Fig. 2,
where the theoretical results from (17) and (19) are also presented. The number of pure noise samples used to compute
the ACF and CCF is 8192. As can be observed, the experimental values closely match the theoretical ones. Two
canonical STFT samples with time distance shorter than
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the window length are actually correlated, rather than independent. The values of the ACF and CCF decrease when
using a Hamming window instead of a rectangular one
due to the smaller window coefficients. The presence of
nonzero values outside the window is due to the finite number of samples used in the computation of the experimental
results.
Since the real and imaginary parts of the canonical
STFT results are not independent, the spectrogram samples
are not strictly chi-square distributed. To evaluate the effect
of this PDF distortion on the false alarm performance of
the canonical STFT-based GoF test, the experimental false
alarm probabilities are obtained using 105 independent tests
and are plotted in Fig. 3. The results provided by the timedomain method are also given. A total of 1024 samples
of an interference-free signal including 5 global positioning system (GPS) signals all with −25 dB signal-to-noise
power ratios (SNRs) are used in each test. The curve for the
time-domain method deviates from the reference straight
line, i.e., experimental Pf a = Pf a , caused by the multiple
GPS signals. Due to the correlated canonical STFT samples, the false alarm probability curves for the TF-domain
GoF test also deviate from the reference. Compared to the
method using a rectangular window, the degradation of the
false alarm performance is partially alleviated if a Hamming window is used since the correlation between two
canonical STFT samples becomes weaker.
An effective solution to solve the above described undesired problem in single antenna GNSS receivers is to
compute the STFT using nonoverlapped samples, which
means that the center sample within each sliding window
is spaced by a delay equal to the window length. Hereafter,
we call this block-wise STFT, which can be written as
L−1
Sb (n, k) =
+ m h[m]e−j 2π mk/N
r nL +
2
m
(20)
where n varies within the range [0, N/L − 1]. x denotes the floor operation generating the largest integer less
than or equal to x. Other parameters are defined as in (9). In
this method, due to nonoverlapped received samples being
used, the real and imaginary parts of the block-wise STFT
samples become independent variables. The experimental
false alarm curve corresponding to a frequency slice away
from the baseband for the block-wise STFT-based method
is also given in Fig. 3, and is shown to be close to the reference. This verifies that the block-wise STFT-based method
alleviates the effect of the presence of GNSS signals in
the received data on the false alarm performance compared
to the time-domain method, and eliminates the false alarm
issue in the canonical STFT-based method.
As can be observed in (20), the window length is a
key parameter in the block-wise STFT-based method. The
increase of window length produces a larger JNR improvement to the evaluated TF-domain samples provided by the
block-wise STFT as shown in (13). However, for a given
number of received samples, the number of TF-domain
Fig. 3. Experimental versus predefined false alarm probabilities for the
time- and TF-domain methods using 1024 interference-free samples
including 5 GPS signals all with −25 dB SNRs.
samples in each frequency slice is decreased with increasing window length, leading to degraded detection performance. Thus, there is a detection performance tradeoff in
terms of window length, which is analyzed in the following
section for both narrow- and wideband interference.
The main computational complexities of the methods
in [21] and [27] are the GoF test and the computation of
the spectrogram, respectively. By contrast, the two versions
of the proposed method would require increased computational complexity due to the computation of the spectrogram and the GoF tests applied to each frequency slice.
The computation complexity (considering only multiplication terms) of the spectrogram can be expressed as [34]
CSP =
N
N
Nf log2 (Nf ) +
Nf
L−q
L−q
(21)
where N is the total number of observed samples and L
is the window length. q denotes the overlap size of windows, which equals to L − 1 and 0 for the canonical and
block-wise STFT, respectively. Nf is the number of frequency bins. For the proposed method, the improved detection capability is at the expense of increased computational
complexity.
V. SIMULATION RESULTS
To verify the effectiveness of the proposed interference
detection method, extensive simulations were carried out
taking into account several detection metrics, including detection probability (Pd ), receiver operating characteristics
(ROC), and Cmin (the minimum distance from the ROC
curve to the upper-left corner). The TF-domain detection
methods using the canonical and block-wise STFT are both
considered in the simulations. As the benchmark against
which the proposed methods will be compared, the detection performance of the GoF test applied to the time-domain
in-phase baseband signal [21] is also shown in the results.
Different window parameters, jammer frequency characteristics including sweep bandwidth and period, as well as
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421
Fig. 4. p-values of 1000 independent tests for the time- and TF-domain
methods using CW interference with −17 dB JNR (rectangular windows
with length 5 and 19 are, respectively, used in the canonical and
block-wise STFT).
sample size have been considered in the subsequent simulations. The number of distinct cells used for each GoF
test is set to 10. CW interference and chirp-type jammers
are chosen as the representative narrow- and wideband interference, respectively. The normalized frequency of CW
interference is fixed at 0.12. A set of five GPS signals all
with −25 dB SNRs is used in the simulations. The sampling
frequency is 32.768 MHz. The predefined significance level
α is fixed at 10−4 except for the simulations relating to ROC
and Cmin . The total number of complex baseband samples
used in each test is 4096 except when investigating the effect of sample size on detection performance. It should be
noted that the detection performance relating to the TFdomain method corresponds to the frequency slice at the
frequency value equal to the CW interference frequency or
median frequency of the chirp sweep bandwidth.
A. ROC Curve
Fig. 4 displays the p-values of 1000 independent tests
for the aforementioned methods using CW interference
with −17 dB JNR. Rectangular windows with lengths 5
and 19 are used in the canonical and block-wise STFT,
respectively. As can be observed from Fig. 4, the method
using the canonical STFT yields the largest number of pvalues below the detection threshold 10−4 and, thus, indicates the presence of interference with the highest detection
probability. The block-wise STFT-based method provides
less p-values below the detection threshold compared to
the canonical STFT-based one. However, the time-domain
method is unable to detect the presence of interference at
−17 dB JNR since almost all the p-values denoted by the
blue curve in Fig. 4 are above the detection threshold.
To show the statistics of the detection results of Fig. 4,
the ROC curves depicting the pair of false alarm and detection probabilities are plotted in Fig. 5. The detection probability is obtained by comparing the p-values with each
predefined false alarm probability, i.e., detection threshold.
422
Fig. 5. ROC curves for the time- and TF-domain methods using CW
interference with −17 dB JNR.
It can be observed that the TF-domain methods using the
canonical and block-wise STFT provide much better detection performance than the existing time-domain method.
The canonical STFT-based method further improves the
detection performance compared to using the block-wise
STFT.
B. Cmin for Narrow- and WideBand Interference
In order to quantitatively show the detection performance improvement of the GoF test applied to the TFdomain samples over time-domain method, another metric C = [Pf2a + (1 − Pd )2 ]1/2 reflecting the distance from
the ROC curve to the upper-left corner of the plot [i.e.,
(Pf a , Pd ) = (0, 1)] is introduced. For each JNR, the minimum value of C, i.e., Cmin , is computed. A smaller Cmin
value indicates that better detection performance can be
achieved at a smaller significance level. A rectangular window with length 5 is used in the canonical STFT. For the
block-wise STFT, rectangular windows with lengths 19 and
3 are used for detecting CW interference and chirp-type
jammer, respectively (based on the simulation results in the
next subsection).
Fig. 6 shows the Cmin curves as a function of JNR for
CW interference and a chirp-type jammer with 10.72 MHz
sweep bandwidth and 8.62 μs sweep period [31]. The simulated JNR is in the range of −25 to −3 dB, i.e., low JNR
environments. As can be observed from Fig. 6, all three GoF
test-based methods are applicable to both narrow- and wideband interference. The time-domain method gives similar
detection performance for narrow- and wideband interference, while the TF-domain method provides slightly worse
performance for wideband interference. The performance
improvement of the TF-domain method over the timedomain method depends on the desired Cmin value. Specifically for Cmin of 0.03 (the horizontal line in Fig. 6), the
minimum detectable JNRs considering CW interference are
−9, −16.5, and −17 dB for time-domain, block-wise, and
canonical STFT-based TF-domain methods, respectively.
Moreover, the three methods provide the minimum
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Fig. 6. Cmin versus JNR for the detection methods using CW
interference and a chirp-type jammer with 10.72-MHz sweep bandwidth
and 8.62-μs sweep period.
detectable JNRs of −9, −13, and −15 dB for chirp-type
jammer, respectively.
As the JNR decreases, the Cmin values for the timedomain and block-wise STFT-based√tests gradually increase and approach the limit value 1/ 2, which is equal to
the distance from the upper-left corner point to the straight
line Pd = Pf a . The Cmin curves for the canonical STFTbased method
√ gradually converge to smaller values compared to 1/ 2, which can be explained by the slight deviation in the distribution of observed samples from the
predefined chi-squared model.
The proposed method is also compared with the common TF peak detection method [26], whose detection results for the same CW interference and chirp-type jammer
are also shown in Fig. 6. The STFT results of the received
signal using a shorter (L1 = 31) and a longer (L2 = 1025)
Hamming windows are both considered. As can be observed, the TF peak detection method using a longer window can provide excellent detection performance for CW
interference, while losing its superiority when dealing with
chirp-type jammer. When a shorter window length is used,
the capabilities for detecting CW interference and chirptype jammer are similar, which are much weaker than those
provided by the proposed method.
C. Detection Probability versus Window Type and
Length
For the TF-domain method, the window type and length
used in the STFT have great impact on the detection performance as formulated in (13). First, the effect of the window
parameters on the detection performance of the canonical
STFT-based method is considered. To avoid serious deterioration of false alarm performance, only two window
lengths (3 and 5) with rectangular and Hamming windows
are used. Fig. 7 depicts the detection probabilities with
respect to JNR using different window types and lengths
in the canonical STFT-based method for CW interference.
The results for the time-domain method are also shown
Fig. 7. Detection probability versus JNR using different window types
and lengths in the canonical STFT-based method for CW interference
(Pf a = 10−4 ).
in Fig. 7 for comparison. The canonical STFT-based TFdomain method using Hamming windows with lengths 3
and 5 provides 3.5 and 7 dB performance improvements
over the time-domain method, respectively, and 7 and 9 dB
improvements can be observed when rectangular windows
with lengths of 3 and 5 are used. An additional 2 dB improvement is obtained compared to the theoretical values
in Table I. This is due to the fact that the TF-domain methods use both the in-phase and quadrature samples, whereas
the time-domain method only uses the in-phase samples.
The close match between the experimental and theoretical
values verifies the effectiveness of the theoretical analysis.
In the block-wise STFT-based method, the setting of
window length that introduces a tradeoff to the detection
performance is then investigated. Since the false alarm
problem is eliminated, a rectangular window is used in
the block-wise STFT. Based on (13) and (20), the jammer
sweep bandwidth and sample size also impact the JNR improvement, and are jointly taken into consideration for this
analysis. As shown in the following subsection, the detection performance is almost insensitive to the jammer sweep
period, which is, thus, fixed at 8.62 μs in this experiment.
Fig. 8 shows the detection probabilities with respect
to window length considering chirp-type jammers with
−15 dB JNR. The sweep bandwidth varies from 2 to
14 MHz. Although outside the sweep bandwidth range
reported in [31], smaller sweep bandwidths are still considered to illustrate the dependence of window length on
the jammer sweep bandwidth. The total sample size used
in each test is 4096. To avoid that the number of evaluated
samples is too small, the window length varies in the range
of 3–39. As can be observed, window length that yields the
best detection probability decreases with increasing jammer
bandwidth. This can be explained by the fact that the increment in JNR improvement brought by increasing window
length used in the block-wise STFT decreases with increasing sweep bandwidth, while the performance degradation
caused by the decreased number of evaluated samples is
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423
Fig. 8. Detection probability versus window length for block-wise
STFT-based method using chirp-type jammers with several sweep
bandwidths and 8.62-μs sweep period (N = 4096, JNR = −15 dB,
Pf a = 10−4 ).
independent of the sweep bandwidth. Hence, a shorter window length should be used for detecting jammers with wider
bandwidths.
Fig. 9 shows the detection probabilities with respect to
window length considering several sample sizes for CW
interference. As can be observed in Fig. 9(a), the detection probabilities for CW interference with −17 dB JNR
using 8192 or 16384 samples in each test are almost 1 for
most values of the window lengths, making it difficult to
obtain the best window length information. Therefore, in
Fig. 9(b)–(d), the JNR values of CW interference are −17,
−19, and −21 dB for N = 4096, 8192, 16384, respectively. It can be observed that the detection probability first
increases with increasing window length. However, if the
window length further increases (L ≥ 21 for N = 4096,
L ≥ 35 for N = 8196, L ≥ 49 for N = 16384), the detection probability begins to change irregularly. The long
window length used leads to a reduced frequency resolution in the STFT, which introduces spectral leakage for
CW interference and, thus, impacts the JNR improvement
brought by the block-wise STFT-based method. Hence, the
case without spectral leakage is also considered through
adjusting the CW interference frequency to exactly coincide with frequency bins. In this case, a best window
length can be obtained for each sample size condition
(L = 27 for N = 4096, L = 39 for N = 8192, L = 57 for
N = 16384). In practical applications, the spectral leakage phenomenon cannot be ignored. Therefore, combining these aforementioned factors, rectangular windows with
lengths 19 and 3 are, respectively, used in the block-wise
STFT-based method for the detection of narrow- and wideband interference when the total sample size is 4096.
D. Detection Probability versus Instantaneous Frequency Parameters for Chirp-Type Jammer
Since the proposed method exploits the TF-domain concentration property of the interference, the detection per424
Fig. 9. Detection probability versus window length for block-wise
STFT-based method using CW interference for several sample sizes
(Pf a = 10−4 ). (a) several sample sizes, JNR = −17 dB. (b) N = 4096,
JNR = −17 dB. (c) N = 8192, JNR = −19 dB. (d) N = 16384,
JNR = −21 dB.
formance is also significantly affected by the time-varying
properties of the interference instantaneous frequency law
characterized by sweep bandwidth and period. Based on
(13), the JNR improvement brought by the TF-domain test
depends highly on the interference sweep bandwidth. The
block-wise STFT-based method using a rectangular window with length 3 to detect a chirp-type jammer with
8.62 μs sweep period is used to demonstrate the results.
The sweep bandwidth is varied between 1 and 20 MHz
with 1 MHz intervals. The detection probabilities as a function of chirp sweep bandwidth for various JNR values are
shown in Fig. 10. As can be seen, the detection probability
decreases with increasing chirp bandwidth. This is due to
the increased frequency residual in the observed TF-domain
samples as formulated in (12).
Sweep period is another important parameter and is
also considered in our evaluation. A chirp-type jammer
with 10.72-MHz sweep bandwidth is used as the interference signal. For consistency with [31], the simulated chirp
sweep period is in the range of 8.62–18.97 μs with 1.48 μs
intervals. A rectangular window with length 3 is used in the
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Fig. 10. Detection probability versus sweep bandwidth for block-wise
STFT-based method using a chirp-type jammer with 8.62-μs sweep
period (a rectangular window with length 3 is used in the block-wise
STFT, Pf a = 10−4 ).
Fig. 12. Detection probability versus JNR using several sample sizes
for the time- and TF-domain methods (a rectangular window with length
5 is used in the canonical and block-wise STFT, Pf a = 10−4 ).
VI. CONCLUSION
Fig. 11. Detection probability versus sweep period for block-wise
STFT-based method using a chirp-type jammer with 10.72-MHz sweep
bandwidth (a rectangular window with length 3 is used in the block-wise
STFT, Pf a = 10−4 ).
block-wise STFT. Fig. 11 depicts the detection probabilities
in terms of sweep period for several JNR values. As can be
seen, the proposed method yields almost constant detection
probabilities for chirp-type jammers with different sweep
periods.
E. Detection Probability versus Sample Size
The size of the observed complex baseband samples,
N is also a critical parameter in the detection method. The
detection probabilities with respect to JNR for CW interference using sample sizes of 1024, 2048, and 4096 are plotted
in Fig. 12. A rectangular window with length 5 is used in
the canonical and block-wise STFT. It can be observed that
the detection performance is improved through increasing
the sample size. Doubling the number of samples yields
nearly 2-dB improvement in terms of detection performance
for all the three tests.
In this paper, we have proposed a precorrelation interference detection algorithm combining the TF and statistical properties of the received GNSS signals. From the
statistical-domain viewpoint, the real and imaginary parts
of the STFT results for an interference-free GNSS signal
can be approximately regarded as independent Gaussian
variables. Thus, the PDF of the spectrogram, which is the
square modulus of the STFT, can be evaluated by the GoF
test applied to each frequency slice in the TF plane. The
accumulative distance between the observed and expected
PDFs is computed as the test statistic, which approximately
follows a chi-square distribution. The quantitative JNR improvement of the evaluated TF-domain samples over the
time-domain ones brought by the accumulation operation
in the STFT is derived and shown to depend on the interference frequency characteristics and window parameters
used in the STFT.
Concerning the tradeoff between detection and false
alarm performance, two versions of the proposed method,
using the canonical and block-wise STFT with the maximum and minimum overlap sizes in the windows, are presented and analyzed. Simulations are conducted to compare the detection performance of the GoF tests applied
to the TF-domain samples and time-domain ones. Results
demonstrate that the canonical and block-wise STFT-based
methods are both applicable to narrow- and wideband interference in low JNR environments, and provide improved detection performance over the existing time-domain method.
In the context of overall system design, the detection
technique and the total sample size in each test need to be
first determined according to the desired detectable interference power and Cmin . If the block-wise STFT-based method
is used, the window length can then be chosen based on the
sample size and anticipated interference types.
The performance gain of the proposed method is tradedoff with an increased computational complexity. However,
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425
due to the constantly growing computational capabilities
of the processors in GNSS receivers, it is an interesting
perspective interference detection technique.
[15]
[16]
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Pai Wang received the B.E. degree in information engineering from the Beijing Institute
of Technology, Beijing, China, in 2012, where she is currently working toward the Ph.D.
degree in information and communication engineering.
From 2015 to 2016, she spent one year as a visiting Ph.D. student in the School of
Electrical Engineering and Telecommunications, University of New South Wales, Sydney, NSW, Australia. Her research interests include the detection, characterization, and
mitigation of interference and jamming signals for GNSS receivers.
Ediz Cetin (S’96–M’02) received the B.Eng. (Hons.) degree in control and computer
engineering, and the Ph.D. degree in unsupervised adaptive signal processing for wireless
receivers from the University of Westminster, London, U.K., in 1996 and 2002, respectively.
From 2002 to 2011, he was with the University of Westminster, initially as a Postdoctoral Research Fellow and subsequently, from 2006 to 2011, as a Senior Lecturer. From
2011 to 2017, he was a Senior Research Associate in the Australian Centre for Space
Engineering Research, University of New South Wales, Sydney, NSW, Australia. He is
currently a Senior Lecturer in the School of Engineering, Macquarie University, Sydney,
NSW, Australia. His research interests include interference detection and localization,
fault-tolerant reconfigurable circuits, adaptive techniques for RF impairment mitigation
for communications and global navigation satellite system receivers, and design and lowpower implementation of digital circuits. To date, he has written or cowritten more than 60
technical publications, a book chapter and holds two patents in the areas of communications
and GNSS receivers.
Dr. Cetin is a Member of the Institution of Engineering and Technology (IET) and
serves as the Chair of the IET New South Wales (NSW) Local Network, as well as the
Chair of the Institute of Electrical and Electronics Engineers (IEEE) NSW Circuits and
Systems/Solid-State Circuits/Photonics/Electron Devices joint chapter.
Andrew G. Dempster (M’92–SM’03) received the B.Eng. and M.Eng.Sc. degrees from
the University of New South Wales (UNSW), Sydney, NSW, Australia, in 1984 and 1992,
respectively, and the Ph.D. degree from the University of Cambridge, Cambridge, U.K.,
in 1995, all in efficient circuits for signal processing arithmetic.
He is currently the Director of the Australian Centre for Space Engineering Research
at UNSW. He was a System Engineer and Project Manager for the first global positioning
system receiver developed in Australia in late 1980s and has been involved in satellite
navigation ever since. He has published in the areas of arithmetic circuits, signal processing,
biomedical image processing, satellite navigation, and space systems. His current research
interests include satellite navigation receiver design and signal processing, areas in which
he has six patents, new location technologies, and space systems.
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427
Yongqing Wang (M’14) received the Ph.D. degree in signal and information processing
from the Beijing Institute of Technology, Beijing, China, in 2008.
He is currently an Associate Professor in the School of Information and Electronics,
Beijing Institute of Technology and leads a research group with more than 20 graduate
students. His research interests include spaceflight TT&C, telecommunication technology,
electronic system simulation and signal simulator, spread spectrum signal processing, and
satellite navigation and positioning technologies.
Siliang Wu received the M.S. degree in automation from the Yanshan University, Qinhuangdao, China, in 1989 and the Ph.D. degree in electromagnetic measuring technology
and instrument from the Harbin Institute of Technology, Harbin, China, in 1995.
From 1996 to 1998, he was in the Beijing Institute of Technology, Beijing, China, as
a Postdoctoral Research Fellow. He is currently a Professor in the School of Information
and Electronics, Beijing Institute of Technology, Beijing, China. His research interests
include radar system design and signal processing, electronic systems simulation and
signal simulator, spread spectrum signal processing, radio monitoring and control, and
satellite navigation and positioning technologies.
428
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