Signal Processing - September 2017 - 86

f (kHz)

Jammers can emit single- or multicominterference mitigation beyond the utiliGNSS receivers are
ponent nonstationary signals that assume
zation of only the signal temporal strucvulnerable to several forms
time-varying frequency characteristics. Each
ture. Most notably, we show that sparsity
of interference, which can
component can be generally described by a
in the TF domain permits CS and sparse
be either intentional or
form of amplitude and frequency modulareconstruction techniques to contribute to
unintentional.
tions. This property yields a highly localGNSS receiver design.
izable signal power in the TF domain, a
by-product of which is a sparse TF representation (TFR).
TFDs
A broad class of jammers emits frequency modulated (FM)
TF analysis is a body of techniques for the representation of
signals, where the IF of each signal component may sweep
the time-varying frequency content of signals [7], [8]. Cona range of several megahertz in a few microseconds [4]-[6].
trary to a single-variable frequency analysis, joint-variable
TF analysis [7], [8], in its linear and bilinear forms, has been
analysis transforms the signal to the TF domain, providing
used to reveal nonstationary jamming signals and separate
a signal TFD or TFR. Among the most common TFRs is the
them from the navigation signals, which have rather uniformly
linear short-time Fourier transform (STFT),
distributed power in the TF domain. In so doing, jamming sig+3
Y (t, f ) = #
h(t l - t) y (t l ) e -i2r ftl dt l,
nals become easily identifiable and detectable and can be effec(1)
-3
tively mitigated [9]-[12].
In this article, we describe the role of TF analysis in GNSS
where y (t) is the signal and h (t) the analysis window, typisignal processing. This role encompasses the enhancement
cally a Hamming, Hanning, or rectangular window. The STFT
of GNSS receiver performance [3], [9], the design of detecis a sliding Fourier transform: at every time t, the windowed
tion units that can be adopted to monitor and protect critisignal h (t - t l ) y (t l ) is formed and then Fourier transformed,
cal infrastructures relying on GNSS signal reception, and
providing the local frequency content of the signal. A disthe assessment of GNSS-derived data [13], [14]. We cast TF
crete-time STFT Z ^n, k h is defined for sampled data, where
n = t /Ts and k = f / f0 , with Ts being the sampling time and f 0
analysis as an important tool for system monitoring through
the characterization of satellite on-board clocks and for the
the frequency resolution [8].
examination of ionospheric scintillation data. Both linear
The STFT is a complex quantity and its visual representaTF transforms and QTFDs are considered. This article
tion is achieved by taking its square magnitude, thus obtainaccentuates recent advances in TF analysis for interference
ing a QTFD known as the spectrogram. Figure 1 shows the
mitigation, which benefits from the emergence of CS and
spectrogram of a nonstationary signal whose frequency conpredicates on the maturity of array signal processing algotent changes with time. The component at f = 4 kHz, beginrithms [15]-[17]. The availability of multiantenna systems
ning at t = 0.5 s and ending at t = 0.8 s, corresponds to a
for GNSS naturally promotes the integration of TFDs with
short-duration sinusoid with constant frequency and amplithe spatial dimension [15], [16]. This improves the jammer
tude. One can think of this component as the representation
IF representation and, as such, offers new possibilities for
of a single-frequency constant amplitude note played over the
time interval 0.5 # t # 0.8 s by an instrument. Light blue
means that energy is present in that TF region. Conversely, a
dark blue color means lack of signal energy. The component
beginning at t = 0 s and f = 5 kHz and ending at t = 1 s and
5
f = 0 kHz is a linear chirp, and it corresponds to a sinusoid
4.5
the frequency of which changes linearly with time. Similarly,
the parabolic component is a higher-order FM signal. The
4
ridge of these components represents an approximation of the
3.5
IF of the corresponding signals [7]. Finally, the TF region
3
0 # t # 1 s, 1 # f # 3 kHz contains numerous components
2.5
with time-varying amplitude and frequency, and it corre2
sponds to a bandpass filtered white Gaussian noise (WGN).
Similarly to frequency analysis, where the spectrum of one
1.5
realization of WGN is made up by frequency components
1
with random amplitudes and phases, the TFR of noise depicts
0.5
random TF behavior. Although the noise components some0
what shadow the linear and quadratic FM signals, the signal
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
respective of TF structures remain pronounced, a property
t (s)
that makes TF analysis an effective tool for the study of weak
signals in noise.
FIGURE 1. TFR of a nonstationary signal. The plot represents the spectroGNSS receivers often have to deal with situations similar
gram of a signal made by the sum of a short-duration sinusoid, a linear
chirp, a higher-order FM signal, and a bandpass noise.
to that depicted in Figure 1 when, for instance, GNSS signals
86

IEEE SIGNAL PROCESSING MAGAZINE

September 2017

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