Signal Processing - September 2017 - 89

Interference
Detection

y [n ]
Input
Samples

TF
Analysis

Y (n , k )

TF
Excision

TFR

-
Y (n , k )
Modified
TFR

TF
Synthesis

y-[n ]
Filtered
Samples

FIGURE 5. A schematic representation of interference mitigation approaches based on TF analysis.

end are brought into a transform domain, where a sparse representation of the interfering signal is evident. The TFR depends
on the selected type of TFD. The usage of the STFT and the
WVD is considered in [9]. The Gabor expansion was investigated in [10], whereas the effectiveness of the Hilbert-Huang
transform was studied in [11].
Independent of the TFR selected, the goal is to obtain a
sparse representation of the interfering signal. Sparse representations allow one better separation of useful and interfering signals and facilitate interference detection, which is
performed by the subsequent processing block. Several papers
[9], [11], [12] investigated interference detection in the TF
domain for GNSS. In the simplest case, the TFR obtained,
Z ^n, k h, is compared with a decision threshold, Th. If at least
one sample of the TFR passes the threshold, the interference
presence is declared. The detection threshold, Th, can be set
using different criteria from detection theory [26] and, e.g.,
the constant false-alarm rate criterion can be adopted. The
statistical properties of the samples of the TFR are analyzed
in the absence of interference, and the threshold is fixed to
obtain a constant false-alarm probability, i.e., the probability
of incorrectly declaring interference present. In the analysis,
the useful GNSS signal is neglected as it is assumed hidden
by the input noise.
When interference is declared present, excision proceeds
with various possible techniques, ranging from the simplest to
most computational demanding approaches. A simple approach
is setting to zero the samples of the TFR that pass the threshold.
Other approaches may require the estimation of the IF of the
interfering signal, which is performed in the TF domain. An
adaptive notch filter can then be used to remove the interfering term directly from the original signal. This approach was
suggested in [27] for spread spectrum communication systems
and applied to GNSS in [9]. The advantage of using an adaptive filter driven by the information extracted from the TFR is
that the filter output is already a time-domain signal and TF
synthesis is not required.
If interference excision is performed in the TF domain, a
modified TFR, Yr (n, k), is obtained, and the corresponding
time-domain signal has to be reconstructed. This operation is
performed by the synthesis block. After excision, Yr (n, k) may
not be a valid TFR, i.e., a signal may not exist where TFR is
equal to Yr (n, k). This implies that time-domain signal deter-

mination has to be performed with respect to a specific reconstruction criterion. For example, the reconstructed signal yr [n]
may be the one with TFR that is the closest to Yr (n, k) in the
minimum mean squared error sense. In the case of linear TFR,
such as the STFT, efficient procedures exist for signal reconstruction [28]. In general, more complex approaches have to be
adopted for QTFDs. TF synthesis is a complex topic; see [8]
for more details on the subject.
The TF interference excision process is illustrated in Figure 6 for the case of GNSS signals corrupted by jamming.
The presence of a jamming signal can be clearly seen from
the histogram of the input samples, which are characterized
by a multimodal, non-Gaussian distribution. The histograms
in Figure 6 have been evaluated by jointly considering the real
and imaginary parts of y [n]. The STFT is used to evaluate
the TFR of y [n] and interference detection is performed by
comparing Y (n, k) 2 with a decision threshold. The samples
of Y (n, k) with a square magnitude greater than Th are excised,
and Yr (n, k) is obtained. A filtered version of y [n] is finally
obtained using the STFT synthesis approach described in [28].
The reconstructed samples yr [n] are characterized by a Gaussian histogram indicating that the jamming signal has been
successfully removed and only Gaussian noise and the weak
GNSS signals are present.
TF excision can introduce distortions to the GNSS signal.
This is because, when excising the jammer, useful signal components may be removed or compromised. The distortions to
the desired signals are, however, minimized by applying appropriate TFDs that lead to high jamming power concentration or
sparse representation in the TF domain. In so doing, only few
TF points are removed, leaving out the rest of the domain intact
where the desired signal resides.

Sparsity-aware processing
Sparsity-based TFR reconstruction
A highly localizable jammer signal in the TF domain is confined to few TF points. This domain vacancy casts the jammer
as a sparse signal in the joint-variable representation and, as
such, creates an opportunity to employ recent advances in CS
and sparse reconstruction techniques for antijam GNSS.
While sparse reconstruction of the jammer can be performed
for both linear and bilinear TFRs [29], we limit our discussion

IEEE SIGNAL PROCESSING MAGAZINE

September 2017

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