IEEE - Aerospace and Electronic Systems - April 2023 - 18

LPI Waveform Recognition Using Adaptive Feature Construction and Convolutional Neural Networks
Figure 3.
Extracted adaptive features from the Frank code and P1 code PSK signal using (a) VMD and (b) EMD.
where f is the original signal and ukðtÞ is the decomposed
IMF with its center frequency wkðtÞ, dðtÞ is the Dirac delta
function, 1
and ðdðtÞþ j
pt is the impulse response of Hilbert transform,
ptÞ ukðtÞ is the analytic signal, which is
defined as that its real part is the original signal and imaginary
part is the Hilbert transform of the original signal.
The solving process of this problem is performed, as
shown in Figure 2 in frequency domain and leads to the
decomposition of the original signal. The IMFs from the
decomposition process are considered as the VMD
features.
Figure 3 displays the five extracted adaptive features as
examples using EMD and VMD from a Frank code phaseshift
key (PSK) signal. In FALPINE, in order to obtain the
unified input of the CNN model, the time domain EMD and
VMD features are combined into one matrix and reshaped to
a specific dimension as the adaptive feature, as shown in
Figure 4(a). As the matrix of the time domain feature is
reshaped into specific dimension, the axis has no specific
meaning for the feature images and is not displayed in the
figures.
PREDEFINEDANALYTICALFEATUREEXTRACTION
(a) WVDfeature. The WVD [1] is widely used in the analysis
of nonstationary signals, such as LPD/LPI signals,
which exhibit the highest signal energy concentration in
the time-frequency plane for modulated signals. This
characteristics of the WVD makes it a promising feature
of detecting and classifying different nonperiodic signals.
The calculation of the WVD is as follows:
Wðt; vÞ¼
18
2p
1
S t
Z
Stþ
t
2
t
2
ejvtdt
(4)
where S is the signal, t is the time lag, and Srepresents
the complex conjugate of the signal. Figure 4(b) shows
the WVD of the Frank code PSK signal with a carrier frequency
of1 kHz and SNR of5 dB.
(b) CWD feature. The CWD [1] uses an exponential
kernel in the time-frequency distributions representation
to minimize the cross-term components that are presented
in the WVD. The attenuation of the cross term in the
time-frequency plane allows the modulation type to be
more readily determined. The CWD feature can be calculated
as follows:
Cfðt; v; fÞ¼
1
2p
ZZZ
ejðmtvtÞfð; tÞAðm; tÞdmdtd
(5)
where fð; tÞ is a kernel function
fð; tÞ¼ e2t2=s
where sðs > 0Þ is a scaling factor, and
Aðm; tÞ¼ x m þ
t
2
x m
t
2
(6)
(7)
where xðmÞ is the time signal and xðmÞ is its complex
conjugate. The CWD of the Frank code PSK signal can be
obtained according to the abovementioned equations, as
shown in Figure 4(c).
(c) Waveletfeature. The wavelet transform is another
popularly used method for signal classification [12]. The
LPI signals often contain numerous nonstationary characteristics.
Hence the wavelet transform is a good choice as
it is proven to be efficient in representing and analyzing
the nonstationary characteristics [26], where the wavelet
features may be appropriate to distinguish the LPI signals.
In order to extract the wavelet features, the discrete
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