Method and system for low-probability-of-intercept radar signal waveform recognition
A method for recognizing a low-probability-of-interception (LPI) radar signal waveform includes: obtaining, by a radar signal receiver, an LPI radar signal s(t), s(t) varying with time t; extracting, by a radar signal processor, an adaptive feature and a pre-defined analytical feature from the LPI radar signal s(t); combining, by the radar signal processor, the adaptive feature with the pre-defined analytical feature to generate a constructed adaptive feature; and applying, by the radar signal processor, a convolutional neural network (CNN) model to classify the constructed adaptive feature to recognize the LPI radar signal waveform.
1. A method for recognizing a low-probability-of-interception (LPI) radar signal waveform, comprising:
obtaining, by a radar signal receiver, an LPI radar signal s(t), s(t) varying with time t;
extracting, by a radar signal processor, an adaptive feature and a pre-defined analytical feature from the LPI radar signal s(t), wherein the pre-defined analytical feature includes a Wigner-Ville Distribution (WVD) feature, a Choi-William Distribution (CWD) feature, and a wavelet feature;
combining, by the radar signal processor, the adaptive feature with the pre-defined analytical feature to generate a constructed adaptive feature according to:
F=ψ{G 1 (F AD ), G 2 (F WVD ), G 3 (F CWD ), G 4 (F WL )}, wherein F is the constructed adaptive feature, F AD is the adaptive feature, F WVD is the WVD feature, F CWD is the CWD feature, F WL is the wavelet feature, G 1 , G 2 , G 3 , G 4 , are linear or non-linear operations, and ψ is a data fusion operation; and
applying, by the radar signal processor, a convolutional neural network (CNN) model to classify the constructed adaptive feature to recognize the LPI radar signal waveform.
2. The method according to claim 1 , wherein:
the adaptive feature includes one or more of an empirical mode decomposition (EMD) feature and a variational mode decomposition (VMD) feature.
3. The method according to claim 2 , wherein extracting the EMD feature includes:
identifying all extrema of s(t);
interpolating all local maxima to form an upper envelop u(t);
interpolating all local minima to form a lower envelop l(t);
calculating a mean envelop m(t), wherein m(t)=(u(t)+l(t))/2;
extracting the mean envelop m(t) from s(t) to obtain h(t), wherein h(t)=s(t)-m(t);
determining whether h(t) is an intrinsic mode function (IMF); and
in response to h(t) being an IMF and a number of obtained IMFs being less than a pre-configured number, iterating all above steps on a residue signal r(t), wherein r(t)=s(t)−h(t), otherwise iterating all above steps on h(t).
4. The method according to claim 2 , wherein extracting the VMD feature includes:
solving:
min
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k
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s
.
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k
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,
wherein u k (t) is a decomposed IMF of s(t) with its center frequency ω k , δ(t) is a Dirac delta function, 1/πt is an impulse response of Hilbert transform,
(
δ
(
t
)
+
j
π
t
)
*
u
k
(
t
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is an analytic signal, a real part of the analytic signal is s(t), an imaginary part of the analytic signal is a Hilbert transform of s(t).
5. The method according to claim 1 , wherein:
the WVD feature is calculated by:
W
(
t
,
ω
)
=
1
2
π
∫
S
(
t
+
τ
2
)
S
*
(
t
-
τ
2
)
e
-
j
ω
t
d
τ
,
wherein S is the LPI radar signal, S* represents complex conjugate of the LPI radar signal, τ is a time lag, and ω is a frequency.
6. The method according to claim 1 , wherein:
the CWD feature is calculated by:
C
s
(
t
,
ω
,
ϕ
)
=
1
2
π
∫
∫
∫
e
j
(
ξ
μ
-
τ
ω
-
ξ
t
)
ϕ
(
ξ
,
τ
)
A
(
μ
,
τ
)
d
μ
d
τ
d
ξ
,
wherein ϕ(ξ, τ) is a kernel function, ϕ(ξ, τ)=e −ξ 2 τ 2 /σ , σ is a scaling factor, σ>1,
A
(
μ
,
τ
)
=
x
(
μ
+
τ
2
)
x
*
(
μ
-
τ
2
)
,
x(μ) is a time signal and x*(μ) is its complex conjugate, μ and τ are axes in an ambiguity domain, τ is a time lag, and ω is a frequency.
7. The method according to claim 1 , wherein:
the wavelet feature is a Morse wavelet feature and is calculated by:
W
(
j
,
k
)
=
∑
j
∑
k
s
(
k
)
ψ
j
k
(
n
)
,
and
ψ
j
k
(
n
)
=
2
-
j
2
ψ
(
2
-
j
n
-
k
)
,
wherein S(k) is the LPI radar signal, W(j, k) are wavelet transform coefficients, ψ jk (n) is a mother wavelet, j is a scale parameter, and k is a shift parameter.
8. The method according to claim 1 , wherein:
G 1 , G 2 , G 3 , G 4 operations include one or more of reshaping, squaring, down-sampling, and maxpooling.
9. The method according to claim 1 , wherein:
the adaptive feature, the WVD feature, the CWD feature, and the wavelet feature are 2-dimensional matrices; and
the data fusion operation includes stacking the 2-dimensional matrices.
10. The method according to claim 1 , wherein:
the CNN model includes three convolution layers.
11. A system for recognizing a low-probability-of-interception (LPI) radar signal waveform, comprising:
a radar signal receiver for obtaining an LPI radar signal s(t), s(t) varying with time t;
a memory storing computer program instructions; and
a radar signal processor coupled to the memory and, when executing the computer program instructions, configured to perform:
extracting an adaptive feature and a pre-defined analytical feature from the LPI radar signal s(t), wherein the pre-defined analytical feature includes a Wigner-Ville Distribution (WVD) feature, a Choi-William Distribution (CWD) feature, and a wavelet feature;
combining the adaptive feature with the pre-defined analytical feature to generate a constructed adaptive feature according to:
F
=
ψ
{
G
1
(
F
A
D
)
,
G
2
(
F
W
V
D
)
,
G
3
(
F
C
W
D
)
,
G
4
(
F
W
L
)
}
,
wherein F is the constructed adaptive feature, F AD is the adaptive feature, F WVD is the WVD feature, F CWD is the CWD feature, F WL is the wavelet feature, G 1 , G 2 , G 3 , G 4 are linear or non-linear operations, and ψ is a data fusion operation; and
applying a convolutional neural network (CNN) model to classify the constructed adaptive feature to recognize the LPI radar signal waveform.
12. The system according to claim 11 , wherein:
the adaptive feature includes one or more of an empirical mode decomposition (EMD) feature and a variational mode decomposition (VMD) feature.
13. The system according to claim 12 , wherein the radar signal processor is further configured to:
identify all extrema of s(t);
interpolate all local maxima to form an upper envelop u(t);
interpolate all local minima to form a lower envelop l(t);
calculate a mean envelop m(t), wherein m(t)=(u(t)+l(t))/2;
extract the mean envelop m(t) from s(t) to obtain h(t), wherein h(t)=s(t)−m(t);
determine whether h(t) is an intrinsic mode function (IMF); and
in response to h(t) being an IMF and a number of obtained IMFs being less than a pre-configured number, iterate all above steps on a residue signal r(t), wherein r(t)=s(t)−h(t), otherwise iterating all above steps on h(t).
14. The system according to claim 12 , wherein the radar signal processor is further configured to:
solve:
min
u
k
,
ω
k
{
∫
∑
k
❘
"\[LeftBracketingBar]"
∂
t
[
(
δ
(
t
)
+
j
π
t
)
*
u
k
(
t
)
]
e
-
j
ω
k
t
❘
"\[RightBracketingBar]"
2
dt
}
s
.
t
.
∑
k
u
k
=
s
(
t
)
,
wherein u k (t) is a decomposed IMF of s(t) with its center frequency ω k , δ(t) is a Dirac delta function, 1/πt is an impulse response of Hilbert transform,
(
δ
(
t
)
+
j
π
t
)
*
u
k
(
t
)
is an analytic signal, a real part of the analytic signal is s(t), an imaginary part of the analytic signal is a Hilbert transform of s(t).
15. The system according to claim 11 , wherein:
the WVD feature is calculated by:
W
(
t
,
ω
)
=
1
2
π
∫
S
(
t
+
τ
2
)
S
*
(
t
-
τ
2
)
e
-
j
ω
t
d
τ
,
wherein S is the LPI radar signal, S* represents complex conjugate of the LPI radar signal, τ is a time lag, and ω is a frequency.
16. The system according to claim 11 , wherein:
the CWD feature is calculated by:
C
s
(
t
,
ω
,
ϕ
)
=
1
2
π
∫
∫
∫
e
j
(
ξ
μ
-
τ
ω
-
ξ
t
)
ϕ
(
ξ
,
τ
)
A
(
μ
,
τ
)
d
μ
d
τ
d
ξ
,
wherein ϕ(ξ, τ) is a kernel function, ϕ(ξ, τ)=e −ξ 2 τ 2 /σ , σ is a scaling factor, σ>1,
A
(
μ
,
τ
)
=
x
(
μ
+
τ
2
)
x
*
(
μ
-
τ
2
)
,
x(μ) is a time signal and x*(μ) is its complex conjugate, μ and τ are axes in an ambiguity domain, τ is a time lag, and ω is a frequency.
17. The system according to claim 11 , wherein:
the wavelet feature is a Morse wavelet feature and is calculated by:
W
(
j
,
k
)
=
∑
j
∑
k
s
(
k
)
ψ
j
k
(
n
)
,
and
ψ
j
k
(
n
)
=
2
-
j
2
ψ
(
2
-
j
n
-
k
)
,
wherein S(k) is the LPI radar signal, W(j, k) are wavelet transform coefficients, ψ jk (n) is a mother wavelet, j is a scale parameter, and k is a shift parameter.