Optimum Weighted Cumulation Target Detection for Resonance Region Multi-carrier Radar Peng Chen School of Information Science and Engineering Southeast University Nanjing, China Email:
[email protected]
Lenan Wu School of Information Science and Engineering Southeast University Nanjing, China Email:
[email protected]
Abstract-This paper presents an optimum weighted cumu lation target detection (OWCTD) algorithm and prewhitening
Prewhitening Processing
Multi-carrier Signal
processing matrix for resonance region multi-carrier radar, and '------
points out that the target detection performance of the wideband multi-carrier pulse radar using OWCTD algorithm is better than
�
the traditional match filter algorithm. In addition, this paper deduces that the target velocity has little effect on the detection performance of the OWCTD algorithm. Finally, the asymptotic
CFAR
target detection probability as SNR tends to infinity is given,
OWCTD
and the OWCTD algorithm performs better than the traditional match filter target detection algorithm at all target velocity.
I.
Fig. 1. a schematic representation of a multi-carrier radar target detection scenano
INTRODUCTION
Resonating radar has obtained wide attention in the study of modern anti-stealth radar [1], such as V HF radar. Due to the feature size of the current combat targets is comparable with the V HF radar wavelength, choosing resonance frequency of the stealth aircraft as the radar working frequency gives radar the anti-stealth capability when electromagnetic wave resonates with the target. This paper mainly considers the target detection of the resonating multi-carrier pulse radar. Orthogonal Frequency Division Multiplexing (OFDM) has been widely used in radar area [2], especially, the multi-carrier phase-coded (MCPC) pulse radars [3]-[6], because of the high waveform bandwidth-time (BT) product [7]. The traditional orthogonal multi-carrier radars pulse compression algorithm that lets the received signal pass through a corresponding match filter directly can obtain the maximum signal to noise ratio (SNR) pulse compression wavefonn [8], [9]. Whereas, when the radar works in the resonance region, the electro magnetic scattering coefficients of different frequencies are inconsistent. The exact values of target scattering coefficients (TSC) can't be obtained, thus it is impossible to obtain the corresponding match filter coefficients, and we must consider the TSC as a random vector in the pulse compression opera tion. Furthermore, the orthogonal condition will be destroyed in the wideband case because of the Doppler effect caused by moving target. This paper considers target detection approach in the circumstance where the distribution of TSC is known and the speed of the target is high. And proposes the optimum weighted cumulation target detection (OWCTD) algorithm to maximize the target detection probability (TDP) based on the idea of diversity gain. The paper is organized as follows: Section 2 gives the OWCTD algorithm with the knowledge of the TSC distribu-
tion; Section 3 presents system simulation results of the multi carrier pulse radar using OWCTD algorithm and compares the TDP with the traditional match filter algorithm; finally, a conclusion is given in Section 4. II.
PROBLEM DESCRIPTION AND MODELING
Figure 1 presents a schematic representation of the problem scenario. We consider a multi-carrier radar system in wideband case [10]. Assuming the MCPC radar pulse signal [10] S (Sl,S2"",SN) is sent, and =
S
=
A Q9 Sml
+
[0, l]MXN
(1
-
A)
Q9
SmO
(1)
where A E is the multi-carrier coding matrix and Q9 is Kronecker product. The number of the sub-carrier is N , and the code length of each sub-carrier is !vI . Sml and SmO are sample vectors of signal 1 and signal a in time domain, respectively.
a
The TSC vector at different frequencies is In order to facilitate the analysis, we assume that the TSC vector follows Gaussian distribution in the resonant region, i.e. , N (m,�) and the mean of the TSC m uniformly distributes within [0, 1] . In order to cancel the complexity caused by the scattering covariance matrix � , the multi-carrier radar signal should be passed through a prewhitening processing matrix firstly. Thus the covariance matrix of is
(al,a2,oo.,aNt.
a
Wa
W
{(Wa) (Wat} W�WT U UT , U is the eigenvectors [;
where � A
=
=
rv
=
A
(2) matrix and
diag {AI,A2,...,AN} is the eigenvalues matrix . In
978-1-4799-0308-5/13/$31.00 © 2013
IEEE
I
order to make the covariance matrix equal prewhitening processing matrix
, we can let the
W=UA'UT where
(3 )
When the distribution of TSC is known, (Sx)T(Sa' +n) in Formula 12 follows Gaussian distribution, therefore the mean and variance are
mHI=E {(Sxf(Sa'+n)}=(Sxf(Sm')
A'=diag {A�,A�,...,A�}
{
(13)
cr�1 =E {[(Sxf(Sa'+n) - mHI]2}
A ° A�= Jc, i # 0, Ai=°
=cr;(Sxf(Sx)+(SxfSSTSx
(14)
Then the TDP can be expressed as The echo signal through prewhitening processing and Gaussian channel is [II]
=SWa+n
(4)
r
where n N (0, cr�I) . Assuming a'= Wa (Wm, I)= N (m', I) . The constant false alarm rate (CFAR) target detection algorithm can be represented as rv
rv
HI
A (x, a') � v (x)
(5)
Ho
where v(x) is the detection threshold, and the pulse compres sion signal A(X, a') can be expressed as
A(X, a')=(Sxf
(6)
r
where x= ( Xl, X2, ... , XN f is the sub-carriers weight vector. The optimum weight vector x, which can maximize the TOP as the distribution of TSC is known, will be given in the following.
Pd(x, a)=
Pja=P ((Sxfn
2
v(x))
(7)
where (Sx)Tn follows Gaussian distribution. And the mean and variance, respectively, are
mHo =E {(Sxfn}=°
(8)
cr�o =E {[(Sxfn]2}=cr;(Sxf(Sx)
(9)
2
( )
v x
Pja=
1,00 G(t, 0, crHeJ dt=Q (v(x) -. ) 2
crHn
( )
v x
where 2
G(X, IL, cr )=
�27W ( 100 J2;=1 Q(x)= V
x
exp
_
exp
(x
-
�)
2cr
2
(10)
)
t2) ( � dt -
The threshold using CFAR target detection can be obtained as (II) When target is appeared, the TOP is
Pd(X, a)=P (A(X, a') 2 v(x) I H ) I =P ((Sxf(Sa'+n) 2 v(x))
(12)
mHI)
crHI
)
cr�(SX)TSX - (Sx)T(Sm')) Q (P Pd(X, a)=Q ( -I jah/ Jcr�(Sx)T(Sx)+(Sx)TSSTSx
(16)
Let
Q (P )Jcr�(Sx)TSx - (Sx)T(Sm') f(x)= -I ja Jcr�(Sx)T(Sx)+(Sx)TSSTSx Owing to monotonically decreasing of Q(. ) function,
(17)
finding the optimum weights vector which can maximize the TOP is equivalent to x=argminf(x) (18) x
The derivation of f(x) is
g(x)
af(x) ax
(19)
B=STS and Q�n [(xTBBx)Bx - (xTBx)BBx] + g(x) vxTBx cr; [(xTBm')Bx - (xTBx)Bm'] + [(xTBm')BBx - (xTBBx)Bm'] (20) Let x=m' , we can get Q (P )cr g(x)= ( -I ja n 1 (m'TBBm'Bm'-m'TBm'BBm') Vm'TBm' (21) The condition which can maximize TOP, i.e., g(x) = 0, is where
_
Thus the probability of false alarm is
-
(15) Substitute Formula 13 and 14 into Formula 15, we can obtain
When target does not exist, the probability of false alarm is
1,00 G(t, mHll crHJ dt=Q (V(X
)
that all nonzero eigenvalues of Bare equal. In the narrowband case, Doppler frequencies are the same among all sub-carriers, i.e., fd = 2vtfcl c , thus the eigenvalues of B are equal when the power of sub-carriers are the same. However, the frequency interval is !:J.f'= (2vt / c+ 1) !:J.f in the wideband case, which means that the orthogonal condition is no longer satisfied. Figure 2 represents the eigenvalues of B when the target velocity is 1000m/ s and the number of sub-carriers is 50. This figure depicts that the nonzero eigenvalues of Bare approximately equal in the case of wideband radar signal in this paper. So x=m' is the optimum target detection weight vector. This conclusion is similar with the maximum ratio combining, but the optimization objective here is the TOP. Finally, the CFAR target detection algorithm in Formula 5 is
HI ,-----(Sm'f � vicr�(Sm')T(Sm') Ho r
(22)
1.0 _,__----,
0.040 -,------, 0.035 -\------1
0.8 +------------7"'O�=----------1
0.030 -\------1 0.025 -\----1 ()) :::>
�
0.020
()) Cl
0.015 -\----1
jjJ
���.�.�.���.��.���.�.�����.�.�\
0.6 -1--------- �/_I_------------1 D o t-
0.4+---------+1----------------1
0.010 -1------1
0.2+--------rr------------------1
0.005 -1------1
�OWCTD ----*- traditional match filter
0.000 +--�----,-�-____r--�-_,__-�-__.-�-___r' 40 o 10 20 30 50
0.0 +--..:.=---�__.�-.--, �---.-�_,:=;::=���===l -50 -40 -20 -10 20 -30 o 10
Index
SNR(dB)
Fig. 2. eigenvalues of the matrix B when the target velocity is and the number of sub-carriers is 50
D o t-
1000m/ s
Fig. 4.
1.0 -,----,
1.0
0.8+--------------'-+;.-, -; ,;.--=------------1
0.9
0.6 +-------------.i'�------------1
0.8
0.4+---------#-----------------1
0.2 +--------;::;1�-----____==========1 _; � OWCTD(N=5)
------- traditional match filter(N=5) ----*- traditional match filter(N=10) N�=1�O'-.----" ::..: T.:;:DlT :.;: C _i_0� W � 0.0 +-�-_,__�-_,__�-_,__�-_,__---,J___,. -50 -40 -30 -20 -10 10 20
Do t-
TDP
when the target velocity is
0.7
0.6
0.5
�OWCTD traditional match filterr
-----A---
1000
2000
l
3000
TDP
when the target velocity is
Fig. 5.
0
By Formula 16 we can obtain the corresponding TOP as
Pd(a)
=
Q (j(m'))
(23 )
where !(x) is defined in Formula 17. Expect the detection probability under the mean of scattering coefficients (24) III.
NUMERICAL RESULTS
In this section, we present the numerical results to demon strate the performance of our proposed OWCTD algorithm, and the simulation parameters are set as follows: the number of orthogonal sub-carriers N 5 , the carrier frequency !e 10MHZ, the radar signal power Ps 1 , the false alarm rate Pia 5%, the sub-carrier frequency interval 6.! 100KHz, the pulse width T 1/6.! 10/.13, the variance of scattering coefficients 1 and the mean of TSC m uniformly distributes within [0, 1] . =
=
5000
6000
7000
8000
9000
10000
asymptotic
TDP
with different target velocity
has little effect on the detection probability and the OWCTD algorithm performs better than the traditional match filter algorithm which doesn't change with the TSC. And the plots also demonstrate the TOP advantage of using more sub carriers. The better performance of the OWCTD algorithm mainly due to extra knowledge of different frequencies TSC is used in the target detection process. Figure 5 illustrates the asymptotic TOP with different target velocity when SNR tends to infinity. We observe the clear advantage of using OWCTD algorithm in target detection, and these results verify the detection algorithm robust for target velocity. One way to explain this phenomenon is that the doppler frequency caused by the moving target has less effect on the TOP compared with target scattering coefficients.
=
=
=
4000
velocity (m/s)
SNR(dB) Fig. 3.
1000m/ s
(J�
=
=
=
Figure 3 and Figure 4 depict the TDP when the target velocity is 0 and 1000rn13, respectively. It is evident from these plots that, the Doppler caused by the moving target
IV.
CONCLUSION
This paper presents the prewhitening processing matrix and OWCTD algorithm under the condition that the TSC follows Gaussian distribution with different means and cor responding covariance matrix, when the radar operates in the resonant region. The target detection probability of the wideband multi-carrier pulse radar using OWCTD algorithm
is better than that of the traditional match fiIter target detection algorithm, when the distribution of target scattering coefficients is known. Furthermore, the target velocity has little effect on the detection performance. Finally, the asymptotic target detection probability as SNR tends to infinity is also given. And OWCTD algorithm performs better than the traditional match filter target detection algorithm at all target velocity. REFERENCES [1]
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