I. INTRODUCTION
Radon-Fourier Transform for Radar Target Detection (III): Optimality and Fast Implementations JI YU JIA XU, Member, IEEE YING-NING PENG, Senior Member, IEEE Tsinghua University XIANG-GEN XIA, Fellow, IEEE University of Delaware As a generalized Doppler filter bank processing, Radon-Fourier transform (RFT) has recently been proposed for long-time coherent integration detection of radar moving targets. The likelihood ratio test (LRT) detector is derived here for rectilinearly moving targets. It is found that the proposed LRT detector has the identical form as the existing RFT detector, which means that the RFT detector is an optimal detector for rectilinearly moving targets under the white Gaussian noise background. For the fast implementations of the RFT detector, instead of the joint 2-D trajectory searching and coherent integration in pulse-range domain, the 1-D fast Fourier transform (FFT)-based frequency bin RFT (FBRFT) method is proposed in the pulse-range frequency domain without loss of integration performance. Moreover, at the cost of a controllable performance loss, a suboptimal approach called subband RFT (SBRFT) is also proposed to reduce the storage memory. It is shown that not only the long-time coherent integration gain can be obtained via the proposed SBRFT, but also the computational complexity and memory cost can be reduced to the level of the conventional Doppler filter banks processing, e.g., moving target detection (MTD). Some numerical experiments are also provided to demonstrate the effectiveness of the proposed methods. Manuscript received December 17, 2009; revised July 16, 2010; released for publication March 17, 2011. IEEE Log No. T-AES/48/2/943799. Refereeing of this contribution was handled by C. Baker. This work was supported in part by China National Science Foundation under Grant 60971087 and 60925005, China Ministry Research Foundation under Grant 9140A07011810JW01 and 9140C130510DZ46, Aerospace Supporting Foundation under Grant J04-2007047, China Aerospace Innovation Fund under Grant CASC200904, and China Aviation Science Foundation under Grant 20080158001. Dr. Xia’s work was supported in part by the Air Force Office of Scientific Research (AFOSR) under Grant FA9550-08-1-0219 and DEPSCoR Grant W911NF-07-1-0422 through ARO. Authors’ addresses: J. Yu, J. Xu, and Y-N. Peng, Department of Electronic Engineering, Tsinghua University, Beijing 100084, P.R. China, E-mail: (
[email protected]); X-G. Xia, Department of Electrical and Computer Engineering, University of Delaware, Newark, DE 19716. c 2012 IEEE 0018-9251/12/$26.00 °
With the compensation of the target’s phase difference among multiple radar pulse samplings, it is known that coherent integration is widely used for coherent search radar, alone or together with incoherent integration, to increase signal-to-noise ratio (SNR) for detection [1, 21—22]. Nevertheless, there are two factors that may affect the coherent integration performance for the conventional radar with either mechanical or electronic scanning beams. First, in order to ensure the radar spatial and temporal overage of the whole region of interest (ROI), the target’s dwell time in a single scanning beam is normally limited. Second, for the conventional Doppler filter bank method, e.g., moving target detection (MTD) [2], the integration is implemented in a single range cell and the integration time is also limited for moving targets. Under the above two factors, only the phase difference, i.e., Doppler frequency, is compensated for the MTD method, while the target’s range walk is ignored among pulse samplings [3]. Hence, the ultimate integration performance is limited. Recently, the rapid development of radar systems makes the first factor no longer a limitation, which utilizes the wide transmitting beam and multiple simultaneous narrow receiving beams to cover the whole ROI [4—6]. All the T/R beams do not need scanning and the dwell time of a moving target may be remarkably increased. However, during the long-time illumination, the moving targets may easily cause the across range unit (ARU) effect. Thus, the envelope shift effect as well as the Doppler phase modulation should be compensated, simultaneously. It therefore becomes urgent to propose the effective methods to overcome the limitations from the above second factor for long-time integration. The early studies of ARU compensation were focused on incoherent integration. In [7], [8], the geometric aspects of long-term incoherent integration were investigated in detail. Also, simultaneous incoherent scan-to-scan integration was proposed for searching radar in [9]—[11] with ARU compensation for multiple targets based on the Hough transform (HT) . Furthermore, the constant false alarm ratio (CFAR) property of the HT detector [12] and its efficient approaches [13] were also studied. Compared with incoherent integration, coherent integration with ARU compensation can deal with both Doppler phase and envelope modulation. Therefore, target detection based on coherent integration is always an important attractive direction in radar signal processing. In [14], integration accounting for ARU in range-Doppler data space was proposed. Besides, a method based on keystone transform (KT) was introduced by Perry, et al. in SAR systems [15, 16] for blind compensation of ARU and then it was introduced into the searching radar detection [17—18]. Though the coherent gain can
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be obtained by KT, the data interpolation operators and the Doppler ambiguity problem [19] may cause some performance loss. We have recently proposed Radon Fourier transform (RFT) [20, 21] to realize the long-time coherent integration with ARU compensation for searching radar. Different from the existing Radon or Fourier transform methods [25, 26], RFT can effectively overcome the internal coupling between the range walk and Doppler phase modulations for coherent radar moving targets. In [20] RFT is proposed as a generalized Doppler filter bank processing based on the comparisons between RFT and the conventional MTD on five aspects, i.e., coherent integration time, filter bank structure, blind speed response, detection performance, and computational complexity integration. As RFT can be regarded as a kind of generalized Doppler filter bank, it is natural that RFT also has a similar and more effective clutter suppression ability than the conventional MTD method [20]. Although one of the four integration expressions, i.e., the form with the polar angle and polar distance, can be regarded as generalized coherent Radon transform [30], the integration form with the velocity and range in [20] has better physical meaning to demonstrate the internal coupling between the range walk and Doppler phase modulations. Also, the property analysis and blind speed sidelobe (BSSL) suppression methods are investigated in [21] for RFT. Therefore, it is expected to employ the RFT-based detector in search radar for long-range, low-SNR, and high-speed air-borne target detection and tracking applications [20]. However, the optimality of this method is not discussed in [20], [21] and the complexity of standard RFT [20, 21] is higher because the filter bank cannot be directly implemented based on fast Fourier transform (FFT). In this paper, the optimality and fast implementation of the RFT detector are investigated. It is shown that a standard RFT processor is a group of likelihood ratio test (LRT) detectors. Besides, two efficient approaches of the RFT detector are proposed to reduce the implementation complexity. The former, called the frequency bin RFT (FBRFT) based detector, can combine the conventional pulse compression (PC) processing into RFT without performance loss. Also, the computational complexity can be reduced to the level of the conventional MTD method based on FFT operations. To reduce the memory demand of FBRFT, the latter, called the subband RFT (SBRFT) based detector, is further proposed, which utilizes the DFT coefficients corresponding to the subband central frequency. It is shown that the SBRFT detector can effectively reduce memory demand with an acceptable performance loss. The remainder of this paper is organized as follows. In Section II the optimality of the RFT 992
detector is verified with the derivation of the LRT detector. In Section III two efficient approaches, i.e., FBRFT detector and SBRFT detector, are proposed in detail. In Section IV, the computational complexities of standard RFT detector, FBRFT detector, and SBRFT detector are discussed based on numerical experiments. In Section V some conclusions are drawn. II.
SIGNAL MODEL AND OPTIMALITY OF FRT-BASED DETECTOR
In this section, the radar target signal model with linear ARU is recalled, and the optimality of the RFT detector is derived. A. 2-D Signal Models for PD Searching Radar Assume that a transmitted baseband signal of coherent radar is p(t) (0 < t < Tp ) where Tp is the pulse duration. The baseband echoes for the mth transmitted pulse are s(t, m) = Ap(t ¡ 2Rm =c) exp(¡j4¼Rm =¸) exp(j'), 0 < t < Tp
(1)
where Tr is the radar pulse repetition interval, c is the light propagation speed, ¸ = c=fc is radar wavelength and fc is the carrier frequency, Rm is the range between target and radar in the mth pulse duration, and ' is the uncertain initial phase of an echo. Although the coherent radar can obtain the relative phase during integration time, the absolute initial phase ' is normally unknown and assumed uniformly distributed [1]. Suppose that the radial velocity of a moving target is v, then the range between the target and radar is Rm = R0 + vmTr (2) where R0 is the slant range between target and radar when m = 0. Suppose radar sampling frequency is fs along range, the sample at t = nTs of the mth pulse with the given initial phase ' can be given as μ ¶ μ ¶ 2R0 2vmTr R0 smn (') = Ap nTs ¡ ¡ exp ¡j4¼ c c ¸ μ ¶ vmTr £ exp ¡j4¼ exp(j') ¸ = Asmn (0) exp(j')
(3)
where Ts = 1=fs . and A is the echo amplitude. Due to the relatively low range resolution for search radar, it is reasonable to assume that the target is an ideal point target and that phase change of echoes is mainly due to the relative geometry of radar and target [1—3]. However, the adverse effect of nonideal factors, such as a 10—20% error in received phase, can lead to the obvious coherent integration loss. We refer the reader interested in this topic to [28], [29].
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Apart from the target’s echoes, the system noise also exists. Herein, the mixture of thermal noise and prewhitened clutter is modeled as white Gaussian noise wmn for simplicity as wmn » N(0, ¾ 2 )
(4)
where ¾2 is the power spectrum density of the mixture noise. Therefore, the radar target detection can be formed as a binary decision problem as xmn = wmn ,
H0
xmn = smn + wmn ,
(5) as L(X) =
(6)
where X ¢ ¢ ¢ = fx11 , x12 , : : : , xMN g is 1 £ MN vector of received samples. Due to the independently identically distributed (IID) assumption of the background noise, the probability density function (pdf) of the vector X under hypothesis H0 is à ! M N 1 1 XX ¤ xmn xmn : p(X; H0 ) = MN 2MN exp ¡ 2 ¼ ¾ ¾ m=1 n=1
(5) H1
p(X; H1 ) H1 ?° p(X; H0 ) H0
(7)
The conditional pdf of hypothesis H1 is
( ) M N 1 1 XX p(X j '; H1 ) = exp ¡ 2 [xmn ¡ Asmn (0) exp(j')][xnm ¡ Asmn (0) exp(j')]¤ : ¼MN ¾2MN ¾
(8)
m=1 n=1
where xmn are the received echo samples at t = nTs of the mth pulse (m = 1, : : : , M; n = 1, : : : , N), H0 and H1 are the hypotheses target and no target, respectively. B. Optimality of RFT Detector In this subsection LRT detectors for target with linear ARU are derived based on (5), and the
L(X) = E
μ
p(X; H1 ) p(X; H0 )
Z
Then the pdf of X under hypothesis H1 is Z p(X; H1 ) = p(X j '; H1 )p' (')d'
(9)
where p' (') is the pdf of initial phase ', which is uniformly distributed in [0, 2¼). Therefore, L(X) can be given as
¶
2¼
p(X j '; H1 ) d' p(X; H0 ) 0 ( ) Z 2¼ M N 1 1 XX ¤ = exp ¡ 2 [(xmn ¡ Asmn (0) exp(j'))(xmn ¡ Asmn (0) exp(j')) ¤ ¡xmn xmn ] d' 2¼ 0 ¾
=
1 2¼
m=1 n=1
1 = 2¼
Z
0
2¼
"
( )! # Ã M N M X N X 1 XX 2A ¤ exp ¡ 2 jAsmn (0)j2 exp ¡ 2 Re exp(¡j') xmn smn (0) d' ¾ ¾ m=1 n=1
"
M
N
1 1 XX = jAsmn (0)j2 exp ¡ 2 2¼ ¾ m=1 n=1
m=1 n=1
#Z
0
2¼
Ã
(
M
N
XX 2A ¤ exp ¡ 2 Re exp(¡j'0 ) jxmn smn (0)j exp(j'0 ) ¾ m=1 n=1
¯M N ¯! # à " M N ¯ 1 2A ¯¯X X 1 XX ¯ 2 ¤ = jAsmn (0)j I0 x s (0) exp ¡ 2 ¯ ¯ mn mn ¯ 2¼ ¾ ¾2 ¯ m=1 n=1
)!
¯ ¯ d' ¯ ¯ 0¯
'0 ='+'0
(10)
m=1 n=1
equivalence is clearly demonstrated between the derived LRT and the proposed RFT in [20], [21]. Since the LRT detector is the optimal detector in statistics, the optimality of the RFT detector follows. The optimal detector based on the likelihood ratio should have the form for the hypothesis test
PN P ¤ where '0 = ] M m=1 n=1 xmn smn (0), ] is the phase angle operator, Ref¢g represents the complex real operator, and I(¢) is the modified Bessel function of the first kind. Because the first term in the right hand side of (10) depends on the constant energy of the target and I(¢) is a monotonic increasing function, the
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equivalent form of L(X) is ¯M N ¯ ¯X X ¯ ¯ ¯ ¤ xmn smn (0)¯ TLRT (R0 , v) = ¯ ¯ ¯ m=1 n=1 ¯M N ¯ ¯X X ¯ H0 ¯ ¯ (mn) =¯ xmn hLRT (R0 , v)¯ ? °LRT (11) ¯ ¯ H1 m=1 n=1
where the °LRT is the threshold and the matching coefficient hmn (R0 , v) is the conjugation of echo smn of the moving target with parameters R0 and v, i.e.,
(mn) ¤ hLRT (R0 , v) = smn (0) μ ¶ μ ¶ 2R 2vmTr R = p¤ nTs ¡ 0 ¡ exp j4¼ 0 c c ¸ μ ¶ vmTr £ exp j4¼ : (12) ¸
n0 (m) = round[(2R0 + 2vmTr )=cTs ] μ ¶ 4¼vmTr (m) hRFT (v) = exp j : ¸
0 smn 0 (m) =
m=1
994
μ
2(r + 2vmTr ) cTs
¶¶
μ
exp j
4¼vmTr ¸
¶
(15)
xmk p¤ [(k ¡ n0 (m))Ts ]:
(16)
Substituting (16) into (13) yields TRFT (R0 , v)
¯ M N ¯ ¯X X ¯ ¯ ¯ (mn) =¯ xmn hRFT (R0 , v)¯ ¯ m=1 n=1 ¯
¯ M N μ ¶¯¯ H ¯X X 4¼vmT ¯ ¯ 0 r =¯ x p¤ f[n ¡ n0 (m)]Ts g exp j ¯ ? °RFT ¸ ¯ m=1 n=1 mn ¯ H1
(17)
(mn) where the 2-D weighting coefficients hRFT (R0 , v) of the RFT detector can be written as μ ¶ (mn) hRFT (R0 , v) = p¤ (nTs ¡ 2R0 =c ¡ 2vmTr =c) exp j
μ
(mn) = hLRT (R, v) exp ¡j
Grv (r, v)
μ
N X k=1
DEFINITION 2 When the line equation rs = r + vt is used, theR continuous standard RFT is defined as 1 Grv (r, v) = ¡1 f(t, r + vt) exp(j2¼"vt)dt, which is called “standard RFT” due to the explicit meaning of parameter r and v. Also, the discrete RFT is given as
srm m, round
(14)
In fact, the delayed and filtered output after PC in (13) can be expressed as
DEFINITION 1 Suppose a 2-D complex function f(t, rs ) 2 C is defined in (t, rs ) plane and a parameterized line equation rs = Á(®1 , ®2 ) + '(®1 , ®2 )t is used for arbitrarily searching lines in the plane, where Á( ) and '( ) are two specific functions to determine a line with parameters ®1 and ®2 , then the continuous RFT is defined as G®1 ®2 (®1 , ®2 ) = R1 ¡1 f(t, Á(®1 , ®2 ) + '(®1 , ®2 )t) exp(j2¼"'(®1 , ®2 )t)dt, where " is a known constant with respect to f(t, rs ).
M X
m=0
0 0 where smn 0 stands for the n th sample in the mth pulse’s echo after PC, and
It is obvious that the above derivation of the LRT is similar to the conventional short-time coherent integration method in [23]. However, the significant difference lies in that the signal model discussed in this paper takes the consideration of the linear ARU effect. Also, the proposed LRT detector as in (11) can compensate the range-Doppler coupling and deal with ARU simultaneously. This 2-D coherent integration indicates that the traditional short-time integration procedure, i.e., PC in range dimension and MTD in pulse dimension, is not an optimal approach in long-time integration with linear ARU effect. As a generalized Doppler filter bank processing, it is shown that RFT [20—21] can obtain the excellent performance of noise suppression and target detection via coherent integration. In this paper we recall the continuous RFT in [20] in the following two definitions.
=
According to the above definitions, the processing procedure of RFT detector for radar echoes is given as follows. In the first step, the conventional PC is implemented for all pulse of echoes. In the second step, the integration result of RFT is calculated for a given range parameter R0 and a given velocity parameter v. Interestingly, by comparing with the above definitions, it is easily found that the optimal LRT detector as in (11) is equivalent to the absolute output of discrete standard RFT detector. To clearly demonstrate the equivalence, we can further rewrite RFT detector in a discrete form as ¯M ¯ ¯X ¯ H0 ¯ ¯ (m) 0 smn0 (m) hRFT (v)¯ ? °RFT (13) TRFT (R0 , v) = ¯ ¯ ¯ H1
:
4¼R0 ¸
¶
:
4¼vmTr ¸
(18)
This shows that the LRT detector weights differ from the RFT weights with only a phase. However, because only the amplitude information is used for the linear law detector, this difference does not affect the
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ultimate result of the detectors, i.e., ¯M N ¯ ¯X X ¯ ¯ ¯ (mn) xmn hLRT (R0 , v)¯ TLRT (R0 , v) = ¯ ¯ ¯ m=1 n=1 ¯M N μ ¶¯ ¯X X 4¼R0 ¯¯ ¯ (mn) =¯ xmn hRFT (R0 , v) exp j ¯ ¯ ¯ ¸ m=1 n=1
= TRFT (R0 , v):
(19)
This equation tells us that the RFT detector as in (13) is an equivalent detector of the LRT of (11). So the RFT detector is the statistically optimal detector for rectilinearly moving targets under a white noise background with an unknown target’s initial phase. As the noise is assumed to be white and Gaussian distributed, the detection threshold of the RFT detector and the RFT detector can be analytically obtained by a given false alarm probability. Under the hypothesis H0 , both real and imaginary parts PN P (mn) of M m=1 n=1 xmn hRFT (R0 , v) are independent and Gaussian distributed. So the testing statistic of the LRT detector, T = TRFT (R0 , v) is under the Rayleigh distribution, i.e., μ ¶ 8 2 < 2T exp ¡ T E¾ 2 pT (T j H0 ) = K¾2 (20) : 0, P PM ¤ where K = N n=1 m=1 snm (0)snm (0) = NM. Then the false alarm probability of (17) can be obtained by ³ ° ´ (21) PFA = exp ¡ RFT 2 : MN¾
However, the signal model here is 2-D and the target ARU effect is considered for RFT. III. EFFICIENT APPROACH OF RFT DETECTOR Under the target model with ARU effect as in (3), it is proven that the RFT detector is also an optimal LRT detector in the above section. However, it should be noted that the range parameter and the velocity parameter of targets are unknown for search radar applications. So the integrated results of the second step should be calculated many times in order to search targets with all possible combinations of range and velocity parameters. It is shown that the complexity of the RFT detector as in (13) is much higher than the conventional PC and MTD methods in [20]. Furthermore, the computational complexity of RFT detector may increase significantly with the increase of pulse number in the long-time integration time. In this section two new efficient approaches are proposed to reduce the computational complexity of the RFT detectors. The first approach, i.e., the FBRFT detector, reduces the computational complexity order of RFT to the level of conventional MTD detectors based FFT operators. In order to further reduce the memory requirement, the suboptimal subband approach, i.e., SBRFT detector, is further proposed. A. Frequency Bin Approach of RFT Detector (FBRFT Detector) According to (13) and (16), the test statistics of RFT after PC are
¯M ¯ μ ¶ ¯X ¯ 4¼vmTr ¯ ¯ ¤ 0 TRFT (R, v) = ¯ exp j IFFTn fFFTn (xmn )FFTn fp [¡(n ¡ n (m))Ts ]gg¯ ¯ ¯ ¸ m=1 ¯M ¯ μ ¶ ¯X ¯ 4¼vmTr ¯ ¯ =¯ exp j IFFTn fFFTn fxmn gFFTn fp¤ (¡nTs )g exp(j2¼fn n0 (m))g¯ ¯ ¯ ¸
(23)
m=1
Therefore, the threshold can be further determined as °RFT =
q ¡MN¾ 2 ln PFA :
(22)
It should be noted that the above derivation of the RFT is similar to the threshold calculation in [23].
where fn is the fast frequency of baseband waveform p(t). The first equation holds due to the relationship between the convolution operator and the FFT operator and the second equation holds due to the time shift property of FFT. Substituting (14) into (23) and ignoring the integer round operation, the test statistics are
¯M ¯ μ ¶ ¯X ¯ 4¼vmTr ¯ ¯ TRFT (R, v) = ¯ exp j IFFTn fFFTn fxmn gFFTn fp¤ (¡nTs )g exp(¡j4¼fn R0 =c) exp(j4¼fn vmTr =c)g¯ ¯ ¯ ¸ m=1 ¯M ¯ μ ¶ ¯X ¯ 4¼vmTr ¯ ¯ ¤ =¯ exp j (24) IFFTn fFFTn fxmn gFFTn fp (¡nTs )g exp(j4¼fn vmTr =c)g¯ : ¯ ¯ ¸ m=1
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Furthermore, by exchanging FFT operation and taking a summation, the detector can be rewritten as
is the compensation coefficient and ¢v is the resolution of velocity. The results of (28) for
¯ (M )¯ μ ¶ ¯ ¯ X fc + fn ¯ ¯ ¤ TRFT (R, v) = ¯IFFTn exp j4¼v mTr FFTn fxmn gFFTn fp (¡nTs )g ¯ : ¯ ¯ c
(25)
m=1
According to (25), the pulse integration does not follow the conventional PC processing in range-pulse dimension. Instead, the pulse integration is implemented in range frequency-pulse domain. Due to the combination of the PC step and the pulse integration step, the computational complexity can be reduced significantly without performance loss. The efficient approach of the RFT detector, called the FBRFT detector, is optimal under the Neyman-Pearson (N-P) principle based on the 2-D signal model and can be implemented among frequency bins as in (25) by 1-D FFT and inverse FFT (IFFT) in range dimension and 1-D DFT in pulse dimension. The benefits of the FBRFT detector in reducing computational complexity will be obvious for searching radar which detects multiple targets with different ranges and velocities in the same time. Suppose that M pulse samplings and Nv £ Nr range-velocity resolution cells of TRFT (R, v) are to be tested, where Nr is the range cell number and Nv is the velocity cell number, the step-by-step of FBRFT detector can be given as follows. Step 1 The FFT operation is used in the range dimension for M pulses, i.e., Xmnˆ = FFTn fxmn g:
(26)
Step 2 The frequency response of range matched filter is multiplied for M pulses, respectively, i.e., Ymnˆ = Hmnˆ Xmnˆ
(27)
where Hmnˆ = FFTn fp¤ (¡nTS )g is the frequency domain matched function of the transmitted baseband waveform p(¢). To further improve the range sidelobe performance, the conventional weighted processing can be used as Hmnˆ = W(n)FFTn fp¤ (¡nTs )g, and W(n) can be the Hamming, Blackman, or other functions. Step 3 The phase difference across pulses is compensated and summed along the pulse dimension for each range frequency cell, i.e., Zknˆ =
M X
Cnˆkm Ymnˆ
m=1
where
996
μ ¶ f + fnˆ Cnˆ = exp j4¼¢v c Tr c
(28)
k = ¡Nv =2 + 1, 2, : : : , Nv =2 can be calculated via fast chirp-Z transform (CZT) [24] as Zknˆ =
M X
(1=2)[m2 +k 2 ¡(m¡k)2 ]
Cnˆ
Ymnˆ
m=1 (1=2)k 2
= Cnˆ
M X
¡(1=2)(m¡k)2
Cmˆ
(1=2)m2
(Cnˆ
Ymnˆ )
m=1 (1=2)k 2
= Cnˆ
¡(1=2)m2
(Cnˆ
(1=2)m2
− (Cnˆ
Ymnˆ ))
(29)
where − is the convolution operation. So the FFT operation can be applied for reducing computational complexity, i.e., (1=2)k2
Zknˆ = Cnˆ
¡(1=2)m2
IFFTm fFFTm fCnˆ
(1=2)m2
g ¢ FFTm fCnˆ
Ymnˆ gg
(30) (1=2)k 2 (k = ¡Nv =2 + 1, 2, : : : , where the coefficients Cnˆ ¡(1=2)m2 max Nv =2, M) and FFTm fCnˆ g (m = ¡Nv =2, : : : ,
Nv =2 + M) can be calculated off-line. Note that the length of the FFT operation should be more than Nv + M to make the multiplication of FFT results in (30) equal to the original convolution result in (29). Step 4 The test statistics are obtained for Nv as Tkn = jIFFTnˆ fZknˆ gj:
(31)
It is noted that step 1, step 2, and step 4 correspond to the steps of the conventional PC, respectively. Furthermore, the complexity of step 3 is of the order of O(J log J), (J = Nv + M). It is much less than the order O(M 2 ) of the standard RFT detectors and close to the order O(M log M) of the conventional FFT-based MTD approach. Though the real computational cost may be different for different processing platforms, the low computational complexity of the proposed method can be normally obtained. The detailed discussion of the computational complexity is given in Section V. B. Suboptimal Subband Approach of RFT Detector (SBRFT Detector) The proposed FBRFT detector can reduce the computational complexity significantly without loss of performance. However, it also causes the larger memory requirement in step 3 as the coefficients differ in different frequency bins. The twiddle factors of J-point FFT and Nr groups of off-line ¡(1=2)m2 compensation coefficients FFTm fCnˆ g should
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be calculated and stored off line. The memory requirement of FBRFT is (Nr + 1)(Nv + M) + Nr (Nv =2 + maxfNv =2, Mg) complex numbers, which is about Nr times of the FFT-based MTD methods. Hence, a suboptimal approach, named subband frequency RFT detector (SBFRT-based detector) is proposed in this paper to reduce the memory requirement. The basic idea of the SBFRT-based detector is to divide the bandwidth B of waveform p(t) into L subbands with bandwidth B=L. So the Nr frequency bins of each pulse after FFT operation (24) are divided into L groups in range frequency domain (let Nr be multiple of L). For all Nr =L frequency bins of the lth group of frequency bins, the same coefficients corresponding to the central frequency, i.e., μ ¶ f ¡ fl Cl = exp j4¼¢v c Tr , c are used for compensation of (28). The above approximation can be expressed as (1=2)k 2
Zknˆ = Cl
¡(1=2)m2
IFFTmˆ fFFTm fCl
(1=2)m2
g ¢ FFTm fCl
Ymnˆ gg
(32) where nˆ = lNr =L, lNr =L + 1, : : : , (l + 1)Nr =L ¡ 1 and l = 0, 1, : : : , L ¡ 1. It is certain that the above approximation can degrade the performance while reducing the number of off-line coefficients by Nr =L times with L subband. When L is equal to Nr , no approximation is introduced. For this case, the SBRFT is identical to the FBRFT and there is no memory requirement reduction. It is evident that the choice of subband number L is a key tradeoff between the performance optimality and the computational reduction. Generally speaking, we should choose L as infrequently as possible under the acceptable performance degradation. Practically, just like the conventional MTD application, the performance degradation may be acceptable when target motion is constrained in one range cell. So the choice of L of (32) should make sure that the possible target motion is within the same range cell for each subband, i.e., vmax MTr < ¢Rsub (L)
(33)
where ¢Rsub (L) is the range resolution of subband wave and vmax is the possible maximum velocity of targets. And if p(t) is the LFM waveform pulse with bandwidth B and duration Tp , ¢Rsub (L) = cL=(2B) and the minimum L leading to the maximum acceptable degradation is » ¼ 2vmax MTr B Lmin = (34) c where d¢e means the cell round operator of integer. The benefits of SBRFT can be obtained when Lmin is less than BTp , i.e., the time-bandwidth product of p(t).
Therefore, the memory reduction ratio ´ of SBRFT can be given as » ¼ 2vmax MTr B c ´ = 1¡ > 0: (35) BTp Furthermore, the above estimation can be simplified as ´ = 1¡2
vmax MTr : c Tp
(36)
It is interesting that the bandwidth does not decide the application condition of the SBRFT detector, which depends on the relation between the ratio of the maximum target velocity and radio speed, i.e., vmax =c, and the duty ratio of pulse duration and integration time, i.e., Tp =Tc . Taking the target with the 3 March maximum velocity for example, the reduction ratio of the memory requirement of step 3 can be as much as 83% for a search radar with Tp = 40 us, B = 5 MHz, and Tc = 1 s. IV. NUMERICAL SIMULATION AND COMPUTATIONAL COMPLEXITY ANALYSIS A. Simulation Results To demonstrate the effectiveness of the FBRFT detector and the SBRFT detector, some numerical simulations are presented. The experimental parameters are given as follows: transmitting LFM pulse duration Tp = 10 us, the bandwidth B = 10 MHz, pulse repetition interval Tr = 1 ms and carrier frequency fc = 3:0 GHz. The radial velocity of the moving target is supposed to be about 1 March, i.e., vtarget = 300 m/s. The range units of the range gate Nr = 512. The integration pulse number is 512, i.e., Tc ¼ 0:5 s. Clearly, the target will move across about 10 range units during the integration time and the ARU effect of the target is obvious. Firstly, the 512-pulse echo signals of such target, which are presented in format 2-D data matrix, are generated according to (3). In order to compare the integration effect of the proposed methods with the conventional MTD method, noise is not contained in echoes . Then the echo data are processed by the conventional MTD, standard RFT, FBRFT, and SBRFT with their own searching parameters, respectively. As shown in Fig. 1, the integration results are obtained in the range-Doppler plane for the MTD method and the range-velocity plane for the other three methods. Fig. 1(a) indicates that the integrated results of the conventional MTD cannot be focused in a single range-Doppler unit and the peak value is only about 5000. It is because MTD is just for the MTD without ARU during the integration time. Fig. 1(b) gives the integrated results of the standard RFT. The searching velocity scope is [¡400 m/s, 400 m/s] with spacing of 0.1 m/s. It is evident that the energy is focused at peak location of
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Fig. 1. (a) Integrated results of conventional MTD method. (b) Integrated results of standard RFT. (c) Integrated results of FBRFT. (d) Integrated results of SBRFT.
(0 m, 300 m/s) with value about 4:4 £ 104 . With the same searching velocity parameters, the integrated results of the FBRFT are illustrated in Fig. 1(c). We can see that the FBRFT result is the same as the standard RFT and the peak value is about 4:9 £ 104 . The output peak value of FBRFT is higher than the standard RFT output due to the error of the rounding operation as in (13). However, the FBRFT can avoid this kind of error to obtain better integration results via the frequency domain processing. In Fig. 1(d) the influence of the approximation of the SBRFT is illustrated. In the simulation the subband number is set to be 16 according to (34), i.e, L = 16. We can see that the output of SBRFT is similar to the outputs of standard RFT and FBRFT. But the peak value is about 3:97 £ 104 and less than those of FBRFT and standard RFT. However, according to (35), more than 90% of coefficient memory is saved in the cost of performance degradation. This advantage makes RFT feasible in some memory-limited systems. From the integration results of the same echoes, it is evident that the integration performances of RFT, FBRFT, and SBRFT are much better than the conventional MTD method when the integration pulse number is large. The echoes of moving targets with the migration of 998
10 range cells can be integrated as well as standard RFT method. Besides, although the integrated peak of SBRFT is less than those of standard RFT and FBRFT, the additional benefit of memory cost is obtained. The detection performances of standard RFT detector, FBRFT detector, and SBRFT detector are further investigated by Monte Carlo trials. The IID Gaussian noises as in (4) are added to the target echoes. 512 pulses are processed and the velocity searching scope is [¡400 m/s, 400 m/s] with spacing of 5 m/s. Other parameters are the same as Fig. 1. The false alarm ratio is set as PFA = 10¡6 . The detections for each trial under different SNR from ¡44 dB to ¡24 dB are performed at the same threshold. Then the detection probability Pd is calculated. The results of Pd are plotted in Fig. 2. The detection probability of a single pulse after PC is also given for a benchmark. The simulation results show that the detection probability of the RBRFT detector is the same as the standard RFT detector, and the SBRFT detector with subband number of 16 performs worse than the standard RFT detector with SNR loss of about 2.5 dB. Furthermore, when Pd = 0:5, the required input SNR of RFT-based and FBRFT-based detectors is about ¡35:5 dB while the required input
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Fig. 2. Detection probability curves as input SNR (pulse number is 512).
Fig. 3. Detection probability of SBRFT detector for different subband number L.
SNR of single pulse detection is about ¡8:7 dB. So the pulse integration gain is about 26.8 dB, which is close to the theoretical SNR gain of 27.1 dB. Furthermore, the performance degradation of the SBRFT detector versus subband number L is illustrated in Fig. 3. With the above system parameters, the detection probabilities Pd = 0:5 of the SBRFT detectors with varied L of 2, 4, 8, 16, 32, 128, 512 are simulated, respectively. It is clearly shown that when L · 16, the required SNR of SBRFT is remarkably higher than that of L = 512. For example, with the same detection probability Pd , the required SNR for L = 8 is about 10 dB higher than that for L = 512 in Fig. 3. That is, the performance loss of the SBRFT detector cannot be acceptable when
L · 16. Fortunately, the required SNR of SBRFT can approach that of L = 512 when L > 16. That is, the SBRFT-detector can perform well in this case. The simulated results match well the result of (34) in Section IVB, i.e., Lmin is about 14 according to simulation parameters. B. Computational Complexity Comparison As discussed in Section IV, the FBRFT detector and the SBRFT detector can reduce the computational complexity of RFT to the level of the conventional FFT-based MTD methods. Due to the same steps of these two approaches, the computational complexities of both approaches are also identical. For a searching radar, suppose that Nv £ Nr range-velocity resolution
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TABLE I Computation Complexity Analysis Step
Processing
Real Operation Number
Eq. (26)
M groups of Nr -point FFT
0:5MNr log2 (Nr )Im + MNr log2 (Nr )Ia
Eq. (27)
MNr -point complex multiplication
Nr MIm
Eq. (30)/(32)
Nr groups of (Nv + M)-point fast CZT
Nr [2J + J log2 (J)]Im + Nr (2L log2 (J))Ia
Eq. (31)
M groups Nr -point FFT operation and Nr Nv -point modular operation
0:5(MNr log2 (Nr ) + Nr Nv )Im + MNr log2 (Nr )Ia
Note: Im : complex multiplication; Ia : complex addition. TABLE II Computational Complexity of All Methods Methods RFT
MTD
FIR-Based Nr
μ
FFT-Based
(M log2 (Nr ) + M + Nv M +
1 2 Nv )Im
+(2M log2 (Nr ) + Nv M ¡ Nv )Ia
MNr
μ¡
log2 (Nr ) +
3 2
¢
+ M Im
+(2 log2 (Nr ) + M ¡ 1)Ia
¶
Nr
ó
M log2 (Nr ) + M + 2J + 4J log2 (J) +
Nv 2
+(2M log2 (Nr ) + 2L log2 (L))Ia
¶
MNr
μ
(log2 (Nr ) +
3 2
+ 12 log2 (M))Im
+(2 log2 (Nr ) + log2 (M))Ia
´ ! Im
¶
TABLE III Complexity of MTD and RFT in Real Operation Methods
FIR-Based
FFT-Based
RFT
Nr (10M log2 Nr + 8MNv + 6M + Nv )
Nr [5J log2 Nr + 18M + 15Nv + 10J log2 (J)]
MTD
Nr M(10 log2 Nr + 8M + 7)
Nr M(10 log2 Nr + 5 log2 M + 9)
cells of TRFT (R, v) should be calculated, where Nr and Nv are the range cell number and velocity cell number, respectively. Then, the step-by-step computational complexity may be given for the FBRFT/SBRFT detector in Table I where J = M + Nv . For comparison, the computational complexity of FBRFT is listed in Table II with the finite-impulse response (FIR)-based RFT detector [20—21], the FIR-based MTD detector, and the FFT-based MTD detector, where the conventional PC is also included, which is different from [20]. Furthermore, the quantity of floating-point operations numbers, which means the real calculation operation number of the given algorithms, is used to evaluate the complexity quantitatively in Table III. In the analysis herein, the real multiplication operation and real addition operation are calculated as one real floating-point operation because both of them need only one cycle for the modern digital signal processors (DSPs) [27]. Therefore, one complex addition is equal to two real floating-point operations while one complex floating-point multiplication is equal to six real floating-points operations. Also, we choose the common 2-radix FFT implementation for all detectors. Fig. 4 illustrates the computational complexities of the above four methods. The range unit number Nr is 1024. The velocity channel number Nv is supposed 1000
to be equal to M. The computation amount curves for different pulse numbers, i.e., M = 16, 32, 64, 128, 256, 512, 1024, are plotted. It is evident that the computational complexities of FIR-based MTD and FIR-based RFT are much larger than other two FFT-based approaches. The FBRFT or the SBRFT method reduces the computational complexity significantly, which is slight larger than the FFT-based MTD method. For M = 1024, the real operation numbers of FBRFT or SBRFT are about 4.25% of the standard RFT detector. For the conventional MTD methods, Nv = M is valid due to the Doppler ambiguity. However, for RFT-based methods, Nv is usually larger than M because different velocities mean different ARU effects on the complex envelope of echoes. For a searching radar using low pulse-repetition frequency (PRF), Nv is often some multiple of M, i.e., K = Nv =M is an integer. Fig. 5 illustrates the computation amount curves in terms of K when M = 1024. We can see that the real operation number increases with K for standard RFT and FBRFT/SBRFT, respectively. In these cases, the complexity of the proposed method is also less than standard RFT, too. V.
CONCLUSION
Based on the derivation of the LRT detector under the 2-D rectilinearly moving target signal model, it is
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Fig. 4. Computation amount versus integration pulse number M.
Fig. 5. Computation amount versus ratio of Nv =M.
shown that the LRT detector has an equivalent form of the standard RFT detector. Furthermore, two efficient approaches, i.e., the FBRFT detector and the SBRFT detector, are proposed to reduce the computational complexity of FIR-based RFT detector. Compared with the conventional MTD and standard RFT methods, the proposed methods may jointly realize the intrapulse PC and inter-pulse integration in an efficient way. The step-by-step complexity analysis shows that the FBRFT detector can reduce the complexity order from O(Nr Nv2 ) to O(Nr J log2 J) without loss of integration performance. The SBRFT detector can further reduce memory with a slight performance
loss. Finally, the numerical experimental results are also provided to demonstrate the effectiveness of the proposed methods. REFERENCES [1]
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Skolnik, M. I. Introduction to Radar Systems (3rd ed.). New York: McGraw-Hill, 2001, 45—46. Mahafza, B. R. Radar Systems Analysis and Design Using Matlab (2nd ed.). Boca Raton, FL: CRC Press, 2005, 155—162. Barton, D. K. Modern Radar System Analysis. Norwood, MA: Artech House, 1988.
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Zhang, S. S., et al. Dim target detection based on keystone transform. In Proceedings of IEEE International Radar Conference, May 2005, 889—894. Perry, R. P., Dipietro, R. C., and Fante, R. L. Coherent integration with range migration using keystone formatting. In Proceedings of IEEE Radar Conference, Apr. 2007. Li, Y., Zeng, T., and Long, T. Range migration compensation and Doppler ambiguity resolution by keystone transform. In Proceedings of the International Conference on Radar, Shanghai, China, 2006. Xu, J., et al. Radon-Fourier transform (RFT) for radar target detection, I: Generalized Doppler filter bank processing. IEEE Transactions on Aerospace and Electronic Systems, 47, 2 (Apr. 2011), 1186—1202. Xu, J., et al. RFT transform: Performance analysis and sidelobe suppression. IEEE Transactions on Aerospace and Electronic Systems, to be published. Kay, S. Fundamentals of Statistical Signal Processing: Detection Theory. Upper Saddle River, NJ: Prentice-Hall, 1998. Richards, M. Fundamentals of Radar Signal Processing. New York: McGraw-Hill, 2005. Oppenheim, A. V. and Schafer, R. W. Discrete-Time Sgnal Processing. Upper Saddle River, NJ: Prentice-Hall, 1989. Easton, Jr., L. R., Ticknor, A. J., and Barrett, H. H. Two-dimensional complex Fourier transform via the Radon transform. Applied Optics, 24 (1985), 3817—3824. Leavers, V. F. Statistical properties of the hybrid Radon Fourier transform. In Proceedings of the British Machine Vision Conference (BMVC2000), 2000. Analog Devices, Inc. ADSP-TS201 TigerSHARC Processor Hardware Reference. [2004-12]. http://analog.coni/en/embedded-processing-dsp/tigersharc/ processors/manuals/resoureces/mdex/hto. Yu, J., Xu, J., Peng, Y. N. Upper bound of coherent integration loss for symmetrically distributed phase noise. IEEE Signal Processing Letters, 15 (2008), 661—664. Richards, M. A. Coherent integration loss due to white Gaussian phase noise. IEEE Signal Processing Letters, 10, 7 (2003), 208—210. Javier, C. M., et al. A coherent Radon transform for small target detection. In Proceedings of the IEEE International Radar Conference, May 2009.
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Ji Yu was born in Jiangxi Province, China, in 1982. He received the B.S. degree from Beijing Normal University, Beijing, China, in 2005. He is a Ph.D. candidate in the Department of Electronic Engineering, Tsinghua University, Beijing, China. His current research interests are in the areas of moving target detection and tracking and array signal processing.
Jia Xu (M’05) was born in Anhui Province, P.R. China, in 1974. He received the B.S. and M.S. degrees from the Radar Academy of Air Force, Wuhan, China in 1995 and 1998, and the Ph.D. degree from Navy Engineering University, Wuhan, China, in 2001. He is an associate professor in the Department of Electronics Engineering, Tsinghua University, China. His current research interests include detection and estimation theory, SAR/ISAR imaging, target recognition, array signal processing, and adaptive signal processing. Dr. Xu received the Outstanding Post-Doctor Honor of Tsinghua University in 2004. He has authored or coauthored more than 80 papers. He is a senior member of the Chinese Institute of Electronics.
Ying-Ning Peng (M’93–SM’97) was born in Sichuan Province, P.R. China, in 1939. He received the B.S. and M.S. degrees from Tsinghua University, Beijing, China, in 1962 and 1965, respectively. Since 1993, he has been with the Department of Electronic Engineering, Tsinghua University, where he is now Professor and Director of the Institute of Signal Detection and Processing. He has worked with real-time signal processing for many years and has published more than 200 papers. His recent research interests include processing, parallel signal processing, and radar polarimetry. Professor Peng is a fellow of the Chinese Institute of Electronics. He has received many awards for his contributions to research and education in China. YU, ET AL.: RADON-FOURIER TRANSFORM FOR RADAR TARGET DETECTION (III)
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Xiang-Gen Xia (M’97–SM’00-7–F’09) received his B.S. degree in mathematics from Nanjing Normal University, Nanjing, China, and his M.S. degree in mathematics from Nankai University, Tianjin, China, and his Ph.D. degree in electrical engineering from the University of Southern California, Los Angeles, in 1983, 1986, and 1992, respectively. He was a senior/research staff member at Hughes Research Laboratories, Malibu, CA, during 1995—1996. In September 1996, he joined the Department of Electrical and Computer Engineering, University of Delaware, Newark, DE, where he is the Charles Black Evans Professor. He was a visiting professor at the Chinese University of Hong Kong during 2002—2003, where he is an adjunct professor. Before 1995, he held visiting positions in a few institutions. His current research interests include space-time coding, MIMO and OFDM systems, digital signal processing, and SAR and ISAR imaging. Dr. Xia has over 190 refereed journal articles published and accepted, and 7 U.S. patents awarded and is the author of the book Modulated Coding for Intersymbol Interference Channels (Marcel Dekker, 2000). He received the National Science Foundation (NSF) Faculty Early Career Development (CAREER) Program Award in 1997, the Office of Naval Research (ONR) Young Investigator Award in 1998, and the Outstanding Overseas Young Investigator Award from the National Nature Science Foundation of China in 2001. He also received the Outstanding Junior Faculty Award of the Engineering School of the University of Delaware in 2001. He is currently an Associate Editor of the IEEE Transactions on Wireless Communications, IEEE Transactions on Signal Processing, Signal Processing (EURASIP), and the Journal of Communications and Networks (JCN). He was a guest editor of Space-Time Coding and Its Applications in the EURASIP Journal of Applied Signal Processing in 2002. He served as an Associate Editor of the IEEE Transactions on Signal Processing from 1996 to 2003, the IEEE Transactions on Mobile Computing from 2001 to 2004, IEEE Transactions on Vehicular Technology from 2005 to 2008, the IEEE Signal Processing Letters from 2003 to 2007, and the EURASIP Journal of Applied Signal Processing from 2001 to 2004. He served as a Member of the Signal Processing for Communications Committee from 2000 to 2005 and is currently a Member of the Sensor Array and Multichannel (SAM) Technical Committee (since 2004) in the IEEE Signal Processing Society. He serves as IEEE Sensors Council Representative of the IEEE Signal Processing Society (since 2002) and served as the Representative of the IEEE Signal Processing Society to the Steering Committee for IEEE Transactions on Mobile Computing from 2005 to 2006. He is Technical Program Chair of the Signal Processing Symposium, Globecom 2007 in Washington D.C. and the General Cochair of ICASSP 2005 in Philadelphia. 1004
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