2010 IEEE Sensor Array and Multichannel Signal Processing Workshop
A Migrating Target Indicator for Wideband Radar Franc¸ois Deudon∗ , Franc¸ois Le Chevalier† , St´ephanie Bidon∗ , Olivier Besson∗ and Laurent Savy‡ ∗ Department Electronics Optronics and Signal, University of Toulouse/ISAE, Toulouse, France
Email: {fdeudon,sbidon,besson}@isae.fr † Thales Syst`emes A´eroport´es, France Email:
[email protected] ‡ ONERA, Radar and Electromagnetism Department, Palaiseau, France Email:
[email protected]
Abstract—The standard way to suppress clutter in narrowband radar is to use Moving Target Indicator (MTI) cancellation techniques. High Range Resolution (HRR) radars are becoming more and more important because they can detect and track targets more accurately. As for such radars the bandwidth is increased, the resolution is decreased and leads to target range migration over the coherent pulse interval (CPI). Due to this range walk, standard low resolution MTI processing is not adapted anymore to HRR MTI radar data. We propose here to extend the principle of the MTI processing to the wideband case. We refer to this method as the Migrating Target Indicator (MiTI), since it eliminates the non-migrating targets from the received signals. Index Terms—MiTI, clutter suppression, wideband radar.
establish a new procedure, the MiTI, which aims at removing nonmigrating targets. Numerical simulations are performed and analyzed in section IV. Conclusions are given in section V.
I. I NTRODUCTION AND PROBLEM STATEMENT
where δR = c/(2B) stands for the range resolution of the radar system, c is the speed of light, M is the number of transmitted pulses, v is the velocity of the target and Tr is the pulse repetition interval (PRI). Moreover, we consider that the Doppler does not affect significantly the complex envelope during one PRI, i.e.,
Future airborne radar signal processing will focus on HRR targets features extraction. Indeed, new technologies allow wider bandwidth for radar systems, inducing smaller resolution, and thus, more accurate tracking of the targets. Nevertheless, as targets are moving, they can migrate significantly in range during the CPI in a wideband context. Hence new processing techniques are needed to track them conveniently. These will have to compensate for range migration on the one hand, and to remove the non-migrating clutter on the other hand. Thus, new data models have been proposed for targets returns in a wideband context. In [1] and [2], the authors derive a wideband data model that takes into account the range walk of the target. Usually, this effect is undesirable for synthetic aperture radar (SAR) imaging [3], [4] but may be exploited for a wideband HRR/MTI mode [5]. Many studies have adressed the issue of optimal processing for wideband radar in white noise. When the signal of interest consists of a single scatterer, the optimal processing is related to the wideband ambiguity function that has been introduced in [1], [6], [7]. The shape of the ambiguity function results in a mainlobe with very proeminent sidelobes spreading over the range-Doppler domain. Clutter suppression is a critical issue for airborne radar processing. The principle of MTI radars has been described in [8]. Their purpose is to reject signals from fixed unwanted targets while retaining signals from moving targets. Given that, for a fixed target, the phase between two consecutive pulses does not change, subtracting from each echo the previous one will cancel immovable targets. On the contrary, as the phase changes between two consecutive received pulses of a moving target, mobiles are retained. In [9]–[11], other optimal filters for clutter suppression in a narrowband context are derived. Nevertheless, in opposition to a conventionnal MTI radar, the range profiles alignments of the targets of interest are destroyed in a HRR/MTI mode whereas unwanted clutter echos around zero Doppler do not migrate. In this paper, we propose an algorithm that attenuates non-migrating components of the signal for wideband radars. The paper is organized as follows. In section II the response from a single point scatterer is derived. This model is used in section III to
978-1-4244-8977-0/10/$26.00 ©2010 IEEE
II. DATA MODEL In this section, the model of a single point scatterer in the slowtime/fast-frequency domain is recalled [1], [4], [5]. A wideband linear frequency modulated (LFM) pulsed waveform with a bandwidth B is considered. It is assumed that the target migrates over the CPI, i.e., vM Tr > δR
(1)
vTr δR .
(2)
Note that to avoid range ambiguities, a low PRF will be considered in this paper. A. Received signal The transmitted signal can be expressed as
!
M −1
u ˜(t) =
X
up (t − mTr )
ej2πF0 t
(3)
m=0
where up (t) is the signal’s complex envelope and F0 is the carrier frequency. The radar return from a moving target can be expressed as s˜(t) = α u ˜(t − τ (t)) (4) where α is the complex amplitude of the echo and τ (t) is the timevarying echo delay of the considered scatterer (note that along all derivation, fixed terms will be absorbed into α ). Because of the target motion, the delay τ (t) is a function of time. Provided that the radial velocity v is constant over the CPI, and that v is negligible compared to the wave propagation velocity c, the delay can be expressed as 2v 2R0 − t (5) c c where R0 is the initial range of the target. After down-conversion, the signal for the mth pulse is 0 2v 0 2v sm (t0 ) = α up 1+ t − τ0 + mTr ej ω˜ D (t +mTr ) (6) c c where t0 = t − mTr denotes the fast-time at the mth pulse, τ0 is the initial round trip delay and ω ˜ D is the Doppler frequency whose expressions are given by
249
τ (t) =
τ0 =
2R0 c
and
ω ˜D 2v = F0 . 2π c
(7)
According to the assumption (2), the term 2(v/c)t0 in (6) can be removed. Nevertheless, as the considered waveform is wideband, we do not neglect the range walk anymore. Consequently, the term 2(v/c)mTr in (6), that accounts for the migration of the target, will not be neglected in the following.
where
B. Pre-processing
The symbol refers to the Hadamard matrix product. Note that the coupling effect between the `th subband and the mth pulse stands for the range migration. The differential Doppler ωd is associated with this coupling effect, and is directly proportional to the Doppler frequency ωd = µωD with µ = B/(LF0 ). (18)
Usually, the received baseband signal is processed by a range matched-filter up (t0 ). The filter output for the mth pulse in the slowtime/fast-time domain is given by ym (t0 ) =
+∞
Z
sm (ξ)u∗p (ξ − t0 )dξ.
(8)
−∞
It is more convenient to express the signal in the slow-time/fastfrequency domain [1], [2] as the range walk can be seen as a coupling effect between the slow-time and the subbands. Consequently, a Fourier transform at the output of the matched filter is applied and the signal response in the fast-frequency domain is
Z
+∞
0
ym (t0 )ejωt dt0 .
Xm (ω) =
(9)
−∞
After some straightforward manipulations as in [1], [5], (9) can be expressed as Xm (ω) = α ej ω˜ D mTr e−jω(τ0 −
2v mT ) r c
ωD ωd ωr
Note that µ does not depend on the target features. III. M IGRATING TARGET I NDICATOR In this section, we briefly introduce the ambiguity function for wideband radars, that is related to the optimal processing for a target in presence of white noise only. Then we describe the MiTI that uses this optimal processing to attenuate non-migrating targets, while preserving other targets responses. A. Ambiguity function Let us consider the data matrix X of size M × L defined by
Up (ω − ω ˜ D )Up∗ (ω)
X = αA + N
where Up (ω) is the Fourier transform of the transmitted pulse up (t). As in [1], it is assumed that the expected range of target Doppler frequencies is negligible compared to the bandwidth B of the signal, i.e., 2v F0 B. (10) c The previous inequality is tantamount to considering that Up (ω) is insensitive to Doppler shift, and (10) reduces to −jω(τ0 − 2v mTr ) c
Xm (ω) = α ej ω˜ D mTr e
|Up (ω)|2 .
2v mT ) r c
.
M −1 L−1
T (˘ ωD , ω ˘r ) =
0 j2πm 2v c Fr
δf
j2πm` 2v c Fr
e
`0
e−j2π` L
In this case, the frequency pair (ˆ ωD , ω ˆ r ) that verifies (ˆ ωD , ω ˆ r ) = arg max |T (˘ ωD , ω ˘ r )| ω ˘ D ,ω ˘r
`0 = Lδf τ0 .
(14)
Therefore, the signature of a single scatterer in the slow-time/rangefrequency domain is given by an M × L matrix A = Ad aD aTr
(15)
with
jm`ωd [Ad ]m,` = e aD = 1 . . . ejωD m jω ` and ωD = 2vF0 /(cFr ), 2π
...
e
(21)
is also the Maximum Likelihood (ML) estimate for (ωD , ωr ) [6]. In the absence of noise, (20) defines the ambiguity function AF M −1 L−1
T (˘ ωD , ω ˘r ) =
XX
e−j`(ω˘ r −ωr ) e−jm(1+`µ)(ω˘ D −ωD )
r
ejωD (M −1)
... ...
jωr
e
ωd = 2vB/(cLFr ), 2π
= AF(˘ ωD − ωD , ω ˘ r − ωr ).
(13)
where `0 is the initial range gate of the target, defined as follows
= 1
(20)
m=0 `=0
F
ar
Xm,` e−j`ω˘ r e−jm(1+`µ)ω˘ D .
m=0 `=0
Xm,` = Xm (2π`δf ) =αe
XX
(12)
The bandwidth B is now sampled into L equispaced subbands of length δf = B/L, so that the signal at the mth pulse and `th subband can be expressed, for ` = 0 . . . L − 1, by [6]
(19)
where A and α respectively denote the signature (15) and the amplitude response of the target. N is an M × L matrix standing for ˜M L 0, σ 2 I . It a white Gaussian noise with power σ 2 , i.e., N ∼ N is well-known that the optimal processing [6] (or matched-filtering) according to (19), is given by
(11)
Moreover, Up (ω) is supposed to be constant over the frequency band [1], i.e., Xm (ω) = α ej ω˜ D mTr e−jω(τ0 −
is the normalized Doppler frequency, is the normalized differential Doppler, is the normalized range-frequency that depends on the initial range gate number `0 of the target.
T
The ambiguity function, shown in Fig.1, has its mainlobe around (ωD , ωr ) with high sidelobes at each target’s velocity modulo the ambiguity velocity va = (cFr )/(2F0 ). B. MiTI We describe in this section an enhanced version of the MTI for wideband radars. The first part of the proposed technique consists of splitting the initial data into two subsets. The signature of a target for the first and second subsets are respectively given by A(0)
(16)
(L−1) T
(1)
A
ωr = −`0 /L (17) 2π
(22)
m,` ¯ m,` ¯
¯ D j`(ωr ) = ej m(1+`µ)ω e
=e
for
250
m ¯ = 0, . . . ,
(23)
jM ωD j m(1+`µ)ω ¯ D j`(ωr +∆ωr ) 2
e
M − 1 and 2
e
` = 0, . . . , L − 1
(24)
0.5
in section III-A, ω ˆ D1 and ω ˆ r1 are the ML estimates for ωD and ωr . ω ˆ D2 and ω ˆ r2 −M µˆ ωD /2 are also estimates for the model parameters. Consequently, this lead us to propose the following expressions
0
0.4 0.3
−5 −10
0.1 0
−15
−0.1
Amplitude (dB)
range frequency
0.2
−20
−0.2
ω ˆ D1 + ω ˆ D2 2 ω ˆ r1 + ω ˆ r2 − M µˆ ωD /2 ω ˆr = 2 and ωr . ω ˆD =
to estimate ωD
(33) (34)
−0.3 −25
IV. S IMULATIONS
−0.4 −0.5
Fig. 1.
−2
−1 0 1 velocity ambiguity va
2
3
In this section, the data sequence X is generated as follows
−30
X=
Ambiguity function for a target at ωD = 0.2, ωr = 0
M µωD . (25) 2 The optimal processing (20) is then applied for each set. The outputs of the first and second subsets are respectively M/2−1 L−1
m=0 ¯
¯ ω ˘ D −j`ω Xm,` e−j m(1+`µ) e ˘r ¯
(26)
`=0
M/2−1 L−1
T1 (˘ ωD , ω ˘r ) =
X X m=0 ¯
¯ ω ˘ D −j`ω Xm+M/2,` e−j m(1+`µ) e ˘ r . (27) ¯
`=0
Provided that the clutter returns consist in targets that do not migrate during the CPI, the signature Ac of unwanted clutter echoes can be obtained by neglecting the term Ad accounting for range migration in relation (15), i.e., Ac = aD (ωD = 0)aTr (ωr ).
(28)
Thus, for a single clutter scatterer, the outputs of the optimal processing verifies the following equality |T1 (˘ ωD , ω ˘ r )| = |T0 (˘ ωD , ω ˘ r )| .
(29)
For a migrating target scatterer, this relation becomes |T1 (˘ ωD , ω ˘ r )| = |T0 (˘ ωD , ω ˘ r + ∆ωr )| .
(30)
Consequently, we have proposed a new algorithm based on the difference between the two images, obtained after optimal processing. This technique is refered to as the MiTI and defined as MiTI(˘ ωD , ω ˘ r ) = |T0 (˘ ωD , ω ˘ r )| − |T1 (˘ ωD , ω ˘ r )| .
(31)
For a single point scatterer, the processing becomes
SNRt = (|αt |2 /σ 2 ).
(32)
In other words, it is the subtraction of two ambiguity functions shifted from ∆ωr . For a non-migrating target, ∆ωr = 0, and the target is effectively suppressed. If the target is fast enough, then mainlobes of the two optimal processing T0 and T1 are not overlapped. Consequently, the output of the MiTI processing for a fast target results in two opposite mainlobes at the same Doppler frequency, with a range-frequency shift ∆ωr . Let (ˆ ωD1 , ω ˆ r1 ) and (ˆ ωD2 , ω ˆ r2 ) denote the position of these two peaks. As highlighted
(36)
Targets 1 and 2 are very slow and can be considered as unwanted echos of clutter. Target 3 is a fast moving target of interest. The rangefrequency of slow targets does not change significantly from the first subset to the second one. This is due to the fact that the target is too slow and remains confined in its initial range bin. On the contrary, fast targets migrate during the CPI. Consequently, as depicted in Fig.2 the outputs of the optimal processor (20) of the two subsets result in two ambiguity functions, shifted over the range-frequency domain. Fig.3 shows the output of the MiTI processing. Fig.4 displays the amplitude of the MiTI processing along the range frequency axis at ωD = 1.5. The two peaks at ωr1 = 0 and ωr2 = ωr1 + M/2µωD are recovered. The following remarks can be made. • Clutter supression. Slow target mainlobes are seriously attenuated as desired. Because they do not move significantly over the CPI, the mainlobes of these targets are located at the same place in the range-Doppler frequency domain. Consequently their lobes are attenuated after MiTI processing. On the contrary, fast targets are moving over the CPI so the lobes are not nulling themselves when subtracting. They even keep their initial height. • Targets position. After MiTI processing, note that fast target signature results in two opposite sidelobes both located at the initial Doppler frequency, and shifted from the initial range frequency. Consequently, it may be possible to estimate target features according to (33) and (34), by locating local minima and maxima at the output of the processing. • Resolution. The mainlobes at the output of the MiTI processing are wider than those of an optimal processing, due to a shorter integration time.
MiTI(˘ ωD , ω ˘ r ) = |AF(ˆ ωD − ωD , ω ˆ r − ωr )| − |AF (ˆ ωD − ωD , ω ˆ r − (ωr + ∆ωr ))| .
(35)
where Nt is the target number, αt and At are respectively the complex amplitude and signature (15) of the t-th target, and N is a white Gaussian noise with power σn2 = 1. Performance of the MiTI is studied according to the scenario given in Table I where the SNR for the t-th target scatterer is defined as
∆ωr =
X X
αt At + N
t=1
where ∆ωr accounts for the range walk during the M/2 pulses of the data set. Note that ∆ωr depends only on the Doppler frequency and can be expressed as
T0 (˘ ωD , ω ˘r ) =
Nt X
V. C ONCLUSION We have proposed a method to suppress clutter in wideband radar. It consists of splitting the data set into two subsets, and then incoherently subtracting the first output of a matched filter in white noise from the second output. Non-migrating targets are then removed, whereas response of fast targets results in two opposite peaks around the intial target range and Doppler frequency. This may allow us to estimate the features of targets after processing. Future work will focus on applying this procedure to multiple subsets in order to improve non-migrating targets attenuation.
251
M σ2
32 1
L µ ωD /(2π) ωr /(2π) SN R
12 10
0.3
8
64 1.6e-3 1 0.09 0.31 10dB
2 0 0 10dB
3 1.5 0 10dB
0.4
12
0.35
10
0.3
8
2 0.15 0 0.1
−2
0.05
−4
0
−6
−0.05
−8
Fig. 3.
−2
−1 0 1 velocity ambiguity va
2
3
−10
MiTI output for the tested scenario
6
2 0.15 0 0.1
−2
0.05
10 8 6 Amplitude (decimal)
4
0.2
Amplitude (dB)
0.25 range frequency
4
0.2
−0.1
−4
0
−6
−0.05
−8
−0.1
6
0.25 range frequency
Scenario Pulses Noise power Processing parameters Number of subbands Migration parameter Target Doppler frequency Range frequency Signal to Noise Ratio
0.4 0.35
Amplitude (dB)
TABLE I S CENARIO
−2
−1 0 1 velocity ambiguity va
2
3
−10
(a)
4 2 0 −2 −4 −6 −8
0.4
12
0.35
10
0.3
8 6 4
0.2
2 0.15 0 0.1
−2
0.05
Fig. 4. Amplitude (dB)
range frequency
0.25
−4
0
−6
−0.05
−8
−0.1
−10 −0.5
−2
−1 0 1 velocity ambiguity va
2
3
−10
(b) Fig. 2. Optimal processing for (a) the first subset and (b) the second subset
0 range frequency
0.5
Cut of the MiTI output at Doppler frequency ωd /(2π) = 1.5
[6] F. Le Chevalier, Principles of Radar and Sonar Signal Processing. Norwood, MA: Artech House, 2002. [7] A. W. Rihaczek, Principles of High-Resolution Radar. Norwood, MA: Artech House, 1996. [8] M. I. Skolnik, Radar Handbook. The McGraw-Hill Companies, 1970. [9] W. D. Rummler, “Clutter suppression by complex weighting of coherent pulse trains,” IEEE Trans. Aerosp. Electron. Syst., vol. 2, no. 6, pp. 689– 699, Nov. 1966. [10] J. K. Hsiao, “On the optimization of MTI clutter rejection,” IEEE Trans. Aerosp. Electron. Syst., vol. 10, no. 5, pp. 622–629, Sep. 1974. [11] G. W. Ewell and A. M. Bush, “Constrained improvement MTI radar processors,” IEEE Trans. Aerosp. Electron. Syst., vol. 11, no. 5, pp. 768–780, Sep. 1975.
ACKNOWLEDGMENT This work was supported by the D´el´egation G´en´erale pour l’Armement (DGA) and by Thales Syst`emes A´eroport´es. The authors are also thankful to ONERA for its contribution. R EFERENCES [1] T. J. Abatzoglou and G. O. Gheen, “Range, radial velocity, and acceleration MLE using radar LFM pulse train,” IEEE Trans. Aerosp. Electron. Syst., vol. 34, no. 4, pp. 1070–1084, Oct. 1998. [2] N. Jiang, R. Wu, and J. Li, “Super resolution feature extraction of moving targets,” IEEE Trans. Aerosp. Electron. Syst., vol. 37, no. 3, pp. 781–793, Jul. 2001. [3] R. P. Perry, R. C. DiPietro, and R. L. Fante, “SAR imaging of moving targets,” IEEE Trans. Aerosp. Electron. Syst., vol. 35, no. 1, pp. 188–200, Jan. 1999. [4] F. Berizzi, M. Martorella, A. Cacciamano, and A. Capria, “A contrastbased algorithm for synthetic range-profile motion compensation,” IEEE Trans. Geosci. Remote Sens., vol. 46, no. 10, pp. 3053–3062, Oct. 2008. [5] F. Deudon, S. Bidon, O. Besson, J. Tourneret, M. Mont´ecot, and F. Le Chevalier, “Modified Capon and APES for spectral estimation of range migrating targets in wideband radar,” in Proc. 2010 IEEE Int. Radar Conf., Washington, DC, USA, May 10–14, 2010.
252
