- Email: [email protected]
Adaptive space—Time radar
J. Franklin Inst. Vol. 335B, No. 1, pp. 1 I L 1998
~
Pergamon
PII:SO016-0032(96)00111-1
Copyright © 1997 The Franklin Institute Published by Elsevier Science Ltd Printed in Great Britain 00164)032/98 $19.00+0.00
Adaptive Space-Time Radar b y RONALD L. FANTE
The M I T R E Corporation, Bedford, M A 01730-1420, U.S.A.
ABSTRACT: Airborne radars have difficulty detectin9 weak taryets because sidelobe clutter enterin9 the radar has Doppler frequencies spread over a wide frequent T range, and this masks weak targets. In this paper we show how adaptive approaches allow one to cancel this clutter, along with any jammers that are present, thus allowin# for taryet detection. 6) 1997 The Franklin Institute.
Published by Elsevier Science Ltd
L Introduction
Unlike a ground-based radar, in which all of the clutter is received at or near zeroDoppler, the clutter return in an airborne radar has Doppler frequencies spread over a band of width 4 I//2, where V is the platform speed and 2 is the radar wavelength. Therefore, conventional moving target indicator (MTI) is ineffective in cancelling this clutter without also cancelling desired targets. Another feature of airborne radar is that there is usually clutter in the same Doppler cell as a target, but it usually arrives from a different azimuth, as illustrated in Fig. 1 for the case of frozen clutter (i.e. no internal
Ground Clutter Directional Antenna+vA~ /~
0"
Beame Angl
,;; ;;" jSideloebe -,,,,~"
90"
Broadside Azimuth
180"
FIG. I. Angle-Doppler plot of clutter and one sidelobe jammer for airborne platform.
2
R.L. Fante Antenna 1
Y
Antenna N
Antenna 2
y
Ul(t)
y
U2(t)
UN(t)
SQO fWN1
NN2
~12
" .."
~
v (t)
FIG. 2. Generic space-time array.
motion). This suggests that adding spatial degrees of freedom to the conventionallyused temporal degrees of freedom should allow us to place a null along the azimuth of the clutter that competes with the target, thus allowing the clutter to be canceled, without canceling the desired target return. In constructing Fig. 1 we assumed that the pulse repetition rate is such that there are no Doppler ambiguities. A typical spacetime processor that combines spatial and temporal degrees of freedom is shown in Fig. 2. The processor is formed by placing a tapped delay line at the output of each of N antennas of the array, with the taps spaced by one pulse repetition interval (PRI)T. This processor has a number of degrees of freedom equal to the number N of antennas times the number K of pulses (taps). The adaptive weights in the processor shown in Fig. 2 are chosen to maximize the signal-to-noise-plus-clutter-plus-jammer ratio S/(N+ C+ J) for a specified desired signal (i.e. specified azimuth and Doppler). That is, the adaptive space-time array is a matched filter that is designed to detect a desired radar signal while rejecting all interference and signals that do not have the properties of the desired signal. The quantity S/(N+ C+J) at the output of the array in Fig. 2 is
S N+C+J-
IwTsl 2 (IwTU[ 2)
(1)
Adaptive S p a c e - T i m e Radar
3
and this is maximized (1) if the weight vector w = [w~l, w~2... WN! W12.-. WN2...]T is given by w=#M-ls
(2)
*,
where # is a constant, M is the covariance matrix defined as M = ( U U +), ( ) denotes an expectation, U is the received voltage vector given by U= [U,(t)...UN(t)
U,(t--T)...UN(t--T)...IT,
and s is the steering vector that describes the voltage that would be received on each element of the array if a target at the desired azimuth and with the desired Doppler were present. In the above equations the superscript T denotes a transpose, * denotes complex conjugate and + denotes conjugate transpose. When w is given by Eq. (1) it can be shown by substituting (2) into (1) that S / ( N + C + J ) after adaptation is given by (3)
-- s T M - I s *.
N+C+J
In an ideal world in which the covariance could be exactly estimated, there is no internal clutter motion, channel match is perfect, the jammers are narrow-band, and there is no jammer multipath it can be shown that S -
-
N + C+J
Isl 2 --,
(4)
r~2
where Isl 2 is the signal power and 0.2 is the thermal noise power. Thus, under ideal conditions the adaptive processor can cancel all clutter and interference, and all that remains after adaptation is thermal noise. Although the architecture shown in Fig. 2 represents an element space configuration, a beam space architecture is equivalent. For example, if a beamformer is placed behind the elements in Fig. 2, the N × 1 voltage vector b at the beamformer output can be written as b = GU, where G is the N x N beamformer matrix. Then, if we note that the new steering vector after beamforming is Sb = Gs and that (bb +) = G ( U U + ) G + = G M G +
we can readily show that
S N~-~-J
)
= s~(bb+)_ls,=
sTM_Is ,
(5)
beam space
so that upon comparing (5) with (3) we see the output signal-to-interference ratio is the same whether an element space or beamspace architecture is used. However, sometimes beamspace may be preferable to element space, because it allows for the development of suboptimum architectures more readily, as will be discussed later.
4
R . L . Fante
]
]
-10
Main Beam Clutter Eigenvalues
At -20
~
I
L
-30
-40 -
W
Sidelobe Clutter
Eigenvalues
...................
~
-50
~
-60 -
T
.
/
. '
.
.
.
T i
.
.
.
.
/
-70 -
. . . . . . . . . '.....................................
...................
-80 -
-90 -
i
0
10
20
'
k
I
IN+KI=52, ' I
40
50
Eigenvalue
i ..............................
,.n.ues
........................... . . . . . . . . . . . . .
i
'
30
/io.
60
l '
70
'
80
'
90
100
Number
FIG. 3. Clutter eigenvalue spread for N = 50, K = 2 (no jammers). II. Understanding the Results in (2) and (3) In practice one must estimate the covariance matrix using a finitet number of samples, but for purposes of understanding we will assume that the number of samples is so large that the ideal covariance matrix is available. For the architecture in Fig. 2 this is an N K × N K square matrix M that can be decomposed into its N K × 1 eigenvectors ek and eigenvalues 2k as (2) NK
M = ~ 2~eke~
(6)
k=l
Likewise M
1 = ~ eke~k~l 2k
(7)
Because M is Hermitian the eigenvalues are real and the eigenvectors are orthogonal. Under ideal conditions (no internal clutter motion, narrowband radar, etc.) there are N + K large eigenvalues associated with the mainbeam and sidelobe clutter [for ambiguous radars there are even more (3, 4)]. A typical eigenvalue strength plot for clutter only (no jammers) is shown in Fig. 3. Strong jammers produce additional large eigenvalues. If there are Nj independent, 1"For an acceptable covariance estimate the number of samples must be at least 2NK and preferably 4NK, where NK is the total number of adaptive degrees of freedom.
Adaptive S p a c e - T i m e Radar
5
80
70
I Jammer Correlation = exp(-B21:2) ]
60 ¢D 7o
=
.................................................
50
~-. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
.........................................
t. . . . . . . . . . . . . .
; .............................
~. . . . . . . . . . . . . . . . . . . . .
~.
~ ,-'~"
!'
;
!
40
.................
.
.
.
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.
.
.
.
.
.
.
.
._ 30 W
El g
.
.
.
.
'. .
10
0 -"~'~
t
S
i~
i
,
.
i
:
Eigenvalue3to 16 i
i
,
0
.
::~
1
~
~
2
.
.
.
.
.
,
i
i
3
4
5
B'C Correlated Uncorrelated
-=
.
6
=
FIG. 4. Effect o f j a m m e r correlation on eigenvalues for two resolved j a m m e r s when N = 16, K=I.
resolved (i.e. spaced in angle by greater than the 3 dB beamwidth of the antenna) jammers there are Nj large jammer eigenvalues. However, unresolved jammers and correlated jammers share a single eigenvalue, as indicated in Figs 4 and 5. Thus, the 70 60
Eigenvalue
I
50 CL
~ 40
LU
:.........
20
i . . . . . . . . . . . . . . . .
//Eigenvalue 2
10 j,
0 ~ '~S
/
-- i
• Eigenvatue3to16 i r
0
1
2
,
i
3
4
5
Angular Separation/3 dB Beamwidth FIG. 5. Effect o f j a m m e r separation 0sep on eigenvalues for N = 16, K = 1, Bz = 2.
6
R . L . Fante
covariance matrix M typically has R = N + K + N j large eigenvalues and N K - R small eigenvalues with magnitudes equal to the variance 0.2 of the system noise. Consequently, Eq. (7) can be rewritten as M _ 1 ~- ~ eke~ 1 NK k= , 2k + ~ k ~+ l eke~
(8)
Next, we recognize that the N K x N K identity matrix I is defined as (2) NK
I = ~ eke + k=l R
NK
= • eke;+
~
k=l
eke~
(9)
k=R+l
where the first term on the right-hand side of Eq. (9) defines the interference (clutter plus jammers) subspace and the second defines the noise subspace. Thus, we can rewrite Eq. (9) as a sum of two projections I = Pll
(10)
+P±
where P~j is the projection into the interference subspace and Pz is the projection orthogonal to the interference subspace. Therefore, Eq. (8) can be rewritten as
M-J = ~ e~e~k=l~-
1
+ ~P~
(ll)
If Eq. (11) is now used in (2) we see that the weight vector w is R
ekeffs,
w = t, ky,' ~ -
- + ~Pj.
=
0.2
(12)
Because, as can be seen from Fig. 3, 1241 >> 0.2 we can neglect the first term on the righthand side of the Eq. (12) so that (13)
w ~- ~ P ± s * . 0 .2
This shows that the desired weight vector is the projection of the steering vector s onto the subspace orthogonal to the interference subspace, and means that as long as the steering vector has a component orthogonal to the interference subspace it is possible to cancel the interference while preserving the desired signal (or at least most of it). It should also be noted that because [2k] >> a 2 we can use Eq. (9) to write Eq. (8) as M- 1
1 NK
1
Adaptive Space-Time Radar
7
This is a useful approach for inverting the covariance matrix when there are only a few dominant eigenvalues, and has been exploited in (6). By using Eq. (14) in (3) it is easy to see how well the limit in (4) is approached.
IlL Sequential Cancelation of Jammers and Clutter In practice, when both jammers and clutter are simultaneously present it is difficult to cancel both because of interactions in the covariance matrix estimate. Thus, it is desirable to cancel the jammers before tackling the clutter. One way to do this is to use the beam space architecture, place spatial nulls in each beam on any jammers in that beam, and then combine those resulting beams to cancel the ground clutter. Let us analyse here how one forms L ~< N beams with nulls on any jammers present. In particular, we need to derive an L × N beamforming matrix F that produces L beams while simultaneously minimizing jammer power. Let J be an N × 1 vector consisting of the voltages on the N antenna elements produced by any jammers present. These jammers then produce a voltage vector at beam ports given by b = FJ. Also, let h~ be an N × 1 vector of the voltages produced on the N antenna elements by an incoming plane wave at an angle 01 relative to the array. Likewise, define h2 as the N x 1 vector after a wave at angle 02, etc. Then, the condition for forming L beams pointed at 0~02... is
Fkhk = Ck
(15)
for k = 1,2 . . . . L, where Fk is the 1 x N vector formed by the kth row of F and the G are specified constraints. Now we desire to minimize the interference power (b+b) subject to the constraints in (15). The solution is
F = flDGR -1
(16)
R = (JJ+)
(17)
where
is the jammer plus noise covariance matrix, G is the L × N beamforming matrix in the absence of jammers, D is a diagonal matrix with the kth diagonal element given by Ck/(h~ R - ~h~) and fl is a constant. Note that if the jammers are absent the jammer plus noise covariance matrix reduces to the noise covariance matrix a2Is, where IN is the N × N identify matrix, so that, as expected, the beamforming matrix F reduces to the jammer-free beamforming matrix G. One may next wonder how one estimates the jammer covariance matrix when both jammers and clutter are present. This can be done by referring to the angle-Doppler plot shown in Fig. l, and recognizing that the jammers spread through all Doppler bins. Thus, if one could choose an angle-Doppler region that is well removed from the peak of the clutter ridge it should be possible to estimate the jammer covariance with little clutter interaction. Of course, for radars that are unambiguoust in Doppler this t Remember that the clutter only occupies the frequency region from + 2 Vj2 so that if the pulse repetition frequency exceeds this value there is a clutter-free region in the Doppler domain.
R. L. Fan te
P°rte-~~ . . ~ ~--Ib-~ ~----~.-~ Beam
C;Swm,
C;Swm
C;Swm, T_, 1
From
Beam p
~.1
(FFT)
DopplerProcessing
From Beamq FIG. 6. Beam-pulse space processor. is a trivial task because there are then regions in Doppler space where the clutter is completely absent, but the jammers are present. Once the jammers have been removed the clutter can then be cancelled either in the beam-pulse domain (also called pre-Doppler) as in Fig. 6 or in the beam-Doppler domain (also called post-Doppler) as in Fig. 7. The architecture in Fig. 7 is what is known as first-order factored processing where the beams in each doppler frequency bin are independently used to cancel clutter. That is, if B(p) is an N x 1 vector of the outputs of the N beam ports for Doppler frequency binp and S(p) is the N x 1 steering vectorS" for the Doppler frequency bin p then the weight vector WR used to combine the beam outputs in Doppler frequency bin p is We = ~ ( B ( p ) B + ( p ) )
'S*(p)
08)
where/~ is a constant. In practice, however, it has been found that combining two Doppler bins produces superior cancellation performance. The reader is referred to (5) for details.
IV. Typical Results There are several experimental programs to test the adaptive cancellation methods discussed in the last few sections. One is called Mountaintop (Lincoln Laboratory) and t S(p) is simply the discrete Fourier transform of the time domain steering vector s.
Adaptive Space-Time Radar Beam
From
Beam
/'~'~ From p-'~M'Beamq
Doppler Bin 1
Output
From
/"~ "~.=~_ From
Beam p ' - ~ . ~
J ~
Beam
Doppler Bin L Output
FIG. 7. Beam-Doppler space processor (first order factored).
is designed to evaluate space-time processing using emulated motion of the radar platform. This program has demonstrated excellent cancellation at UHF of emulated airborne clutter at the White Sands Missile Range. Another is called MCARM (Northrup Grumman & Rome Laboratory) and consists of actual BAC-111 flights of a 1.25 GHz adaptive array with 22 spatial degrees of freedom. In Fig. 8 we show the MCARM signal-to-clutter plus noise ratio (there were no jammers present) before and after adaptation when the processing architecture in Fig. 7 is used (7). Note that up to 40 dB of clutter cancellation has been achieved. The clutter is not cancelled completely for a number of reasons including the fact the first order factored processing (as in Fig. 7) is suboptimal and that the clutter is spatially inhomogeneous, so that the covariance matrix estimate is not accurate for all ranges. Additional results are available, but will be omitted for conciseness.
V. Additional Considerations In this brief exposition we have ignored many of the implementation problems involved in the actual performance of a space-time adaptive array. Some of these are: how does one obtain sufficient samples to estimate the covariance matrix and what are the effects of inhomogeneous clutter on its value?; what are the effects of channel mismatch, radar-aircraft crabbing and scatter from obstacles such as airplane wings that are in the near-field of the array?; is the computational load reasonable?; and finally how does jammer multipath affect the interference cancellation? All of these points have been investigated to some degree. Jammer multipath is especially serious
l0
R. L. Fante
30
~-'0
-40
........
. ........
. ........
; .......
; ........
! ........
7........
i. .......
. .... t
t ....... "-~........
:~........
:i .......
~: .......
:i
:~
:"
~
:: .......
i ........
i ........
- .......
i .......
._...
....
i
250
•
. .i ........
300
350
400 450 Range Bin
500
550
600
FIG. 8. Improvement in signal to interference ratio for a particular Doppler bin and receive azimuth using first order factored processing. The clutter free SNR = 20 dB.
because the radiation of an airborne sidelobe jammer can be scattered off the ground into the mainbeam (or its elevation sidelobes) of the radar thus giving the appearance of multiple mainbeam jammers. Because the interference is spread in all azimuths across the main beam it cannot be canceled, without simultaneously canceling the target signal, using spatial nulls. Thus, time domain cancellation techniques are required (8), but these are beyond the scope of this paper.
VI. Summary We have given a brief exposition of how a space-time adaptive canceller can remove the deleterious effects of both ground clutter and jammers, thus enabling an airborne radar to detect weak targets. In particular we have shown how the processor accepts target signals, but rejects anything that does not look like a target, as specified by the target's azimuth and Doppler frequency. Thus, in principle, all interference except system noise can be eliminated.
Adaptive Space-Time Radar
11
References (1) L. Brennan and I. Reed, "Theory of adaptive radar", IEEE Trans, Vol. AES-9, pp. 237252, 1973. (2) B. Friedman "Principle and Techniques of Applied Mathematics", Wiley, New York, 1962. (3) J. Ward, "Space-time adaptive processing for airborne radar," MIT Lincoln Lab Report 1015, 1994. (4) L. Brennan and F. Staudaher, "Subclutter visibility demonstration", Adaptive Sensors Inc. Tech. Report RL-TR-92-21, 1992. (5) R. DiPietro, "Extended factored space-time processing for airborne radar systems", "'Proc. of the 26th Asilomer Conference on Siynals, Systems and Computing", pp. 425~430, 1992. (6) A. Haimovich, M. Puch and M. Baren, "Training and signal cancellation in adaptive radar", Diyest of the 1996 IEEE Radar Conference, pp. 124-128, Ann Arbor, 1996. (7) B. Suresh-Babu, J. Torres and W. Melvin, "Processing and Evaluation of Multichannel Airborne Radar Measurements (MCARM)", MTR96B0000053, The MITRE Corporation, Bedford, MA, 1996. (8) R. Fante and J. Tortes, "Cancellation of diffuse jammer multipath by an airborne adaptive radar", IEEE Trans, Vol. AES-31, pp. 805-820, 1995.
Bibliography E. Barile, R. Fante and J. Torres, "Some limitations on the effectiveness of airborne adaptive radar", IEEE Trans, Vol. AES-28, pp. 1015-1032, October 1992. L. Brennan, J. Mallett and I. Reed, "Adaptive arrays in airborne MTI radar", IEEE Trans, Vol. AP-24, pp. 607-615, September 1976. H. Wang and L. Cai, "On adaptive spatial-temporal processing for airborne surveillance radar systems", IEEE Trans, Vol. AES-30, pp. 660-669, July 1994. R. Klemm, "Adaptive clutter for airborne phased array radars", lEE Proceedings, Parts F and H, Vol. 130, pp. 125-132, 1983. R. Monzingo and T. Miller, "Introduction to Adaptive Arrays", Wiley, New York, 1980. R. Compton, "Adaptive Antennas", Prentice Hall, Englewood Cliffs, New Jersey, 1988. A. Farina, "Antenna-based Signal Processing Techniques for Radar Systems", Artech House, Norwood, Massachusetts, 1992.
~
Pergamon
PII:SO016-0032(96)00111-1
Copyright © 1997 The Franklin Institute Published by Elsevier Science Ltd Printed in Great Britain 00164)032/98 $19.00+0.00
Adaptive Space-Time Radar b y RONALD L. FANTE
The M I T R E Corporation, Bedford, M A 01730-1420, U.S.A.
ABSTRACT: Airborne radars have difficulty detectin9 weak taryets because sidelobe clutter enterin9 the radar has Doppler frequencies spread over a wide frequent T range, and this masks weak targets. In this paper we show how adaptive approaches allow one to cancel this clutter, along with any jammers that are present, thus allowin# for taryet detection. 6) 1997 The Franklin Institute.
Published by Elsevier Science Ltd
L Introduction
Unlike a ground-based radar, in which all of the clutter is received at or near zeroDoppler, the clutter return in an airborne radar has Doppler frequencies spread over a band of width 4 I//2, where V is the platform speed and 2 is the radar wavelength. Therefore, conventional moving target indicator (MTI) is ineffective in cancelling this clutter without also cancelling desired targets. Another feature of airborne radar is that there is usually clutter in the same Doppler cell as a target, but it usually arrives from a different azimuth, as illustrated in Fig. 1 for the case of frozen clutter (i.e. no internal
Ground Clutter Directional Antenna+vA~ /~
0"
Beame Angl
,;; ;;" jSideloebe -,,,,~"
90"
Broadside Azimuth
180"
FIG. I. Angle-Doppler plot of clutter and one sidelobe jammer for airborne platform.
2
R.L. Fante Antenna 1
Y
Antenna N
Antenna 2
y
Ul(t)
y
U2(t)
UN(t)
SQO fWN1
NN2
~12
" .."
~
v (t)
FIG. 2. Generic space-time array.
motion). This suggests that adding spatial degrees of freedom to the conventionallyused temporal degrees of freedom should allow us to place a null along the azimuth of the clutter that competes with the target, thus allowing the clutter to be canceled, without canceling the desired target return. In constructing Fig. 1 we assumed that the pulse repetition rate is such that there are no Doppler ambiguities. A typical spacetime processor that combines spatial and temporal degrees of freedom is shown in Fig. 2. The processor is formed by placing a tapped delay line at the output of each of N antennas of the array, with the taps spaced by one pulse repetition interval (PRI)T. This processor has a number of degrees of freedom equal to the number N of antennas times the number K of pulses (taps). The adaptive weights in the processor shown in Fig. 2 are chosen to maximize the signal-to-noise-plus-clutter-plus-jammer ratio S/(N+ C+ J) for a specified desired signal (i.e. specified azimuth and Doppler). That is, the adaptive space-time array is a matched filter that is designed to detect a desired radar signal while rejecting all interference and signals that do not have the properties of the desired signal. The quantity S/(N+ C+J) at the output of the array in Fig. 2 is
S N+C+J-
IwTsl 2 (IwTU[ 2)
(1)
Adaptive S p a c e - T i m e Radar
3
and this is maximized (1) if the weight vector w = [w~l, w~2... WN! W12.-. WN2...]T is given by w=#M-ls
(2)
*,
where # is a constant, M is the covariance matrix defined as M = ( U U +), ( ) denotes an expectation, U is the received voltage vector given by U= [U,(t)...UN(t)
U,(t--T)...UN(t--T)...IT,
and s is the steering vector that describes the voltage that would be received on each element of the array if a target at the desired azimuth and with the desired Doppler were present. In the above equations the superscript T denotes a transpose, * denotes complex conjugate and + denotes conjugate transpose. When w is given by Eq. (1) it can be shown by substituting (2) into (1) that S / ( N + C + J ) after adaptation is given by (3)
-- s T M - I s *.
N+C+J
In an ideal world in which the covariance could be exactly estimated, there is no internal clutter motion, channel match is perfect, the jammers are narrow-band, and there is no jammer multipath it can be shown that S -
-
N + C+J
Isl 2 --,
(4)
r~2
where Isl 2 is the signal power and 0.2 is the thermal noise power. Thus, under ideal conditions the adaptive processor can cancel all clutter and interference, and all that remains after adaptation is thermal noise. Although the architecture shown in Fig. 2 represents an element space configuration, a beam space architecture is equivalent. For example, if a beamformer is placed behind the elements in Fig. 2, the N × 1 voltage vector b at the beamformer output can be written as b = GU, where G is the N x N beamformer matrix. Then, if we note that the new steering vector after beamforming is Sb = Gs and that (bb +) = G ( U U + ) G + = G M G +
we can readily show that
S N~-~-J
)
= s~(bb+)_ls,=
sTM_Is ,
(5)
beam space
so that upon comparing (5) with (3) we see the output signal-to-interference ratio is the same whether an element space or beamspace architecture is used. However, sometimes beamspace may be preferable to element space, because it allows for the development of suboptimum architectures more readily, as will be discussed later.
4
R . L . Fante
]
]
-10
Main Beam Clutter Eigenvalues
At -20
~
I
L
-30
-40 -
W
Sidelobe Clutter
Eigenvalues
...................
~
-50
~
-60 -
T
.
/
. '
.
.
.
T i
.
.
.
.
/
-70 -
. . . . . . . . . '.....................................
...................
-80 -
-90 -
i
0
10
20
'
k
I
IN+KI=52, ' I
40
50
Eigenvalue
i ..............................
,.n.ues
........................... . . . . . . . . . . . . .
i
'
30
/io.
60
l '
70
'
80
'
90
100
Number
FIG. 3. Clutter eigenvalue spread for N = 50, K = 2 (no jammers). II. Understanding the Results in (2) and (3) In practice one must estimate the covariance matrix using a finitet number of samples, but for purposes of understanding we will assume that the number of samples is so large that the ideal covariance matrix is available. For the architecture in Fig. 2 this is an N K × N K square matrix M that can be decomposed into its N K × 1 eigenvectors ek and eigenvalues 2k as (2) NK
M = ~ 2~eke~
(6)
k=l
Likewise M
1 = ~ eke~k~l 2k
(7)
Because M is Hermitian the eigenvalues are real and the eigenvectors are orthogonal. Under ideal conditions (no internal clutter motion, narrowband radar, etc.) there are N + K large eigenvalues associated with the mainbeam and sidelobe clutter [for ambiguous radars there are even more (3, 4)]. A typical eigenvalue strength plot for clutter only (no jammers) is shown in Fig. 3. Strong jammers produce additional large eigenvalues. If there are Nj independent, 1"For an acceptable covariance estimate the number of samples must be at least 2NK and preferably 4NK, where NK is the total number of adaptive degrees of freedom.
Adaptive S p a c e - T i m e Radar
5
80
70
I Jammer Correlation = exp(-B21:2) ]
60 ¢D 7o
=
.................................................
50
~-. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
.........................................
t. . . . . . . . . . . . . .
; .............................
~. . . . . . . . . . . . . . . . . . . . .
~.
~ ,-'~"
!'
;
!
40
.................
.
.
.
.
.
.
.
.
.
.
.
.
.
._ 30 W
El g
.
.
.
.
'. .
10
0 -"~'~
t
S
i~
i
,
.
i
:
Eigenvalue3to 16 i
i
,
0
.
::~
1
~
~
2
.
.
.
.
.
,
i
i
3
4
5
B'C Correlated Uncorrelated
-=
.
6
=
FIG. 4. Effect o f j a m m e r correlation on eigenvalues for two resolved j a m m e r s when N = 16, K=I.
resolved (i.e. spaced in angle by greater than the 3 dB beamwidth of the antenna) jammers there are Nj large jammer eigenvalues. However, unresolved jammers and correlated jammers share a single eigenvalue, as indicated in Figs 4 and 5. Thus, the 70 60
Eigenvalue
I
50 CL
~ 40
LU
:.........
20
i . . . . . . . . . . . . . . . .
//Eigenvalue 2
10 j,
0 ~ '~S
/
-- i
• Eigenvatue3to16 i r
0
1
2
,
i
3
4
5
Angular Separation/3 dB Beamwidth FIG. 5. Effect o f j a m m e r separation 0sep on eigenvalues for N = 16, K = 1, Bz = 2.
6
R . L . Fante
covariance matrix M typically has R = N + K + N j large eigenvalues and N K - R small eigenvalues with magnitudes equal to the variance 0.2 of the system noise. Consequently, Eq. (7) can be rewritten as M _ 1 ~- ~ eke~ 1 NK k= , 2k + ~ k ~+ l eke~
(8)
Next, we recognize that the N K x N K identity matrix I is defined as (2) NK
I = ~ eke + k=l R
NK
= • eke;+
~
k=l
eke~
(9)
k=R+l
where the first term on the right-hand side of Eq. (9) defines the interference (clutter plus jammers) subspace and the second defines the noise subspace. Thus, we can rewrite Eq. (9) as a sum of two projections I = Pll
(10)
+P±
where P~j is the projection into the interference subspace and Pz is the projection orthogonal to the interference subspace. Therefore, Eq. (8) can be rewritten as
M-J = ~ e~e~k=l~-
1
+ ~P~
(ll)
If Eq. (11) is now used in (2) we see that the weight vector w is R
ekeffs,
w = t, ky,' ~ -
- + ~Pj.
=
0.2
(12)
Because, as can be seen from Fig. 3, 1241 >> 0.2 we can neglect the first term on the righthand side of the Eq. (12) so that (13)
w ~- ~ P ± s * . 0 .2
This shows that the desired weight vector is the projection of the steering vector s onto the subspace orthogonal to the interference subspace, and means that as long as the steering vector has a component orthogonal to the interference subspace it is possible to cancel the interference while preserving the desired signal (or at least most of it). It should also be noted that because [2k] >> a 2 we can use Eq. (9) to write Eq. (8) as M- 1
1 NK
1
Adaptive Space-Time Radar
7
This is a useful approach for inverting the covariance matrix when there are only a few dominant eigenvalues, and has been exploited in (6). By using Eq. (14) in (3) it is easy to see how well the limit in (4) is approached.
IlL Sequential Cancelation of Jammers and Clutter In practice, when both jammers and clutter are simultaneously present it is difficult to cancel both because of interactions in the covariance matrix estimate. Thus, it is desirable to cancel the jammers before tackling the clutter. One way to do this is to use the beam space architecture, place spatial nulls in each beam on any jammers in that beam, and then combine those resulting beams to cancel the ground clutter. Let us analyse here how one forms L ~< N beams with nulls on any jammers present. In particular, we need to derive an L × N beamforming matrix F that produces L beams while simultaneously minimizing jammer power. Let J be an N × 1 vector consisting of the voltages on the N antenna elements produced by any jammers present. These jammers then produce a voltage vector at beam ports given by b = FJ. Also, let h~ be an N × 1 vector of the voltages produced on the N antenna elements by an incoming plane wave at an angle 01 relative to the array. Likewise, define h2 as the N x 1 vector after a wave at angle 02, etc. Then, the condition for forming L beams pointed at 0~02... is
Fkhk = Ck
(15)
for k = 1,2 . . . . L, where Fk is the 1 x N vector formed by the kth row of F and the G are specified constraints. Now we desire to minimize the interference power (b+b) subject to the constraints in (15). The solution is
F = flDGR -1
(16)
R = (JJ+)
(17)
where
is the jammer plus noise covariance matrix, G is the L × N beamforming matrix in the absence of jammers, D is a diagonal matrix with the kth diagonal element given by Ck/(h~ R - ~h~) and fl is a constant. Note that if the jammers are absent the jammer plus noise covariance matrix reduces to the noise covariance matrix a2Is, where IN is the N × N identify matrix, so that, as expected, the beamforming matrix F reduces to the jammer-free beamforming matrix G. One may next wonder how one estimates the jammer covariance matrix when both jammers and clutter are present. This can be done by referring to the angle-Doppler plot shown in Fig. l, and recognizing that the jammers spread through all Doppler bins. Thus, if one could choose an angle-Doppler region that is well removed from the peak of the clutter ridge it should be possible to estimate the jammer covariance with little clutter interaction. Of course, for radars that are unambiguoust in Doppler this t Remember that the clutter only occupies the frequency region from + 2 Vj2 so that if the pulse repetition frequency exceeds this value there is a clutter-free region in the Doppler domain.
R. L. Fan te
P°rte-~~ . . ~ ~--Ib-~ ~----~.-~ Beam
C;Swm,
C;Swm
C;Swm, T_, 1
From
Beam p
~.1
(FFT)
DopplerProcessing
From Beamq FIG. 6. Beam-pulse space processor. is a trivial task because there are then regions in Doppler space where the clutter is completely absent, but the jammers are present. Once the jammers have been removed the clutter can then be cancelled either in the beam-pulse domain (also called pre-Doppler) as in Fig. 6 or in the beam-Doppler domain (also called post-Doppler) as in Fig. 7. The architecture in Fig. 7 is what is known as first-order factored processing where the beams in each doppler frequency bin are independently used to cancel clutter. That is, if B(p) is an N x 1 vector of the outputs of the N beam ports for Doppler frequency binp and S(p) is the N x 1 steering vectorS" for the Doppler frequency bin p then the weight vector WR used to combine the beam outputs in Doppler frequency bin p is We = ~ ( B ( p ) B + ( p ) )
'S*(p)
08)
where/~ is a constant. In practice, however, it has been found that combining two Doppler bins produces superior cancellation performance. The reader is referred to (5) for details.
IV. Typical Results There are several experimental programs to test the adaptive cancellation methods discussed in the last few sections. One is called Mountaintop (Lincoln Laboratory) and t S(p) is simply the discrete Fourier transform of the time domain steering vector s.
Adaptive Space-Time Radar Beam
From
Beam
/'~'~ From p-'~M'Beamq
Doppler Bin 1
Output
From
/"~ "~.=~_ From
Beam p ' - ~ . ~
J ~
Beam
Doppler Bin L Output
FIG. 7. Beam-Doppler space processor (first order factored).
is designed to evaluate space-time processing using emulated motion of the radar platform. This program has demonstrated excellent cancellation at UHF of emulated airborne clutter at the White Sands Missile Range. Another is called MCARM (Northrup Grumman & Rome Laboratory) and consists of actual BAC-111 flights of a 1.25 GHz adaptive array with 22 spatial degrees of freedom. In Fig. 8 we show the MCARM signal-to-clutter plus noise ratio (there were no jammers present) before and after adaptation when the processing architecture in Fig. 7 is used (7). Note that up to 40 dB of clutter cancellation has been achieved. The clutter is not cancelled completely for a number of reasons including the fact the first order factored processing (as in Fig. 7) is suboptimal and that the clutter is spatially inhomogeneous, so that the covariance matrix estimate is not accurate for all ranges. Additional results are available, but will be omitted for conciseness.
V. Additional Considerations In this brief exposition we have ignored many of the implementation problems involved in the actual performance of a space-time adaptive array. Some of these are: how does one obtain sufficient samples to estimate the covariance matrix and what are the effects of inhomogeneous clutter on its value?; what are the effects of channel mismatch, radar-aircraft crabbing and scatter from obstacles such as airplane wings that are in the near-field of the array?; is the computational load reasonable?; and finally how does jammer multipath affect the interference cancellation? All of these points have been investigated to some degree. Jammer multipath is especially serious
l0
R. L. Fante
30
~-'0
-40
........
. ........
. ........
; .......
; ........
! ........
7........
i. .......
. .... t
t ....... "-~........
:~........
:i .......
~: .......
:i
:~
:"
~
:: .......
i ........
i ........
- .......
i .......
._...
....
i
250
•
. .i ........
300
350
400 450 Range Bin
500
550
600
FIG. 8. Improvement in signal to interference ratio for a particular Doppler bin and receive azimuth using first order factored processing. The clutter free SNR = 20 dB.
because the radiation of an airborne sidelobe jammer can be scattered off the ground into the mainbeam (or its elevation sidelobes) of the radar thus giving the appearance of multiple mainbeam jammers. Because the interference is spread in all azimuths across the main beam it cannot be canceled, without simultaneously canceling the target signal, using spatial nulls. Thus, time domain cancellation techniques are required (8), but these are beyond the scope of this paper.
VI. Summary We have given a brief exposition of how a space-time adaptive canceller can remove the deleterious effects of both ground clutter and jammers, thus enabling an airborne radar to detect weak targets. In particular we have shown how the processor accepts target signals, but rejects anything that does not look like a target, as specified by the target's azimuth and Doppler frequency. Thus, in principle, all interference except system noise can be eliminated.
Adaptive Space-Time Radar
11
References (1) L. Brennan and I. Reed, "Theory of adaptive radar", IEEE Trans, Vol. AES-9, pp. 237252, 1973. (2) B. Friedman "Principle and Techniques of Applied Mathematics", Wiley, New York, 1962. (3) J. Ward, "Space-time adaptive processing for airborne radar," MIT Lincoln Lab Report 1015, 1994. (4) L. Brennan and F. Staudaher, "Subclutter visibility demonstration", Adaptive Sensors Inc. Tech. Report RL-TR-92-21, 1992. (5) R. DiPietro, "Extended factored space-time processing for airborne radar systems", "'Proc. of the 26th Asilomer Conference on Siynals, Systems and Computing", pp. 425~430, 1992. (6) A. Haimovich, M. Puch and M. Baren, "Training and signal cancellation in adaptive radar", Diyest of the 1996 IEEE Radar Conference, pp. 124-128, Ann Arbor, 1996. (7) B. Suresh-Babu, J. Torres and W. Melvin, "Processing and Evaluation of Multichannel Airborne Radar Measurements (MCARM)", MTR96B0000053, The MITRE Corporation, Bedford, MA, 1996. (8) R. Fante and J. Tortes, "Cancellation of diffuse jammer multipath by an airborne adaptive radar", IEEE Trans, Vol. AES-31, pp. 805-820, 1995.
Bibliography E. Barile, R. Fante and J. Torres, "Some limitations on the effectiveness of airborne adaptive radar", IEEE Trans, Vol. AES-28, pp. 1015-1032, October 1992. L. Brennan, J. Mallett and I. Reed, "Adaptive arrays in airborne MTI radar", IEEE Trans, Vol. AP-24, pp. 607-615, September 1976. H. Wang and L. Cai, "On adaptive spatial-temporal processing for airborne surveillance radar systems", IEEE Trans, Vol. AES-30, pp. 660-669, July 1994. R. Klemm, "Adaptive clutter for airborne phased array radars", lEE Proceedings, Parts F and H, Vol. 130, pp. 125-132, 1983. R. Monzingo and T. Miller, "Introduction to Adaptive Arrays", Wiley, New York, 1980. R. Compton, "Adaptive Antennas", Prentice Hall, Englewood Cliffs, New Jersey, 1988. A. Farina, "Antenna-based Signal Processing Techniques for Radar Systems", Artech House, Norwood, Massachusetts, 1992.