Parametric Clutter
Rejection for Space-Time
Adaptive Processing
A. Lee Swindlehurst and Peter Parker
Brigham Young University
Dept. of Electrical & Computer Engineering
Provo, UT 84602
voice: (801) 378-4343
fax: (801) 378-6586
email: swindle@ee.byu.edu
A. Lee Swindlehurst
Department of Electrical & Computer Engineering
Brigham Young University
Provo, UT
84602
Motivation:
Motivation: Circular
Circular
Array
Array STAP
STAP
• Small sample-support interference rejection
techniques needed for non-stationary clutter
• Offset need for larger sample-support due to
increased clutter rank
• Use small sample-support plus parametric modeling
to reduce computation
A. Lee Swindlehurst
Department of Electrical & Computer Engineering
Brigham Young University
Provo, UT
84602
Data Model
•
M antennas, N pulses
•
Target in primary range bin p
x p (t ) = ba(θ )e jωt + c p (t ),
spatial steering vector
•
t = 0, 1, 2, L, N − 1
clutter, jammer, noise, etc.
Space-Time Slice
[
X p = x p (0 )
x p (1) L
x p (N − 1)]
= ba(θ )v T (ω ) + C p
[
temporal
Mω/7 V
=
1
H
steering vector
A. Lee Swindlehurst
Department of Electrical & Computer Engineering
L H M (1 −1) ω/7V
Brigham Young University
]
Provo, UT
84602
Data Model (cont.)
•
Vectorized Forms
1. [ S = YHF ( ; S )
= E Y ( ω) ⊗ D ( θ ) + F S
2. [ S = YHF ( ; 7S )
= E D (θ ) ⊗ Y (ω ) + F S
•
Secondary Data
target-free
range bins
A. Lee Swindlehurst
{F N } N = 1, L , 1 V
N ≠S
( (F N ) = 0 , ( (F N F N* ) = 5
Department of Electrical & Computer Engineering
interference
covariance
Brigham Young University
Provo, UT
84602
Space-Time
Autoregressive Modeling
L
z ) = ∑ H i z −i
−1
• Define
i =0
M ‘ x M matrices
• Model: for some L,
( z −1 )ck (t ) = H0ck (t ) + H1ck (t − 1) + L + HLck (t − L)
= k (t )
is spatially and temporally white
• To estimate
(z −1 ) , solve
Ns
min
H 0 ,L , H L
A. Lee Swindlehurst
closed-form
least-squares
solution
N
∑ ∑
k =1
i = L +1
( z )ck (i )
−1
Department of Electrical & Computer Engineering
2
Brigham Young University
Provo, UT
84602
Least-Squares Solution
• Define
H * = [H 0 L H L ]
• Then
Ns
N
k =1
i = L +1
∑ ∑
c k (N ) 
ck (L + 1)

Ck = 
M
L
M

 ck (1)
ck (N − L )
( z −1 )c k ( i )
2
[
= H * C1 L C Ns
]
2
• Solution: columns of H are M’ least dominant
singular vectors of:
[C
A. Lee Swindlehurst
1
L C Ns
]
M(L+1) x Ns (N-L) matrix
Department of Electrical & Computer Engineering
Brigham Young University
Provo, UT
84602
Prior Work
• Vector AR models used previously for clutter
modeling by Michels, Rangaswamy, etc.
• Used to develop GLRT and AMF detectors for
Gaussian and non-Gaussian assumptions
• Here, we use AR model to generate structured
subspace orthogonal to clutter subspace
• Clutter is then projected out of primary data prior
to space-time filtering
A. Lee Swindlehurst
Department of Electrical & Computer Engineering
Brigham Young University
Provo, UT
84602
An Interpretation
STAR filter attempts to minimize clutter power:
dimension
M‘(N-L) x 1
k
0L 0
 HL HL−1 L H0


O
M
 c =+ c
=0 O
k
k
0
M

O
O


L
L
0
0
H
H
H
0
L
L−1

Least squares algorithm finds block Toeplitz +that minimizes
Ns
∑
k =1
2
k
Ns
= ∑ + ck
2
k =1
Span(+ ) orthogonal to clutter subspace if it dominates white noise:
R = + ⊥Q +
⊥
+
clutter & jamming
A. Lee Swindlehurst
Department of Electrical & Computer Engineering
2
I
white (thermal) noise
Brigham Young University
Provo, UT
84602
Filtering the Primary Data
Clutter lies in span of + , so we project it out:
(
)
x p = + ++
+ xp
123
/
banded block Toeplitz
Like an eigencanceller, but here subspace is structured.
Occasionally, a more computationally efficient approach is:
• Use SVD to find orthogonal complement of +:
++
⊥
=0
dimension MN x (M-M’)N+M’L
• Filter data as follows:
(
x p/ = I − + ⊥+
A. Lee Swindlehurst
Department of Electrical & Computer Engineering
⊥
)x
p
Brigham Young University
Provo, UT
84602
Algorithm Summary
[
1. Use SVD on secondary data to solve for H0 H1 L HL
computational order:
2. Form +and filter data:
(
)
]
O Ns M 2 (L + 1) (N − L )
2
P+ x p
(
)
computational order: O M / M 2 (L + 1)2 (N − L )
3. Perform regular beam and Doppler filtering for detection
computational order:
negligible
−1
Resultant test statistic is (v ⊗ a ) P+ * x p not (v ⊗ a ) R x p
*
A. Lee Swindlehurst
Department of Electrical & Computer Engineering
*
Brigham Young University
Provo, UT
84602
STAR
STAR Filter
Filter Example
Example
18 element ULA, 18 pulses
SJR = -40dB SCR = -40dB SNR = 10dB
Fully adaptive STAP requires ~60 secondary snapshots for similar performance
A. Lee Swindlehurst
Department of Electrical & Computer Engineering
Brigham Young University
Provo, UT
84602
Jammer
Jammer Obscured
Obscured Target
Target
18 element ULA, 18 pulses
SJR = -40dB SCR = -40dB SNR = 10dB
Fully adaptive STAP requires ~70 secondary snapshots for similar performance
A. Lee Swindlehurst
Department of Electrical & Computer Engineering
Brigham Young University
Provo, UT
84602
Computational Comparison
Some typical numbers: M = 20, N = 18, L = 2, M / = 15
• Proposed approach:
O(57,600Ns ) + O(864,000)
• Fully adaptive algorithm:
(
) (
)
O Ns M 2N 2 + O M 2N 2 = O(130,000Ns ) + O(9,700,000)
rank of clutter covariance ≅ 75
• PRI staggered algorithm:
(
) (
O N s M 2K 2 N + O
# of sub-CPIs = 3
A. Lee Swindlehurst
)
M 2K 2N = O (65,000Ns ) + O( 2,900,000 )
rank of sub-CPI covariance ≅ 45
Department of Electrical & Computer Engineering
Brigham Young University
Provo, UT
84602
Model Order Selection
When correctly specified, structure of +implies clutter rank
equals MN − M / (N − L ) = 360 − (18 − L ) M /
Approach for simulations:
1. Pick “arbitrary” value for M /
penalty term
2. Choose L using MDL:
L = arg min
p
Ns
∑
k =1
(p )
k
2
 M / M (p + 1) − M / (M / + 1) / 2 

+ log2 
Ns MN


prediction error when L = p
For M / = 15, L = 1-2 for long ranges, 2-3 for short ranges
A. Lee Swindlehurst
Department of Electrical & Computer Engineering
Brigham Young University
Provo, UT
84602
Model Order Sensitivity
50 km Range
300 km Range
20
16
18
M’=5
M’=10
M’=15
M’=20
M’=30
12
Average SINR loss (dB)
Average SINR loss (dB)
16
M’=5
M’=10
M’=15
M’=20
M’=30
14
14
12
10
8
6
10
8
6
4
4
2
2
0
0
2
4
6
8
10
12
14
0
0
L
A. Lee Swindlehurst
Department of Electrical & Computer Engineering
2
4
6
8
10
12
L
Brigham Young University
Provo, UT
84602
14
Sample SINR Loss
50 km Range, Training length 30 samples
30
20
SINR (dB)
10
0
−10
Optimum
STAR
PRI
FULL
−20
−30
−40
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
Normalized Doppler
A. Lee Swindlehurst
Department of Electrical & Computer Engineering
Brigham Young University
Provo, UT
84602
Average SINR Loss
25
20
15
SINR (dB)
10
5
Ave. SINR loss is
area between curves
0
−5
−10
−15
−20
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
Normalized Doppler
A. Lee Swindlehurst
Department of Electrical & Computer Engineering
Brigham Young University
Provo, UT
84602
Performance at Short Ranges
20 km Range
50 km Range
20
20
M’=15
M’=20
PRI
FULL
M’=15
M’=20
PRI
FULL
16
Average SINR loss (dB)
Average SINR loss (dB)
18
15
10
14
12
10
8
6
4
2
5
1
10
2
10
0
1
10
2
10
Training length
Training length
No jammer present
A. Lee Swindlehurst
Department of Electrical & Computer Engineering
Brigham Young University
Provo, UT
84602
Range-Varying STAR Filter
• To improve performance at short ranges, use linearly
varying matrix coefficients:
Ns 

H i (k ) = H i +  k −
 + i , k = 1,L, Ns
2 

coefficient i for secondary snapshot k
• Analogous to ESMI technique of Hayward
• Solution found from SVD of



A. Lee Swindlehurst
C1
L
Ns
2
L
( )C
1−
1
CNs −1
( )C
Ns
−1
2
Department of Electrical & Computer Engineering
Ns −1
CN s 

Ns
CNs 
2
Brigham Young University
Provo, UT
84602
Performance with Range-Varying Weights
30 km Range L=4
30 km Range L=3
12
12
ESMI−−15
ESMI−−20
STAR−−15
STAR−−20
PRI
FULL
10
Average SINR loss (dB)
Average SINR loss (dB)
10
8
6
4
ESMI−−15
ESMI−−20
STAR−−15
STAR−−20
PRI
FULL
8
6
4
2
2
0
1
10
2
10
0
1
10
2
10
Training Length
Training Length
No jammer present
A. Lee Swindlehurst
Department of Electrical & Computer Engineering
Brigham Young University
Provo, UT
84602
Performance at Long Range
300 km Range
8
7
M’=15
M’=20
PRI
FULL
Average SINR loss (dB)
6
5
4
3
2
1
0
1
10
2
10
Training length
A. Lee Swindlehurst
Department of Electrical & Computer Engineering
Brigham Young University
Provo, UT
84602
Performance vs. Range with Jammer
20km Range
300km Range
40
20
18
M’=15
M’=20
PRI
FULL
16
30
25
20
15
M’=15
M’=20
PRI
FULL
10
5
Average SINR loss (dB)
Average SINR loss (dB)
35
14
12
10
8
6
4
2
0
1
10
2
10
0
1
10
Training data length
2
10
Training data length
JCR = 10 dB, Jammer DOA 30 deg., SINR Loss at 0 deg.
A. Lee Swindlehurst
Department of Electrical & Computer Engineering
Brigham Young University
Provo, UT
84602
Effect of Sample Support
Training length 30 samples
20
18
18
M’=15
M’=20
PRI
FULL
14
12
10
8
6
14
12
10
8
6
4
4
2
2
0
50
100
150
200
250
M’=15
M’=20
PRI
FULL
16
Average SINR loss (dB)
16
Average SINR loss (dB)
Training length 100 samples
20
300
350
0
0
50
Range (km)
100
150
200
250
300
350
Range (km)
No jammer present
A. Lee Swindlehurst
Department of Electrical & Computer Engineering
Brigham Young University
Provo, UT
84602
400
Performance vs. Sample Support
with Jammer
100 secondary data vectors
30 secondary data vectors
25
M’=15
M’=20
PRI
FULL
20
Average SINR loss (dB)
Average SINR loss (dB)
25
15
10
5
0
M’=15
M’=20
PRI
FULL
20
15
10
5
0
50
100
150
200
250
300
350
0
0
Range
50
100
150
200
250
300
350
Range
JCR = 10 dB, Jammer DOA 30 deg., SINR Loss at 0 deg.
A. Lee Swindlehurst
Department of Electrical & Computer Engineering
Brigham Young University
Provo, UT
84602
3-D STAR Filter for
Hot Clutter
Idea: add matrix taps to exploit correlation in fast-time:
P
min ∑
Ns
N
∑ ∑
p =0 k =1 i =L +1
p
( z )ck − p (i )
−1
2
For example, with P=2, solution involves SVD of
C2 L CNs −1 CNs 


C1 L CNs −2 CNs −1 
Low computation, sample support make full 3-D STAR
solution feasible for larger P as well
A. Lee Swindlehurst
Department of Electrical & Computer Engineering
Brigham Young University
Provo, UT
84602
Hot Clutter Examples
30 km Range L=2
30 km Range L=1
30
30
3D−−15
3D−−20
STAR−−15
STAR−−20
3D−−15
3D−−20
STAR−−15
STAR−−20
25
Average SINR loss (dB)
Average SINR loss (dB)
25
20
15
10
20
15
10
5
5
0
1
10
2
10
0
1
10
Training Length
A. Lee Swindlehurst
Department of Electrical & Computer Engineering
2
10
Training Length
Brigham Young University
Provo, UT
84602
Conclusions
• STAR filtering ideal for STAP problems that
require small secondary sample support
• Structured model yields computationally efficient
solution
• Simulations with realistic circular array data show
promising performance
• Easily extended to handle hot or range-varying
clutter models
A. Lee Swindlehurst
Department of Electrical & Computer Engineering
Brigham Young University
Provo, UT
84602