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by Dimitri Nion* & Nicholas D. Sidiropoulos* * Technical - - PowerPoint PPT Presentation
by Dimitri Nion* & Nicholas D. Sidiropoulos* * Technical - - PowerPoint PPT Presentation
A PARAFAC-based technique for detection and localization of multiple targets in MIMO radar systems by Dimitri Nion* & Nicholas D. Sidiropoulos* * Technical University of Crete, Chania, Crete, Greece (E-mails: nikos@telecom.tuc.gr ,
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Roadmap
- I. Introduction (problem statement + data model)
- II. State of the art (Localization via Capon beamforming and MUSIC)
- III. Localization via PARAFAC
- IV. Conclusion and perspectives
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- I. Introduction: problem statement
Tx Rx
K targets in the same range-bin Transmitter Tx and receiver Rx equipped with closely spaced antennas Target = a point source in the far field
= target
Problem : estimate the number of targets and their DODs and DOAS
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- I. Introduction: parameters
Mt transmit colocated antennas Mr receive colocated antennas K targets in the range-bin of interest A(θ)=[a(θ1), …, a(θK) ] the MtxK transmit steering matrix B(α)=[b(α1), …, b(αK) ] the MrxK receive steering matrix S=[s1(t); s2(t); …; sMt(t)] is MtxL, holds Mt mutually orthogonal transmitted pulse waveforms, with L samples per pulse period Q consecutive pulses are transmitted βkq RCS reflection coeff. of target k during pulse q
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- I. Introduction: data model
Assumption : Swerling case II target model « Receive and Transmit steering matrices B(α) and A(θ) constant over the duration of Q pulses while the target reflection coefficients βkq are varying independently from pulse to pulse».
Q q
q T q q
,..., 1 , ) ( ) ( = + Σ = Z A B Y θ α Q q diag
q T Kq q q
,..., 1 , ) ( ]) ,..., ([ ) (
1
= + = W S A B X θ β β α
q
Σ =
Mr x L received data
Times of arrival known (targets in the same range-bin). Right multiply by (1/L)SH and simplify (1/L)SSH = I
Mr x Mt received data after matched filtering
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- II. State of the art: single-pulse radar-imaging
Beamforming techniques [Xu, Li & Stoica]. Example: Capon Beamforming . Suppose colocated arrays (α=θ). Radar-imaging techniques working on per-pulse basis:
, )] ( ) ( )][ ( ) ( [ ) ( ) ( ) ( ˆ
* 1 * 1
θ θ θ θ θ θ θ β a a b R b a S X R b
T XX H H q XX H
L
− −
=
Q q
q T q q
,..., 1 , ) ( ) ( = + Σ = W S A B X θ α
H q q XX
L X X R 1 =
MUSIC estimator.
XX
R E b E E b
- f
rs eigenvecto noise = =
w H w w H MUSIC
P , ) ( ) ( 1 ) ( θ θ θ
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- II. State of the art: single-pulse radar-imaging
Typical Capon and MUSIC spectra for a given pulse
Widely spaced targets (-30°,10°,40°) Closely spaced targets (-30°,-25°,-20°)
Problem 1: single lobe occurs for closely located targets Problem 2: update spectrum for each new pulse scintillation due to fading (fluctuations of RCS coeff. from pulse to pulse)
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- II. State of the art: multiple-pulses radar-imaging
Q : Mitigate RCS fluctuations? first need a multi-pulse data model
) (α B = ) (θ
T
A
q
Σ
q
Z +
q
Y
q T qK q K K q
z b a b a y + ⊗ ⊗ = ] ,..., [ )] ( ) ( ),..., ( ) ( [
1 1 1
β β α θ α θ
.
) (θ A ) (α B =
T q
c =
Q pulses (concatenation) vectorize
Y
.
) ( [ θ A )] (α B =
T
C Z +
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- II. State of the art: multiple-pulses radar-imaging
Radar-imaging techniques working on a multi-pulse basis:
Y
.
) ( [ θ A )] (α B =
T
C
Capon beamforming [Yan, Li, Liao]
)) ( ) ( ( )) ( ) ( ( 1 ) , (
1
α θ α θ α θ b a R b a ⊗ ⊗ =
− YY H Capon
P
MUSIC
)) ( ) ( ( )) ( ) ( ( 1 ) , ( α θ α θ α θ b a E E b a ⊗ ⊗ =
H w w H MUSIC
P
YY w
R E
- f
rs eigenvecto noise =
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- II. State of the art: multiple-pulses radar-imaging
MUSIC, Mt=Mr=4 MUSIC, Mt=Mr=9 Capon, Mt=Mr=9 Capon, Mt=Mr=4
{ } { } { } { }
° − ° ° ° ° = ° ° − ° ° ° = = 45 , 50 , 30 , 25 , 20 , 65 , 40 , 30 , 35 , 40 , 5
k k
K α θ
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- III. Localization via PARAFAC: overview
Problems: Capon and MUSIC 2D-imaging work on multi-pulse basis but fail if no distinguishable lobe for each target (e.g. closely located targets) Capon and MUSIC spectra have to be computed for each pair of angles time consuming for dense angular grid Our contribution: starting from the same data model,
Y
.
) ( [ θ A )] (α B =
T
C
exploitation of the algebraic structure of Y is sufficient for blind estimation of A(θ), B(α) and C. Indeed Y follows the well-known PARAFAC model.
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- III. Localization via PARAFAC: model
Mr x Mt matrix observed Q times, q=1,…,Q. B(α) and A(θ) fixed over Q pulses.
) (α B = ) (θ
T
A
q
Σ
q
Z +
q
Y
=
( ) θ
T
A ( ) α B
Y
Mr Mt Q Mt Mr Q K K
QxK matrix C, [C]qk=βqk cK
+
c1 b(α1)
+ …
= a(θ1) b(αK) a(θK) PARAFAC decomposition: Y =Sum of K rank-1 tensors. Each target contribution is a rank-1 tensor
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- III. Localization via PARAFAC: summary
Given the (MrxMtxQ) tensor Y , compute its PARAFAC decomposition in K terms to estimate A(θ), B(α) and C. Several algorithms in the literature (e.g. Alternating Least Squares (ALS), ALS+Enhanced Line Search, Levenberg-Marquardt, Simultaneous Diagonalization, …) Key point: under some conditions (next slide), PARAFAC is unique up to trivial indeterminacies: Columns of A(θ), B(α) and C arbitrarily permuted (same permutation) Columns of A(θ), B(α) and C arbitrarily scaled (scaling factor removed by recovering the known array manifold structure on the steering matrices estimates, after which the DODs and DOAs are extracted).
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- III. Localization via PARAFAC: uniqueness
Condition 1: A(θ) and B(α) full rank and C full-column rank. If
) 1 ( 2 1 1 2 − ≥ − − ≥ K K ) (M )M (M M K
r r t t
and
then uniqueness is guaranteed a.s. [De Lathauwer]. Condition 2: A(θ) and B(α) are full rank Vandermonde matrices and C full- column rank. If
K M M M M M M
r t r t r t
≥ − ≥ ) , min( 3 ) , max( and
then uniqueness is guaranteed a.s. [Jiang, Sidiropoulos, Ten Berge]. Mt=Mr 3 4 5 6 7 8 Kmax condition 1 4 9 14 21 30 40 Kmax condition 2 6 12 20 30 42 56
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- III. Localization via PARAFAC: simulations
CAPON MUSIC PARAFAC
Mt=Mr=4 Mt=Mr=9 Mt=Mr=4 Mt=Mr=9 Mt=Mr=9 Mt=Mr=4
{ } { } { } { }
° − ° ° ° ° = ° ° − ° ° ° = = 45 , 50 , 30 , 25 , 20 , 65 , 40 , 30 , 35 , 40 , 5
k k
K α θ
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- III. Localization via PARAFAC: simulations
K=4 targets, Mt = Mr = 4 and Mt =Mr =6, 100 Monte-Carlo runs Angles randomly generated for each run (with minimum inter-target spacing of 5°)
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- IV. Conclusion
PARAFAC = deterministic alternative to radar-imaging (Capon, MUSIC, etc) Guaranteed identifiability RCS fluctuations from pulse to pulse = time diversity = 1 dimension of the observed tensor PARAFAC outperforms MUSIC and Capon Peak detection in radar-imaging fails for closely located targets PARAFAC = estimation based on exploitation of strong algebraic structure of
- bserved data.
Extension (work in progress): Generalization to the case of multiple sufficiently spaced transmit and receive sub-arrays.
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Appendix: Target tracking via adaptive PARAFAC
« Adaptive algorithms to track the PARAFAC decomposition » [Nion & Sidiropoulos 2009]
( 1) t + A
( 1) t + Y
I J+1 K
PARAFAC
( 1) t + B ( 1) t + C
I J+1 R R R K New Slice
( ) t A
( ) t Y
I J K
PARAFAC
( ) t B ( ) t C
I J K R R R Time
LINK = adaptive algorithms to track the PARAFAC decomposition
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