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:
[email protected] ,
[email protected] ) Conference ICASSP 2009, April 19-24, 2009, Taipei, Taiwan
Content of the talk Context:
MIMO radar system.
Problem:
Detection and localization of multiple targets present in the same range-bin.
State of the art: Radar-imaging localization methods (e.g. Capon, MUSIC) Limits: Radar-imaging fails for closely spaced targets + sensitivity to Radar Cross Section (RCS) fluctuations Contribution: Novel method, deterministic, exploits multilinear algebraic structure of received data Æ PARAFAC Decomposition of an observed tensor 2
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
3
I. Introduction: problem statement = target
Tx
Rx
Æ K targets in the same range-bin Æ Transmitter Tx and receiver Rx equipped with closely spaced antennas Æ Target = a point source in the far field Problem : estimate the number of targets and their DODs and DOAS 4
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 5
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».
X q = B(α )diag ([ β1q ,..., β Kq ]) A (θ )S + Wq , q = 1,..., Q T
Mr x L received data
= Σq
ÆTimes of arrival known (targets in the same range-bin). Æ Right multiply by (1/L)SH and simplify (1/L)SSH = I
Yq = B(α )Σ q A T (θ ) + Z q , q = 1,..., Q Mr x Mt received data after matched filtering
6
II. State of the art: single-pulse radar-imaging Radar-imaging techniques working on per-pulse basis:
X q = B(α )Σ q A (θ )S + Wq , q = 1,..., Q T
Æ Beamforming techniques [Xu, Li & Stoica]. Example: Capon Beamforming . Suppose colocated arrays (α=θ). H −1 H * ( ) b θ R X S a (θ ) XX q ˆ , β (θ ) = H −1 T * L[b (θ )R XX b(θ )][a (θ )a (θ )]
R XX
1 H = X qX q L
Æ MUSIC estimator.
PMUSIC (θ ) =
1 , Ew = noise eigenvectors of R XX H H b (θ )EwEw b(θ ) 7
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 Æ scintillation due to fading 8 (fluctuations of RCS coeff. from pulse to pulse)
II. State of the art: multiple-pulses radar-imaging Q : Mitigate RCS fluctuations? Æ first need a multi-pulse data model
Yq
= B(α )
A (θ ) +
Σq
T
Zq
vectorize
y q = [a(θ1 ) ⊗ b(α1 ),..., a(θ K ) ⊗ b(α K )][ β q1 ,..., β qK ] + z q T
= A (θ )
.
B(α )
= cTq
Q pulses (concatenation)
Y = [ A (θ )
.
B (α )] C T + Z 9
II. State of the art: multiple-pulses radar-imaging Radar-imaging techniques working on a multi-pulse basis:
. [ A ( θ ) B(α )] CT = Y Capon beamforming [Yan, Li, Liao]
PCapon
1 (θ , α ) = −1 ( a (θ ) ⊗ b (α )) H R YY ( a (θ ) ⊗ b (α ))
MUSIC
PMUSIC
1 (θ , α ) = H H ( a (θ ) ⊗ b (α )) E w E w ( a (θ ) ⊗ b (α ))
E w = noise eigenvecto rs of R YY 10
II. State of the art: multiple-pulses radar-imaging
K = 5, {θ k } = {40 °,35 °,30 °, − 40 °,65 °}, {α k } = {20 °,25 °,30 °,50 °, − 45 °} Capon, Mt=Mr=4
Capon, Mt=Mr=9
MUSIC, Mt=Mr=4
MUSIC, Mt=Mr=9
11
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 Æ time consuming for dense angular grid
Æ Our contribution: starting from the same data model,
Y = [ A (θ )
. B (α )]
C
T
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.
12
III. Localization via PARAFAC: model
Yq
Σq
= B(α )
A (θ ) + T
Zq
Mr x Mt matrix observed Q times, q=1,…,Q. B(α) and A(θ) fixed over Q pulses. Mt
QxK matrix C, [C]qk=βqk
Q Q K K
=
Mr
Y
Mr
B(α) c1
=
a(θ1) b(α1)
Mt
cK + … +
AT (θ )
a(θK) b(αK)
PARAFAC decomposition: Y =Sum of K rank-1 tensors. Each target contribution is a rank-1 tensor 13
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. Æ 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: ¾ Columns of A(θ), B(α) and C arbitrarily permuted (same permutation) ¾ 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). 14
III. Localization via PARAFAC: uniqueness Condition 1: A(θ) and B(α) full rank and C full-column rank. If
K ≥ 2 and M t(M t − 1 )M r(M r − 1 ) ≥ 2 K ( K − 1) then uniqueness is guaranteed a.s. [De Lathauwer]. Condition 2: A(θ) and B(α) are full rank Vandermonde matrices and C fullcolumn rank. If
max( M t , M r ) ≥ 3 and M t M r − min( M t , M r ) ≥ K 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 15
III. Localization via PARAFAC: simulations
K = 5, {θ k } = {40 °,35 °,30 °, − 40 °,65 °}, {α k } = {20 °,25 °,30 °,50 °, − 45 °}
Mt=Mr=4 CAPON
Mt=Mr=9
Mt=Mr=4
Mt=Mr=4
MUSIC
PARAFAC
Mt=Mr=9
Mt=Mr=9
16
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°)
17
IV. Conclusion PARAFAC = deterministic alternative to radar-imaging (Capon, MUSIC, etc) ¾ Guaranteed identifiability ¾ RCS fluctuations from pulse to pulse = time diversity = 1 dimension of the observed tensor PARAFAC outperforms MUSIC and Capon ¾ Peak detection in radar-imaging fails for closely located targets ¾ PARAFAC = estimation based on exploitation of strong algebraic structure of observed data.
Extension (work in progress): Generalization to the case of multiple sufficiently spaced transmit and receive sub-arrays. 18
Appendix: Target tracking via adaptive PARAFAC « Adaptive algorithms to track the PARAFAC decomposition » [Nion & Sidiropoulos 2009] J
K I
R
R
R
PARAFAC
Y(t )
I
K
J
C(t )
A(t ) Time
B(t )
LINK = adaptive algorithms to track the PARAFAC decomposition J+1
K I
R
Y(t + 1)
I
PARAFAC New Slice
R J+1
A(t + 1)
R K
C(t + 1)
B(t + 1)
19
Appendix: Target tracking via adaptive PARAFAC 5 moving targets. Estimated trajectories. Comparison between Batch PARAFAC (applied repeatedly) and PARAFAC-RLST (« Recursive Least Squares Tracking »)
20
