Candecomp/Parafac based Array Processing David Brie CRAN UMR 7039 - PowerPoint PPT Presentation

Candecomp/Parafac based Array Processing David Brie CRAN UMR 7039 - Universit´ e de Lorraine - CNRS Winter school : Search for Latent Variables : ICA, Tensors and NMF 2 – 4 February 2015, Villard de Lans 1

Candecomp/Parafac Decompositions CP based Array Processing CP-Based Vector Sensor Array Processing 2

Candecomp/Parafac Decompositions 3

Useful matrix operations A = [ a 1 · · · a R ] , ( I × R ) B = [ b 1 · · · b R ] , ( J × R ) – Kronnecker product   A (1 , 1) B A (1 , 2) B · · · A (2 , 1) B A (2 , 2) B · · ·  , ( IJ × R 2 )   A ⊗ B =  . . . – Khatri-Rao product A ⊙ B = [ a 1 ⊗ b 1 · · · a R ⊗ b R ] , ( IJ × R ) – Outer product a ◦ b ( i, j ) = a ( i ) b ( j ) i.e. a ◦ b = ab T , rank 1 matrix. a ◦ b , ( I × J ) , a ◦ b ◦ c , ( I × J × K ) , a ◦ b ◦ c ( i, j, k ) = a ( i ) b ( j ) c ( k ) i.e. rank 1 tensor. 4

CP Decomposition of an order 3 tensor CP : Candecomp/Parafac [Harshman, 1970-1972], [Carroll - Chang, 1970] R � X ≈ a r ◦ b r ◦ c r = [ [ A , B , C ] ] r =1 A ( I × R ) , B ( J × R ) , C ( K × R ) c 1 c R b 1 b R C σ 1 σ R B a 1 a R X = = · · · A + + Tensor rank : minimal number of rank 1 tensors to represent X . Alternative writing Slice : X k = AD k ( C ) B T , k = 1 , · · · , K Unfolding : X ( KJ × I ) = ( B ⊙ C ) A T 5

CP based Array Processing Antenna arrays 6

b Direction Of Arrival (DOA) z S k Far field EM wave ⇒ plane wave θ Direction cosine k   m sin( θ ) cos( φ ) × × × × × y k = sin( θ ) sin( φ )   φ cos( θ ) x Narrowband assumption Received signal on sensor 1 : s ( t ) e j 2 π λ t λ ( t − t m ) ≈ s ( t ) e j 2 π e j 2 π Received signal on sensor m : s ( t − t m ) e j 2 π λ t m λ t � �� � � �� � complex envelope carrier λ k T d m where d m is the position of the m th sensor Phase factor : e j 2 π λ t m = e j 2 π 7

Multiple Sources P narrowband sources Steering vector of the p th source : a ( k p ) = [ e j 2 π λ k T d 1 · · · e j 2 π λ k T d M ] T X = AS T + N dim( X ) = ( M × K ) � � A = a ( k 1 ) , . . . , a ( k P ) ( M × P ) � � S = s 1 , s 2 , . . . , s P ( K × P ) DOA estimation ⇒ estimation of A ⇒ estimation of k p , p = 1 , · · · , P Unambiguous estimation ⇒ inter-element spacing ≤ λ/ 2 Eigen decomposition approaches (Music, Esprit) 8

bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc Rotational Invariance Basic idea of Esprit : the array can be decomposed into 2 translated but otherwise identical subarrays ⇒ GEVD of stacked data corresponding to each subarray Two non-overlapping subarrays Two overlapping subarrays The idea of Esprit is difficult to extend to multiple invariances ! 9

bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc CP-based Array Processing Seminal work of Sidiropoulos, Bro and Giannakis (2000) ⇒ Handling multiple invariances ⇒ Joint estimation of both DOA and sources An array with multiple invariances S T + N � � X = A 1 ⊙ A 2 10

bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc Multi-scale Array Level 1 Level 2 Level 3 N � d n d l 1 ,l 2 ,...,l N = l n n =1 11

Data Model (1) Phase factor for one narrowband source : N N j 2 π j 2 π � � � k T d n � � λ k T d n � a l 1 ,l 2 ,...,l N ( k ) = exp = exp . l n l n λ n =1 n =1 Array manifold for the entire sensor array a ( k ) = a 1 ( k ) ⊗ · · · ⊗ a N ( k ) , with   e j (2 π/λ ) k T d n 1 .   . a n ( k ) =   .   e j (2 π/λ ) k T d n Ln ⊗ Kronecker product 12

Data Model (2) P narrowband sources � � A 1 = a 1 ( k 1 ) , . . . , a 1 ( k P ) ( L 1 × P ) . . . � � = a N ( k 1 ) , . . . , a N ( k P ) ( L N × P ) A N � � = s 1 , s 2 , . . . , s P ( K × P ) S � � S T + N , X = A 1 ⊙ · · · ⊙ A N ⇒ CP Model of order N + 1 Single snapshot ⇒ CP Model of order N 13

Model identifiability – [Sidiropoulos and Bro 2000] : sufficient condition for the uniqueness of CP decomposition for N -way arrays – P sources with distinct DOAs and not fully correlated – Number of snapshots greater than number of sources ( K > P ) N � min( L n , P ) ≥ P + N. n =1 – Case of a single snapshot N � min( L n , P ) ≥ 2 P + N − 1 . n =1 14

Parameter estimation (1) Estimation of the steering vectors by CP decomposition of the data a p n : n th level estimated steering vector for the p th source ˆ Estimating the DOA parameters for the p th source ⇒ minimization of the following criterion : N � I N ( k p ) = J n ( k p ) . n =1 a p n − a n ( k p ) � 2 , n = 1 , . . . , N, J n ( k p ) = � ˆ 15

b b b Parameter estimation (2) I N ( k p ) : non-convex criterion highly non-linear ⇒ Sequential strategy similar to a GNC approach I 1 ( k p ) , ∆ ≤ λ/ 2 I 2 ( k p ) I 3 ( k p ) ⇒ The ordering of the sub-arrays is important ! 16

Parameter estimation (3) First Stage : Estimate A 1 , . . . , A N by CP decomposition of the data. Second Stage : ◮ For p = 1 , . . . , P and for n = 1 , . . . , N compute k ∗ p,n = argmin I n ( k p ) . k p ◮ Output : The estimated parameters for the P sources : ˆ k p = (ˆ u p , ˆ v p , ˆ w p ) = k ∗ p,N with p = 1 , . . . , P . 17

bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc Simulation results (1) Niveau 3 λ 2 Y 10 λ O X – Two scales L 1 = 5 , L 2 = 4 – Three scales L 1 = 5 , L 2 = 2 , L 3 = 2 ⇒ single snapshot case – Two sources with distinct DOAs – Comparison with ESPRIT [Wong, Zoltowski 1998] 18

Simulation results (2) 10 0 10 -2 R.M.S.E for ( u, v ) estimation (in radians) CP-based method for Source #1 R.M.S.E for ( u, v ) estimation (in radians) CP-based method for Source #2 MUSIC-ESPRIT for Source #1 MUSIC-ESPRIT for Source #2 10 -1 Dual-Size ESPRIT for Source #1 Dual-Size ESPRIT for Source #2 CRB for Source #1 10 -3 CRB for Source #2 10 -2 10 -3 10 -4 CP-based method for Source #1 CP-based method for Source #2 MUSIC-ESPRIT for Source #1 10 -4 MUSIC-ESPRIT for Source #2 Dual-Size ESPRIT for Source #1 Dual-Size ESPRIT for Source #2 CRB for Source #1 CRB for Source #2 10 -5 10 -5 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 SNR dB SNR dB CRMSE , K= 5 CRMSE , K=20 10 -1 10 0 R.M.S.E for ( u, v ) estimation (in radians) CP-based method for Source #1 R.M.S.E for ( u, v ) estimation (in radians) CP-based method for Source #2 MUSIC-ESPRIT for Source #1 10 -1 MUSIC-ESPRIT for Source #2 Dual-Size ESPRIT for Source #1 Dual-Size ESPRIT for Source #2 CRB for Source #1 10 -2 10 -2 CRB for Source #2 10 -3 10 -3 10 -4 CP-based method for Source #1 CP-based method for Source #2 MUSIC-ESPRIT for Source #1 MUSIC-ESPRIT for Source #2 10 -5 Dual-Size ESPRIT for Source #1 Dual-Size ESPRIT for Source #2 CRB for Source #1 CRB for Source #2 10 -4 10 -6 10 0 10 1 10 2 0 10 20 30 40 50 60 70 SNR dB # of snapshots CRMSE , SNR = 15 dB CRMSE , K=1 19

Simulation results (3) – Four scales L 1 = L 2 = L 3 = L 4 = 3 – Two sources with distinct DOAs – Comparison with ESPRIT [Wong, Zoltowski 1998] and Tensor-ESPRIT [Haardt et al. 2008] 20

Simulation results (4) 10 -2 CP-based method for Source #1 R.M.S.E for ( u, v ) estimation (in radians) R.M.S.E for ( u, v ) estimation (in radians) CP-based method for Source #1 CP-based method for Source #2 CP-based method for Source #2 Standard Tensor-ESPRIT for Source #1 Standard Tensor-ESPRIT for Source #1 Standard Tensor-ESPRIT for Source #2 Standard Tensor-ESPRIT for Source #2 Unitary Tensor-ESPRIT for Source #1 10 -3 Unitary Tensor-ESPRIT for Source #1 Unitary Tensor-ESPRIT for Source #2 Unitary Tensor-ESPRIT for Source #2 Dual-Size ESPRIT for Source #1 Dual-Size ESPRIT for Source #1 Dual-Size ESPRIT for Source #2 Dual-Size ESPRIT for Source #2 CRB for Source #1 CRB for Source #1 CRB for Source #2 CRB for Source #2 10 -3 10 -4 10 -4 0 5 10 15 20 25 0 5 10 15 20 25 SNR dB SNR dB CRMSE , K= 10 CRMSE , K=50 10 -1 10 -1 CP-based method for Source #1 CP-based method for Source #1 R.M.S.E for ( u, v ) estimation (in radians) CP-based method for Source #2 CP-based method for Source #2 Standard Tensor-ESPRIT for Source #1 Standard Tensor-ESPRIT for Source #1 R.M.S.E for ( u, v ) estimation (in radians) Standard Tensor-ESPRIT for Source #2 Standard Tensor-ESPRIT for Source #2 Unitary Tensor-ESPRIT for Source #1 Unitary Tensor-ESPRIT for Source #1 Unitary Tensor-ESPRIT for Source #2 Unitary Tensor-ESPRIT for Source #2 Dual-Size ESPRIT for Source #1 Dual-Size ESPRIT for Source #1 Dual-Size ESPRIT for Source #2 10 -2 Dual-Size ESPRIT for Source #2 CRB for Source #1 10 -2 CRB for Source #1 CRB for Source #2 CRB for Source #2 10 -3 10 -3 10 -4 10 -4 10 0 10 1 10 2 0 5 10 15 Inter-grid spacing, in units of half-wavelengths # of snapshots CRMSE , SNR = 6 dB CRMSE , K=5, SNR = 6 dB 21

CP-Based Vector Sensor Array Processing 22