MLSP 2006 Subspace-based Blind Identification of IIR Systems Outline Motivation Contributions omez 1 Juan Carlos G´ Problem Formulation < jcgomez@fceia.unr.edu.ar > Blind ID Enrique Baeyens 2 Simulation Results < enrbae@eis.uva.es > Conclusions 1 Laboratory for System Dynamics and Signal Processing FCEIA, Universidad Nacional de Rosario, Argentina 2 Department of Systems Engineering and Automatic Control ETSII, Universidad de Valladolid, Spain September 6-8, 2006 MLSP 2006 (Maynooth,Ireland) September 6-8, 2006 1 / 28
Outline MLSP 2006 Motivation for Blind ID 1 Outline Motivation Contributions 2 Contributions Problem Formulation Problem Formulation 3 Blind ID Simulation Results Blind Identification of Basis Coefficients 4 Conclusions Simulation Results 5 Conclusions 6 MLSP 2006 (Maynooth,Ireland) September 6-8, 2006 2 / 28
Motivation for Blind ID Blind Identification: based only on output data MLSP 2006 In the digital communication area, blind channel Outline identification avoids the need for transmission of training Motivation sequences and, in this way, a more efficient use of the Contributions available bandwidth can be achieved. Problem Formulation IIR models are more suitable than FIR models for Blind ID representing systems whose impulse responses have slow Simulation Results decays. Conclusions SIMO model structures appear when a source is transmitted through a continuous channel and several measurements are performed at the receiver due to multiple sensors, or oversampling of the received signal to ensure identifiability. MLSP 2006 (Maynooth,Ireland) September 6-8, 2006 3 / 28
Contributions A new subspace technique for the blind identification of MLSP 2006 IIR, SIMO systems represented using orthonormal bases Outline with fixed poles is presented. Motivation Basis coefficients are estimated in closed form by first Contributions computing the column space of the output Hankel matrix, Problem Formulation using Singular Value Decomposition, and then solving a Blind ID Least Squares Problem through an Eigenvalue Simulation Decomposition. Results Conclusions The method presents improvements in the estimation accuracy when compared to similar techniques based on FIR or Laguerre models. MLSP 2006 (Maynooth,Ireland) September 6-8, 2006 4 / 28
Problem Formulation The LTI, SIMO, IIR system is represented using orthonormal MLSP 2006 bases as follows Outline � p − 1 � Motivation � Contributions Y ( z ) = b ℓ B ℓ ( z ) U ( z ) Problem ℓ =0 Formulation � �� � Blind ID G ( z ) Simulation Results where Conclusions � � T ∈ R m unknown basis coefficients b 1 ℓ , b 2 ℓ , · · · , b m b ℓ = ℓ B ℓ ( z ) rational orthonormal bases on H 2 ( T ) . MLSP 2006 (Maynooth,Ireland) September 6-8, 2006 5 / 28
Problem Formulation (cont.) The Orthonormal Bases with Fixed Poles (OBFP) MLSP 2006 described in (Ninness et al., 1997), (Gomez, 1998) are Outline considered. Motivation The OBFP are defined as Contributions � Problem � 1 − ξ i z � 1 − | ξ ℓ | 2 ℓ − 1 � Formulation B ℓ ( z ) = , ℓ ≥ 1 Blind ID z − ξ ℓ z − ξ i i =0 Simulation � Results 1 − | ξ 0 | 2 Conclusions B 0 ( z ) = z − ξ 0 The OBFP can incorporate an arbitrary number of stable poles, and they have the more common FIR and Laguerre model structures as special cases. MLSP 2006 (Maynooth,Ireland) September 6-8, 2006 6 / 28
Problem Formulation (cont.) MLSP 2006 Outline Motivation Contributions Problem Formulation Blind ID Simulation Results Conclusions Fig. 1: IIR Filter Structure using OBFP. MLSP 2006 (Maynooth,Ireland) September 6-8, 2006 7 / 28
Problem Formulation (cont.) MLSP 2006 1 − ξ ℓ z Outline A ℓ ( z ) � ( all-pass section ) , z − ξ ℓ Motivation � 1 − | ξ ℓ | 2 Contributions L ℓ ( z ) � ( low-pass section ) z − ξ ℓ Problem Formulation Blind ID A state-space realization of the system can be obtained by Simulation giving each all-pass section and each low-pass section a one Results dimensional state-space realization. It is not difficult to see Conclusions that the resulting state-space realization will be non minimal . MLSP 2006 (Maynooth,Ireland) September 6-8, 2006 8 / 28
Problem Formulation (cont.) MLSP 2006 Outline Motivation Contributions Problem Formulation Blind ID Simulation Results Conclusions Fig. 2: Input-Output equivalent IIR filter structure. MLSP 2006 (Maynooth,Ireland) September 6-8, 2006 9 / 28
Problem Formulation (cont.) 1 Q 0 ( z ) � z − ξ 0 MLSP 2006 1 − ξ ℓ − 1 z Q ℓ ( z ) � ℓ ≥ 1 , Outline z − ξ ℓ Motivation Contributions A minimal state-space realization for the Q ℓ ( z ) , ( ℓ ≥ 1) Problem blocks is given by Formulation B ℓ A ℓ Blind ID � �� � ���� Simulation x ℓ ξ ℓ x ℓ (1 − ξ ℓ ξ ℓ − 1 ) u ℓ = k + Results k +1 k y ℓ x ℓ u ℓ � = 1 k − ξ ℓ − 1 Conclusions ���� k k � �� � C ℓ D ℓ while for the Q 0 ( z ) blocks is given by x 0 ξ 0 x 0 = k + u k k +1 y 0 x 0 � = k k MLSP 2006 (Maynooth,Ireland) September 6-8, 2006 10 / 28
Problem Formulation (cont.) Defining � � T MLSP 2006 k , · · · , x p − 1 x 0 k , x 1 � x k k � � T Outline y p − 1 y 0 y 1 � y k � � k , � k , · · · , � Motivation k � � T Contributions � y 1 k , y 2 k , · · · , y m y k k Problem � [ b 0 , b 1 , · · · , b p − 1 ] b Formulation Blind ID a minimal state-space realization of the IIR model is given Simulation Results by Conclusions x k +1 = Ax k + Bu k = Cx k + Du k y k MLSP 2006 (Maynooth,Ireland) September 6-8, 2006 11 / 28
Problem Formulation (cont.) ✷ ✸ A 0 0 0 MLSP 2006 · · · ✻ B 1 C 0 A 1 ✼ 0 ✻ · · · ✼ ✻ ✼ B 2 D 1 C 0 B 2 C 1 0 ✻ ✼ · · · Outline ✻ ✼ B 3 D 2 D 1 C 0 B 3 D 2 C 1 0 A = ✻ ✼ , · · · ✻ ✼ ✻ ✼ Motivation . . . ... ✻ ✼ . . . ✹ ✺ . . . Contributions B p − 1 D p − 2 · · · D 1 C 0 B p − 1 D p − 2 · · · D 2 C 1 A p − 1 · · · Problem B = [1 , 0 , · · · , 0] T , C = b Λ ❡ C, D = 0 Formulation ✷ ✸ Blind ID C 0 0 0 · · · ✻ ✼ D 1 C 0 C 1 0 ✻ ✼ · · · Simulation ✻ D 2 D 1 C 0 D 2 C 1 ✼ 0 ✻ ✼ · · · Results ✻ ✼ ❡ D 3 D 2 D 1 C 0 D 3 D 2 C 1 C = ✻ 0 ✼ , · · · ✻ ✼ ✻ ✼ Conclusions . . . ... ✻ ✼ . . . ✹ ✺ . . . D p − 1 D p − 2 · · · D 1 C 0 D p − 1 D p − 2 · · · D 2 C 1 C p − 1 · · · q ✷ ✸ 1 − ξ 2 0 0 0 · · · q ✻ ✼ ✻ ✼ 1 − ξ 2 0 0 ✻ ✼ 1 · · · ✻ ✼ Λ � ✻ ✼ , ... ✻ ✼ ✻ ✼ 0 0 0 ✹ ✺ q 1 − ξ 2 0 0 · · · p − 1 MLSP 2006 (Maynooth,Ireland) September 6-8, 2006 12 / 28
Problem Formulation (cont.) Considering that there are N + α − 1 measurements available, MLSP 2006 the system’s input-output equation can be written in matrix Outline form as Motivation � � X � � Contributions Y α = Γ α X + H α U α = Γ α H α U α Problem Formulation Blind ID where Simulation ✷ ✸ ✷ ✸ u 1 u 2 u N y 1 y 2 y N Results · · · · · · ✻ u 2 u 3 u N +1 ✼ ✻ y 2 y 3 y N +1 ✼ · · · · · · ✻ ✼ ✻ ✼ Conclusions U α � ✻ ✼ , Y α � ✻ ✼ . . . . . . ... ... ✻ ✼ ✻ ✼ . . . . . . ✹ ✺ ✹ ✺ . . . . . . u α u α +1 u N + α − 1 y α y α +1 y N + α − 1 · · · · · · ✷ ✸ ✷ ✸ D 0 0 0 · · · C 0 ✻ CB D ✼ · · · · · · ✻ CA ✼ ✻ ✼ 0 ✻ ✼ ✻ CAB CB D ✼ · · · ✻ ✼ ✻ ✼ Γ α � , H α � . , ✻ ✼ ✻ ✼ . ... ✹ ✺ ✻ ✼ . ✹ ✺ 0 CA α − 1 CA α − 2 B CA α − 3 B D · · · · · · ✂ x 1 ✄ X � x 2 x N · · · MLSP 2006 (Maynooth,Ireland) September 6-8, 2006 13 / 28
Problem Formulation (cont.) Considering that C = b Λ � C , the system equation becomes MLSP 2006 � X � Outline Y α = ( I α ⊗ b ) H Fundamental equation U α Motivation Contributions where Problem Formulation ✂❡ ✄ H � ❡ Blind ID Γ α H α ✷ ✸ ✷ ✸ 0 0 0 · · · 0 Simulation Λ ❡ C ✻ Λ ❡ ✼ Results CB 0 · · · · · · 0 ✻ ✼ ✻ ✼ Λ ❡ CA ✻ ✼ ✻ ✼ Λ ❡ Λ ❡ CAB CB 0 · · · 0 Conclusions Γ α � ❡ ✻ ✼ H α � ❡ ✻ ✼ , . ✻ ✼ ✻ ✼ . ... ✻ ✼ ✹ ✺ . ✹ ✺ 0 Λ ❡ CA α − 1 Λ ❡ Λ ❡ CA α − 2 B CA α − 3 B · · · · · · 0 MLSP 2006 (Maynooth,Ireland) September 6-8, 2006 14 / 28
Blind Identification of basis coefficients Blind ID Problem: given the (possibly noisy) output MLSP 2006 measurements Y α and the basis poles, find an estimate of the Outline parameter matrix b , and possibly of the structured matrices X Motivation and U α . Contributions � X � Problem Formulation Y α = ( I α ⊗ b ) H Fundamental equation Blind ID U α Simulation Results Provided that αm > p + α , N > p + α , and that matrices � X � Conclusions ( I α ⊗ b ) H and have full column rank, then the following U α subspace relationships hold �� X �� col ( Y α ) = col (( I α ⊗ b ) H ) , row ( Y α ) = row U α MLSP 2006 (Maynooth,Ireland) September 6-8, 2006 15 / 28
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