Subspace-based Blind Identification of IIR Systems Outline - - PowerPoint PPT Presentation

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Subspace-based Blind Identification of IIR Systems Outline - - PowerPoint PPT Presentation

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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 <

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MLSP 2006 Outline Motivation Contributions Problem Formulation Blind ID Simulation Results Conclusions

Subspace-based Blind Identification of IIR Systems

Juan Carlos G´

  • mez 1

<jcgomez@fceia.unr.edu.ar>

Enrique Baeyens 2

<enrbae@eis.uva.es>

1Laboratory for System Dynamics and Signal Processing

FCEIA, Universidad Nacional de Rosario, Argentina

2Department 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

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MLSP 2006 Outline Motivation Contributions Problem Formulation Blind ID Simulation Results Conclusions

Outline

1

Motivation for Blind ID

2

Contributions

3

Problem Formulation

4

Blind Identification of Basis Coefficients

5

Simulation Results

6

Conclusions

MLSP 2006 (Maynooth,Ireland) September 6-8, 2006 2 / 28

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MLSP 2006 Outline Motivation Contributions Problem Formulation Blind ID Simulation Results Conclusions

Motivation for Blind ID

Blind Identification: based only on output data In the digital communication area, blind channel identification avoids the need for transmission of training sequences and, in this way, a more efficient use of the available bandwidth can be achieved. IIR models are more suitable than FIR models for representing systems whose impulse responses have slow decays. 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

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MLSP 2006 Outline Motivation Contributions Problem Formulation Blind ID Simulation Results Conclusions

Contributions

A new subspace technique for the blind identification of IIR, SIMO systems represented using orthonormal bases with fixed poles is presented. Basis coefficients are estimated in closed form by first computing the column space of the output Hankel matrix, using Singular Value Decomposition, and then solving a Least Squares Problem through an Eigenvalue Decomposition. 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

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MLSP 2006 Outline Motivation Contributions Problem Formulation Blind ID Simulation Results Conclusions

Problem Formulation

The LTI, SIMO, IIR system is represented using orthonormal bases as follows Y (z) = p−1

  • ℓ=0

bℓBℓ(z)

  • G(z)

U(z) where bℓ =

  • b1

ℓ, b2 ℓ, · · · , bm ℓ

T ∈ Rm unknown basis coefficients Bℓ(z) rational orthonormal bases on H2(T).

MLSP 2006 (Maynooth,Ireland) September 6-8, 2006 5 / 28

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MLSP 2006 Outline Motivation Contributions Problem Formulation Blind ID Simulation Results Conclusions

Problem Formulation (cont.)

The Orthonormal Bases with Fixed Poles (OBFP) described in (Ninness et al., 1997), (Gomez, 1998) are considered. The OBFP are defined as Bℓ(z) =  

  • 1 − |ξℓ|2

z − ξℓ  

ℓ−1

  • i=0

1 − ξiz z − ξi

  • , ℓ ≥ 1

B0(z) =

  • 1 − |ξ0|2

z − ξ0 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

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MLSP 2006 Outline Motivation Contributions Problem Formulation Blind ID Simulation Results Conclusions

Problem Formulation (cont.)

  • Fig. 1: IIR Filter Structure using OBFP.

MLSP 2006 (Maynooth,Ireland) September 6-8, 2006 7 / 28

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MLSP 2006 Outline Motivation Contributions Problem Formulation Blind ID Simulation Results Conclusions

Problem Formulation (cont.)

Aℓ(z) 1 − ξℓz z − ξℓ (all-pass section), Lℓ(z)

  • 1 − |ξℓ|2

z − ξℓ (low-pass section) A state-space realization of the system can be obtained by giving each all-pass section and each low-pass section a one dimensional state-space realization. It is not difficult to see that the resulting state-space realization will be non minimal.

MLSP 2006 (Maynooth,Ireland) September 6-8, 2006 8 / 28

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MLSP 2006 Outline Motivation Contributions Problem Formulation Blind ID Simulation Results Conclusions

Problem Formulation (cont.)

  • Fig. 2: Input-Output equivalent IIR filter structure.

MLSP 2006 (Maynooth,Ireland) September 6-8, 2006 9 / 28

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MLSP 2006 Outline Motivation Contributions Problem Formulation Blind ID Simulation Results Conclusions

Problem Formulation (cont.)

Q0(z) 1 z − ξ0 Qℓ(z) 1 − ξℓ−1z z − ξℓ , ℓ ≥ 1 A minimal state-space realization for the Qℓ(z), (ℓ ≥ 1) blocks is given by xℓ

k+1

=

Aℓ

  • ξℓ xℓ

k + Bℓ

  • (1 − ξℓξℓ−1) uℓ

k

  • yℓ

k

= 1

  • Cℓ

xℓ

k −ξℓ−1 Dℓ

uℓ

k

while for the Q0(z) blocks is given by x0

k+1

= ξ0x0

k + uk

  • y0

k

= x0

k

MLSP 2006 (Maynooth,Ireland) September 6-8, 2006 10 / 28

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MLSP 2006 Outline Motivation Contributions Problem Formulation Blind ID Simulation Results Conclusions

Problem Formulation (cont.)

Defining xk

  • x0

k, x1 k, · · · , xp−1 k

T

  • yk
  • y0

k,

y1

k, · · · ,

yp−1

k

T yk

  • y1

k, y2 k, · · · , ym k

T b

  • [b0, b1, · · · , bp−1]

a minimal state-space realization of the IIR model is given by xk+1 = Axk + Buk yk = Cxk + Duk

MLSP 2006 (Maynooth,Ireland) September 6-8, 2006 11 / 28

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MLSP 2006 Outline Motivation Contributions Problem Formulation Blind ID Simulation Results Conclusions

Problem Formulation (cont.)

A = ✷ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✹ A0 · · · B1C0 A1 · · · B2D1C0 B2C1 · · · B3D2D1C0 B3D2C1 · · · . . . . . . ... . . . Bp−1Dp−2 · · · D1C0 Bp−1Dp−2 · · · D2C1 · · · Ap−1 ✸ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✺ , B = [1, 0, · · · , 0]T , C = bΛ ❡ C, D = 0 ❡ C = ✷ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✹ C0 · · · D1C0 C1 · · · D2D1C0 D2C1 · · · D3D2D1C0 D3D2C1 · · · . . . . . . ... . . . Dp−1Dp−2 · · · D1C0 Dp−1Dp−2 · · · D2C1 · · · Cp−1 ✸ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✺ , Λ ✷ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✹ q 1 − ξ2 · · · q 1 − ξ2

1

· · · ... · · · q 1 − ξ2

p−1

✸ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✺ , MLSP 2006 (Maynooth,Ireland) September 6-8, 2006 12 / 28

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MLSP 2006 Outline Motivation Contributions Problem Formulation Blind ID Simulation Results Conclusions

Problem Formulation (cont.)

Considering that there are N + α − 1 measurements available, the system’s input-output equation can be written in matrix form as Yα = ΓαX + HαUα =

  • Γα

Hα X Uα

  • where

Uα ✷ ✻ ✻ ✻ ✻ ✹ u1 u2 · · · uN u2 u3 · · · uN+1 . . . . . . ... . . . uα uα+1 · · · uN+α−1 ✸ ✼ ✼ ✼ ✼ ✺ , Yα ✷ ✻ ✻ ✻ ✻ ✹ y1 y2 · · · yN y2 y3 · · · yN+1 . . . . . . ... . . . yα yα+1 · · · yN+α−1 ✸ ✼ ✼ ✼ ✼ ✺ Γα ✷ ✻ ✻ ✻ ✻ ✹ C CA . . . CAα−1 ✸ ✼ ✼ ✼ ✼ ✺ , Hα ✷ ✻ ✻ ✻ ✻ ✻ ✻ ✹ D · · · CB D · · · · · · CAB CB D · · · ... CAα−2B CAα−3B · · · · · · D ✸ ✼ ✼ ✼ ✼ ✼ ✼ ✺ , X ✂x1 x2 · · · xN ✄ MLSP 2006 (Maynooth,Ireland) September 6-8, 2006 13 / 28

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MLSP 2006 Outline Motivation Contributions Problem Formulation Blind ID Simulation Results Conclusions

Problem Formulation (cont.)

Considering that C = bΛ C, the system equation becomes Yα = (Iα ⊗ b) H X Uα

  • Fundamental equation

where

H ✂❡ Γα ❡ Hα ✄ ❡ Γα ✷ ✻ ✻ ✻ ✻ ✹ Λ ❡ C Λ ❡ CA . . . Λ ❡ CAα−1 ✸ ✼ ✼ ✼ ✼ ✺ , ❡ Hα ✷ ✻ ✻ ✻ ✻ ✻ ✻ ✹ · · · Λ ❡ CB · · · · · · Λ ❡ CAB Λ ❡ CB · · · ... Λ ❡ CAα−2B Λ ❡ CAα−3B · · · · · · ✸ ✼ ✼ ✼ ✼ ✼ ✼ ✺

MLSP 2006 (Maynooth,Ireland) September 6-8, 2006 14 / 28

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MLSP 2006 Outline Motivation Contributions Problem Formulation Blind ID Simulation Results Conclusions

Blind Identification of basis coefficients

Blind ID Problem: given the (possibly noisy) output measurements Yα and the basis poles, find an estimate of the parameter matrix b, and possibly of the structured matrices X and Uα. Yα = (Iα ⊗ b) H X Uα

  • Fundamental equation

Provided that αm > p + α, N > p + α, and that matrices (Iα ⊗ b) H and X Uα

  • have full column rank, then the following

subspace relationships hold

col (Yα) = col ((Iα ⊗ b) H) , row (Yα) = row X Uα

  • MLSP 2006 (Maynooth,Ireland)

September 6-8, 2006 15 / 28

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MLSP 2006 Outline Motivation Contributions Problem Formulation Blind ID Simulation Results Conclusions

Blind Identification of basis coefficients (cont.)

The parameter matrix b can then be estimated by computing the column space of the output Hankel matrix Yα and exploiting the Kronecker structure of the system. Assuming Yα has rank (p + α), and performing its SVD yields Yα = ΦΣΨT =

  • Φ1

Φ2

  • Φ

Σ1 O O Σ2

  • Σ

ΨT

1

ΨT

2

  • ΨT

≈ Φ1Σ1ΨT

1 ,

Then, the following subspace relationship holds col (Yα) = col ((Iα ⊗ b) H) = col (Φ1)

MLSP 2006 (Maynooth,Ireland) September 6-8, 2006 16 / 28

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MLSP 2006 Outline Motivation Contributions Problem Formulation Blind ID Simulation Results Conclusions

Blind Identification of basis coefficients (cont.)

There must exist a nonsingular matrix T such that the columns

  • f Φ1 can be converted into the columns of matrix (Iα ⊗ b) H.

Matrices b and T can then be estimated by solving the Least Squares Problem ( b, T) = arg min

T,b2=1

(Iα ⊗ b)H − Φ1T2

F

  • r equivalently
  • b

= arg min

b2=1

(I − Φ1(ΦT

1 Φ1)−1ΦT 1 )(Iα ⊗ b)H2 F

  • T

= (ΦT

1 Φ1)−1ΦT 1 (Iα ⊗

b)H

MLSP 2006 (Maynooth,Ireland) September 6-8, 2006 17 / 28

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MLSP 2006 Outline Motivation Contributions Problem Formulation Blind ID Simulation Results Conclusions

Blind Identification of basis coefficients (cont.)

Remark: The constraint b2 = 1 has been imposed to avoid the trivial solution (b = 0) inherent to this least squares

  • problem. There is no loss of generality since the parameter

matrix b is identifiable only up to scalar factor. The estimate of matrix b can be written as

  • b = arg min

b2=1

Tr

  • HT (Iα ⊗ bT )Φ⊥

1 (Iα ⊗ b)H

  • Or equivalently
  • b = arg min

b2=1

bT Gb Quadratic Form

MLSP 2006 (Maynooth,Ireland) September 6-8, 2006 18 / 28

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MLSP 2006 Outline Motivation Contributions Problem Formulation Blind ID Simulation Results Conclusions

Blind Identification of basis coefficients (cont.)

where b vec (b), and the entries of matrix G are given by Gij = Tr

  • HT (Iα ⊗ eT

j )Φ⊥ 1 (Iα ⊗ ei)H

  • where ei vec−1 (ei), with ei being a vector of zeros, except

for a one in the i-th position. Solution: By invoking Rayleigh-Ritz theorem, the estimate of vector b is given by the unit length eigenvector (vmin) corresponding to the smallest eigenvalue (λmin) of matrix G.

MLSP 2006 (Maynooth,Ireland) September 6-8, 2006 19 / 28

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MLSP 2006 Outline Motivation Contributions Problem Formulation Blind ID Simulation Results Conclusions

Blind Identification of basis coefficients (cont.)

Blind Identification Algorithm Step 1: Compute the output Hankel matrix Yα from ouput measurements. Step 2: Perform the SVD of Yα. Step 3: Compute matrix G. Step 4: Compute the EVD of matrix G. Step 5: Compute the estimate b as

  • b = vec−1 (vmin)

MLSP 2006 (Maynooth,Ireland) September 6-8, 2006 20 / 28

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MLSP 2006 Outline Motivation Contributions Problem Formulation Blind ID Simulation Results Conclusions

Blind Identification of basis coefficients (cont.)

Remark: The EVD in Step 4 of the Blind ID Algorithm can actually be solved by performing the SVD of the square root of matrix G, which is guaranteed to exist since G is a symmetric positive definite matrix.

MLSP 2006 (Maynooth,Ireland) September 6-8, 2006 21 / 28

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MLSP 2006 Outline Motivation Contributions Problem Formulation Blind ID Simulation Results Conclusions

Simulation results

  • Fig. 3: True system.

m = 3 outputs p = 5 internal states Poles at: (0.1, 0.3, 0.6, 0.8, 0.9)

F1(z) = 0.3365z4 − 0.8025z3 + 0.8448z2 − 0.5488z + 0.1832 z5 − 2.700z4 + 2.6900z3 − 1.1970z2 + 0.2250z − 0.0130 F2(z) = 0.1581z4 − 0.2805z3 + 0.2140z2 − 0.1530z + 0.0780 z5 − 2.700z4 + 2.6900z3 − 1.1970z2 + 0.2250z − 0.0130 F3(z) = 0.3026z4 − 0.7236z3 + 0.7939z2 − 0.5348z + 0.1776 z5 − 2.700z4 + 2.6900z3 − 1.1970z2 + 0.2250z − 0.0130 MLSP 2006 (Maynooth,Ireland) September 6-8, 2006 22 / 28

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MLSP 2006 Outline Motivation Contributions Problem Formulation Blind ID Simulation Results Conclusions

Simulation results (cont.)

The outputs were corrupted with zero mean, colored, independent noise signals obtained by filtering Gaussian white noise with variance σ2 . The resulting noise power spectral density is given by Φ(ω) = 0.64σ2 1.04 − 0.4 cos(ω) 4000 samples were collected α = 20 (number of rows in Hankel matrices) SNR

PN+α−1

k=1

yk2 mσ2(N+α−1)

RMSE m

i=1 Mi −

Mi2, where Mi is the vector of numerator polynomial coefficients of the transfer function between the input and the i-th output, and Mi is its corresponding estimate.

MLSP 2006 (Maynooth,Ireland) September 6-8, 2006 23 / 28

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MLSP 2006 Outline Motivation Contributions Problem Formulation Blind ID Simulation Results Conclusions

Simulation results (cont.)

Location of bases poles: FIR bases: poles at z = 0 Laguerre bases: poles at z = 0.5 OBFP-exact: poles at z = (0.1, 0.3, 0.6, 0.8, 0.9) OBFP-approx.: poles at z = (0.15, 0.35, 0.65, 0.85, 0.95)

MLSP 2006 (Maynooth,Ireland) September 6-8, 2006 24 / 28

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Simulation results (cont.)

  • Fig. 4: RMSE vs. SNR for different locations of the basis poles.

MLSP 2006 (Maynooth,Ireland) September 6-8, 2006 25 / 28

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MLSP 2006 Outline Motivation Contributions Problem Formulation Blind ID Simulation Results Conclusions

Simulation results (cont.)

  • Fig. 5: Measured (solid blue line) and estimated (dashed green line)
  • utputs (SNR = 35.5646 dB, FIT = 87.62 %). .

MLSP 2006 (Maynooth,Ireland) September 6-8, 2006 26 / 28

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MLSP 2006 Outline Motivation Contributions Problem Formulation Blind ID Simulation Results Conclusions

Conclusions

A new subspace technique for the blind identification of IIR, SIMO systems represented using orthonormal bases with fixed poles has been presented. Basis coefficients are estimated in closed form by first computing the column space of the output Hankel matrix, using SVD, and then solving a Least Squares Problem via an Eigenvalue Decomposition. 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 27 / 28

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MLSP 2006 Outline Motivation Contributions Problem Formulation Blind ID Simulation Results Conclusions

Many Thanks !!!

Subspace-based Blind Identification of IIR Systems

Juan Carlos G´

  • mez 1

<jcgomez@fceia.unr.edu.ar>

Enrique Baeyens 2

<enrbae@eis.uva.es>

1Laboratory for System Dynamics and Signal Processing

FCEIA, Universidad Nacional de Rosario, Argentina

2Department of Systems Engineering and Automatic Control

ETSII, Universidad de Valladolid, Spain September 6-8, 2006 MLSP 2006 (Maynooth,Ireland) September 6-8, 2006 28 / 28