Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 1 Subspace based Multi-Dimensional Model Order Selection in Colored Noise Scenarios

Universidade de BrasíliaLaboratório de Processamento de Sinais em Arranjos 1

Subspace based Multi-DimensionalSubspace based Multi-DimensionalModel Order SelectionModel Order Selection

in Colored Noise Scenariosin Colored Noise Scenarios

João Paulo C. Lustosa da Costa, Florian Roemer, Dominik Schulz, and Rafael Timóteo de Sousa Jr.

University of Brasília (UnB)Department of Electrical Engineering (ENE)Laboratory of Array Signal Processing

PO Box 4386Zip Code 70.919-970, Brasília - DF

Homepage: http://www.pgea.unb.br/~lasp

Universidade de BrasíliaLaboratório de Processamento de Sinais em Arranjos

OutlineOutline

Motivation Data Model and Goal ESTER R-D ESTER Simulations Conclusions

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OutlineOutline


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MotivationMotivation

The model order selection (MOS) problem is encountered in a variety of signal processing applications including radar, sonar, communications, channel modeling, malicious activity

detection in computer networks [1], medical imaging, and the estimation of the parameters of the dominant multipath components from

MIMO channel measurements. Not only for signal processing applications, but also in several science fields, e.g.,

chemistry, food industry, stock markets, pharmacy and psychometrics, the MOS problem is investigated.

It has been shown in [2] that for data contaminated by white noise, the R-D Exponential Fitting Test (R-D EFT) outperforms all the other model order selection schemes proposed in the literature. Since for data contaminated by colored noise, the R-D EFT is not applicable, the Closed-Form PARAFAC based Model Order Selection (CFP-MOS) scheme has been proposed [2] .

[1]: B. M. David et al., "Blind Automatic Malicious Activity Detection in Honeypot Data," The International Conference on Forensic Computer Science (ICoFCS) 2011.

[2]: J. P. C. L. da Costa, F. Roemer, M. Haardt, and R. T. de Souza Jr., ``Multi-Dimensional Model Order Selection,'' EURASIP Journal on Advances in Signal Processing 2011:26, July 2011, Springer publisher.


MotivationMotivation

The computational complexity of CFP-MOS is prohibitive for several applications. Therefore, another multidimensional model order selection scheme for colored noise scenarios is required.

In this paper, we propose a multi-dimensional extension of Estimation Error (ESTER) [3]. Since in [4] ESTER is outperformed by Subspace based Automatic Model Order Selection (SAMOS) scheme, we also compare it to our proposed R-D ESTER.

[3]: R. Badeau, B. David, and G. Richard, “Selecting the modeling order for the ESPRIT high resolution method an alternative approach”, In Proc. IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP 2004), Montreal, Canada May 2004.

[4]: J.-M. Papy, L. de Lathauwer, and S. Van Huffel, “A shift invariance-based order-selection technique for exponential data modeling”, in IEEE Signal Processing Letters, Vol. 14, pp. 473-476, Jul. 2007.


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Data Model and GoalData Model and Goal Noiseless case

Our objective is to estimate Our objective is to estimate dd from the noisy observations . from the noisy observations .

Matrix data model

+ +=


Colored noise model

Data Model and GoalData Model and Goal


Receive array: 1-D or 2-D

Frequency

Time

Transmit array: 1-D or 2-D

Direction of Arrival (DOA)

Delay

Doppler shift

Direction of Departure (DOD)

Example of MultiExample of Multi-Dimensional DataDimensional Data Channel model

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Data Model and GoalData Model and Goal

Noiseless data representationNoiseless data representation

ProblemProblem

where is the colored noise tensor.

= ++

Our objective is to estimate Our objective is to estimate dd from the noisy observations . from the noisy observations .

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OutlineOutline


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ESTER (1)ESTER (1)

Estimation of Signal Parameters via Rotation Invariance Techniques

ESPRIT

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dd

Shift Invariance Equation


ESTER (2)ESTER (2)

Estimation of Signal Parameters via Rotation Invariance Techniques

ESPRIT

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The steering matrix A and the first d eigenvectors U of the covariance matrix generate the same subspace

Note that Us is related to the low-rank approximation.

Spatial frequencies


ESTER (3)ESTER (3)

ESTER Algorithm [3][3]

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OutlineOutline


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Operations on TensorsOperations on Tensors(Multidimensional Arrays)(Multidimensional Arrays)

• Unfoldings

• Concatenation


Operations on Tensors and MatricesOperations on Tensors and Matrices

• n-mode product

i.e., all the n-mode vectors multiplied from the left-hand-side by

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Review: The SVD of MatricesReview: The SVD of Matrices

“Full SVD”

“Economy size SVD”

Low-rank approximation


Extension to the HOSVD of TensorsExtension to the HOSVD of Tensors

“Full HOSVD”

Low-rank approximation (truncated HOSVD)

“Economy size HOSVD”


R-D ESTER (1)R-D ESTER (1)

R-D ESTER Algorithm

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OutlineOutline


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SimulationsSimulations

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SimulationsSimulations

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ConclusionsConclusions

In this work, we have proposed a multi-dimensional extension of the ESTER (R-D ESTER) scheme.

By taking into account the multi-dimensional structure of the data, the R-D ESTER outperforms significantly showing its matrix based version.

Similarly to the R-D EFT, the R-D ESTER is also based on the HOSVD.

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Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 1 Subspace based Multi-Dimensional Model Order Selection in Colored Noise Scenarios