Ilmenau University of Technology Communications Research Laboratory 1  A new multi-dimensional model order selection technique called closed- form PARAFAC based model order selection (CFP-MOS) scheme  based on the multiple estimates of the closed-form PARAFAC [4]  suitable for applications with PARAFAC data model  For the estimation of spatial frequencies, we propose to apply the closed-form PARAFAC based parameter estimator (CFP-PE)  for arrays without shift invariance property in conjunction with the Peak Search based estimator  due to the decoupling of dimensions robust against modeling errors  increase of the maximum model order via merging of dimensions separation via Least Squares Khatri-Rao Factorization (LSKRF) [8] Main Contributions

Ilmenau University of Technology Communications Research Laboratory 2 R-D Parameter Estimation  Encountered in a variety of applications  mobile communications, spectroscopy, multi-dimensional medical imaging, and  the estimation of the parameters of the dominant multipath components from MIMO channel sounder measurements  Traditional approaches require  stacking the dimensions into one highly structured matrix, since the measured data is multi-dimensional  For the R-D parameter estimation  the model order selection, i.e., the estimation of the number of principal components, is required  parameters can be extracted from the main components using the estimated model order and assuming some structure of the data For instance, in MIMO applications, the main components are represented by the superposition of undamped complex exponentials, where each vector of complex exponentials is mapped to a certain spatial frequency.

Ilmenau University of Technology Communications Research Laboratory 3  In [1], we have proposed the R-Dimensional Exponential Fitting Test (R-D EFT)  a multi-dimensional extension of the Modified Exponential Fitting Test (M-EFT)  is based on the HOSVD of the measurement tensor  is superior to other schemes in the literature [2,3]  restricted to applications in the presence of white Gaussian noise  Since colored noise is common in many applications, we propose the closed-form PARAFAC based model order selection (CFP-MOS) scheme.  Once the model order is estimated, the extraction of the spatial frequencies from the main components can be performed.  In general, for this task, closed-form schemes like R-D ESPRIT-type algorithms [5] are applied, since their performance is close to the Cramér-Rao lower bound (CRLB).  In [6], the 3-D and 4-D versions of the Multi-linear Alternating Least Squares (MALS)  decompose the measurement tensor into factors  Easy to obtain the spatial frequencies via shift invariance or peak search based estimator  We propose to apply a closed-form PARAFAC based parameter estimator (CFP-PE). State of the Art

Ilmenau University of Technology Communications Research Laboratory 4 Tensor Algebra  3-D tensor = 3-way array   n-mode products between and   Unfoldings M1M1 M2M2 M3M3 “1-mode vectors” “2-mode vectors” “3-mode vectors” i.e., all the n-mode vectors multiplied from the left-hand-side by1 2

Ilmenau University of Technology Communications Research Laboratory 5  For the estimation of the spatial frequencies, we assume a data model where Data Model Noiseless data representation Problem where is the colored noise tensor. Therefore, our objective is to estimate the model order d and the spatial frequencies. the elements of the vector can be mapped into a certain spatial frequency for and

Ilmenau University of Technology Communications Research Laboratory 6 SVD and PARAFAC ++ = ++ =   Another way to look at the SVD   PARAFAC Decomposition   The task of the PARAFAC analysis: Given (noisy) measurements and the model order d, find such that Here is the higher-order Frobenius norm (square root of the sum of the squared magnitude of all elements).

Ilmenau University of Technology Communications Research Laboratory 7 Closed-form PARAFAC based Model Order Selection   Our approach: based on simultaneous matrix diagonalizations (“closed-form”).   By applying the closed-form PARAFAC (CFP) [4]   R*(R-1) simultaneous matrix diagonalizations (SMD) are possible;   R*(R-1) estimates for each factor are possible;   selection of the best solution by different heuristics (residuals of the SMD) is done Residuals (k,l,i)(k,l,i)(k,l,i)(k,l,i) Residuals k and l are the tuples of the SMD and i indicates left or right factor matrix to be estimated [4]. b Sorting based on the residuals   Residuals  the Frobenius norm of the off-diagonal elements of the diagonalized matrices. In practice, the small residuals means small error in the estimated parameters.   The index b sorts the factors based on the residuals, which gives us B lim

Ilmenau University of Technology Communications Research Laboratory 8 Closed-form PARAFAC based Model Order Selection   For P = 2, i.e., P < d   We assume d = 3 and we consider only solutions with the two smallest residuals of the SMD, i.e., b = 1 and 2.   Due to the permutation ambiguities, the components of different tensors are ordered using the amplitude based approach proposed in [7].   For P = 4, i.e., P > d + = + = = + = + P

Ilmenau University of Technology Communications Research Laboratory 9 Closed-form PARAFAC based Model Order Selection   For instance, let us consider the estimated factor where P is the candidate value for the model order d and b is the index of the ordered multiple factors according to the assumed heuristics.   We define the following error function   Taking into account all the components in the cost function   As proposed in the paper, another expression for the cost function is possible by using the spatial frequencies instead of the angular distances.

Ilmenau University of Technology Communications Research Laboratory 10 Comparing the performance of CFP-MOS Simulations     White Gaussian noise       Colored Gaussian noise  

Ilmenau University of Technology Communications Research Laboratory 11 Closed-form PARAFAC based Parameter Estimation   Let us consider Merging dimensions   We can stack dimensions and obtain where   With merging Least Squares Khatri-Rao Factorization Factorization [8]   Given, we desire and   Reshaping the merged vector   Since the product should be a rank-one matrix, we can apply the SVD-based rank-one approximation   Therefore,   Without merging in both cases, we assume that

Ilmenau University of Technology Communications Research Laboratory 12 Closed-form PARAFAC based Parameter Estimation Peak search based estimator   Given from the CFP decomposition, we can compute the respective spatial frequency via Simulations          

Ilmenau University of Technology Communications Research Laboratory 13 Comparing the performance of CFP-PE Simulations            

Ilmenau University of Technology Communications Research Laboratory 14 Comparing the performance of CFP-PE Simulations          

Ilmenau University of Technology Communications Research Laboratory 15 [1] J. P. C. L. da Costa, M. Haardt, F. Roemer, and G. Del Galdo, “Enhanced model order estimation using higher-order arrays”, in Proc. 41-st Asilomar Conf. on Signals, Systems, and Computers, Pacific Grove, CA, USA, Nov. 2007, invited paper. [2] J. P. C. L. da Costa, M. Haardt, and F. Roemer, “Robust methods based on the HOSVD for estimating the model order in PARAFAC models,” in Proc. 5-th IEEE Sensor Array and Multichannel Signal Processing Workshop (SAM 2008), (Darmstadt, Germany), pp , July [3] J. P. C. L. da Costa, A. Thakre, F. Roemer, and M. Haardt, “Comparison of model order selection techniques for high- resolution parameter estimation algorithms,” in Proc. 54th International Scientific Colloquium (IWK), (Ilmenau, Germany), Sept [4]F. Roemer and M. Haardt, “A closed-form solution for multilinear PARAFAC decompositions,” in Proc. 5-th IEEE Sensor Array and Multichannel Signal Processing Workshop, (Darmstadt, Germany), pp , July [5] M. Haardt, F. Roemer, and G. Del Galdo, “Higher-order SVD based subspace estimation to improve the parameter estimation accuracy in multi-dimensional harmonic retrieval problems,” IEEE Trans. Signal Processing, vol. 56, pp , July [6] X. Liu and N. Sidiropoulos, “PARAFAC techniques for high-resolution array processing”, in High-Resolution and Robust Signal Processing, Y. Hua, A. Gershman, and Q. Chen, Eds., Marcel Dekker, 2004, Chapter 3. [7] M. Weis, F. Roemer, M. Haardt, D. Jannek, and P. Husar, “Multi-dimensional Space-Time-Frequency component analysis of event-related EEG data using closed-form PARAFAC”, in Proc. IEEE Int. Conf. Acoustic, Speech, and Signal Processing (ICASSP), Taipei, Taiwan, pp , Apr [8] F. Roemer and M. Haardt, “Tensor-Based channel estimation (TENCE) for Two-Way relaying with multiple antennas and spatial reuse”, in Proc. IEEE Int. Conf. Acoust. Speech, and Signal Processing (ICASSP), Taipei, Taiwan, pp , Apr. 2009, invited paper. [9] E. Radoi and A. Quinquis, “A new method for estimating the number of harmonic components in noise with application in high resolution RADAR”, EURASIP Journal on App. Sig. Proc., pp , References