Abstract

The computation of a structured canonical polyadic decomposition (CPD) is useful to address several important modeling problems in real-world applications. In this paper, we consider the identification of a nonlinear system by means of a Wiener-Hammerstein model, assuming a high-order Volterra kernel of that system has been previously estimated. Such a kernel, viewed as a tensor, admits a CPD with banded circulant factors which comprise the model parameters. To estimate them, we formulate specialized estimators based on recently proposed algorithms for the computation of structured CPDs. Then, considering the presence of additive white Gaussian noise, we derive a closed-form expression for the Cramer-Rao bound (CRB) associated with this estimation problem. Finally, we assess the statistical performance of the proposed estimators via Monte Carlo simulations, by comparing their mean-square error with the CRB.

Highlights

  • The canonical polyadic decomposition (CPD), which can be seen as one possible extension of the SVD to higher-order tensors [1], is a well-established mathematical tool utilized in many scientific disciplines [2]

  • Y); (ii) the estimator CP Toeplitz (CPTOEP), described in Section 3.3.2; (iii) the estimator CPTOEP-constrained ALS (CALS), which corresponds to refining the CPTOEP estimate by applying the CALS algorithm; (iv) the estimator CPTOEP-Broyden–Fletcher– Goldfarb–Shanno (BFGS), in which a similar refinement is obtained by minimizing a least-squares criterion

  • For each estimate η(y), we compute εη = η − η(y) 2. This procedure is repeated for 100 realizations of N and εη is averaged for each level of σ2, yielding a mean-square error estimate denoted by MSEη

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Summary

INTRODUCTION

The canonical polyadic decomposition (CPD), which can be seen as one possible extension of the SVD to higher-order tensors [1], is a well-established mathematical tool utilized in many scientific disciplines [2] As it requires only mild assumptions for being essentially unique, the CPD provides means for blindly and jointly identifying the components of multilinear models, which arise in many real-world applications; see [1,2,3] for some examples. The assessment of the statistical performance of CPD computation algorithms via comparison with the Cramer-Rao bound (CRB) [10] is of practical interest, since it can guide the choice for an appropriate algorithm in application domains It can provide valuable information for the study and development of such algorithms.

Tensors and the CP decomposition
The WH model and its equivalent Volterra model
CPD-based WH model identification
ANALYTICAL CRB FOR CPD-BASED WH ESTIMATION ALGORITHMS
Identifiability
Circulant-constrained ALS algorithm
CPTOEP algorithm
Closed-form expression for the CRB
SIMULATION RESULTS
CONCLUSION

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