Multilinear Algebra Based Fitting of a Sum of Exponentials to Oversampled Data Lieven De Lathauwer Jean-Michel Papy Sabine Van Huffel K.U.Leuven ∗ FMTC † /K.U.Leuven ∗ K.U.Leuven ∗ ∗ Catholic University of Leuven † Flander’s Mechatronics Technology Center (Leuven) L. De Lathauwer 1
Overview Overview ■ Introduction Introduction Decimative method ◆ The use of the complex exponential model; Tensor approach ◆ The complex exponential model; Simulation Results Conclusions ◆ Standard Matrix approach; ■ Decimative method ■ Tensor approach ■ Simulation results ■ Conclusions L. De Lathauwer 2
Overview Overview ■ Introduction Introduction Decimative method ■ Decimative method Tensor approach Simulation Results ◆ Decimation; Conclusions ◆ More structure; ■ Tensor approach ■ Simulation results ■ Conclusions L. De Lathauwer 2
Overview Overview ■ Introduction Introduction Decimative method ■ Decimative method Tensor approach Simulation Results ■ Tensor approach Conclusions ◆ Construction of a third-order tensor; ◆ The higher-order VDMD; ◆ The Tucker Decomposition or Higher-Order SVD; ◆ Dimensionality reduction; ◆ Ill-conditioning issue ◆ Unsymmetric tensor approximation ◆ Summary ■ Simulation results ■ Conclusions L. De Lathauwer 2
Overview Overview ■ Introduction Introduction Decimative method ■ Decimative method Tensor approach Simulation Results ■ Tensor approach Conclusions ■ Simulation results ◆ Two-peak, damped signal ◆ Two-peak, undamped signal ◆ Five-peak, damped signal ■ Conclusions L. De Lathauwer 2
Overview Overview ■ Introduction Introduction Decimative method ■ Decimative method Tensor approach Simulation Results ■ Tensor approach Conclusions ■ Simulation results ■ Conclusions L. De Lathauwer 2
The use of the complex exponential model This model is ubiquitous in digital signal processing applications: Overview Introduction The use of the complex exponential model ■ Nuclear magnetic resonance (NMR) spectroscopy, The complex exponential model Standard Matrix ■ audio processing, approach Decimative method ■ speech processing, Tensor approach Simulation Results ■ material health monitoring, Conclusions ■ shape from moments L. De Lathauwer 3
The complex exponential model x t The discrete-time model has the following form: Overview Introduction The use of the K � complex exponential model x n = a k exp { jϕ k } exp { ( α k +2 jπν k ) n. ∆ t } + b n n = 0 , . . . , N − 1 , The complex exponential model k =1 Standard Matrix (1) frequency wgn phase approach amplitude damping Decimative method sampling time interval factor Tensor approach K � Simulation Results c k z n x n = k + b n n = 0 , . . . , N − 1 , (2) Conclusions k =1 ■ c k = a k exp { jϕ k } : complex amplitudes , ■ z k = exp { ( α k + 2 jπν k )∆ t } : signal poles . ν k f 0 − 1 1 2∆ t 2∆ t L. De Lathauwer 4
Standard Matrix approach Overview Starting point for a subspace representation = ⇒ { x n } is arranged in Introduction a Hankel matrix: The use of the complex exponential model · · · x 0 x 1 x 2 x M − 1 The complex exponential model . ... . Standard Matrix · · · . x 1 x 2 approach . ... ... . H = Decimative method · · · . x 2 Tensor approach . . . . . . . . . . . . x N − 2 Simulation Results · · · · · · x L − 1 x N − 2 x N − 1 Conclusions L. De Lathauwer 5
Standard Matrix approach Overview H can be factorized as follows (noise-free case): Introduction The use of the complex exponential H = model The complex 1 · · · 1 exponential model Standard Matrix z 1 z 1 z M − 1 z 1 z 2 · · · 1 · · · c 1 approach 0 1 K 1 1 1 . . . . z 2 z 2 ... · · · Decimative method . . . . 1 K . . . · · · . . . . 0 Tensor approach . . . z M − 1 z 1 z 2 . . . c K 1 · · · K K K Simulation Results z L − 1 z L − 1 · · · Conclusions 1 K = SCT T (3) rank-K matrix = ⇒ Vandermonde decomposition Due to the structure of the noise-free model x n , H is rank deficient The rank equals the number of signal poles (model order) L. De Lathauwer 5
Standard Matrix approach Overview H can be factorized as follows (noise-free case): Introduction The use of the complex exponential H = model The complex 1 · · · 1 exponential model Standard Matrix z 1 z 1 z M − 1 z 1 z 2 · · · 1 · · · c 1 approach 0 1 K 1 1 1 . . . . z 2 z 2 ... · · · Decimative method . . . . 1 K . . . · · · . . . . 0 Tensor approach . . . z M − 1 z 1 z 2 . . . c K 1 · · · K K K Simulation Results z L − 1 z L − 1 · · · Conclusions 1 K = SCT T (3) subspace-of-interest rank-K matrix = ⇒ Vandermonde decomposition Due to the structure of the noise-free model x n , H is rank deficient The rank equals the number of signal poles (model order) L. De Lathauwer 5
Standard Matrix approach In the noise-free case, this rank deficiency is reflected by the SVD of Overview Introduction H : The use of the complex exponential V H model λ 1 1 0 The complex . . . · · · · · · ... 0 . . . exponential model . . . Standard Matrix 0 H = V H · · · · · · U 1 U K U L λ K approach K . . . . . . · · · · · · Decimative method . . . 0 0 V H Tensor approach M Simulation Results � � � � � � � Conclusions H � 0 Σ V H � = � U � Σ � = U U 0 V V H 0 0 0 L. De Lathauwer 5
Standard Matrix approach In the presence of noise H is a full rank matrix: Overview Introduction The use of the complex exponential V H model λ 1 1 0 The complex . . . · · · · · · ... 0 . . . exponential model . . . Standard Matrix 0 H = V H · · · · · · U 1 U K U L λ K approach K . . . . . . · · · · · · Decimative method . . . 0 Σ 0 V H Tensor approach M Simulation Results � � � � � � � Conclusions H � 0 Σ V H � = � U � Σ � = U U 0 V V H 0 Σ 0 0 L. De Lathauwer 5
Problem Statement Overview If the signal poles are close (closely spaced peaks), the Vandermonde vectors are almost dependent. Therefore, in the presence of noise, � Introduction U Decimative method might yield a very poor estimate of the column space of S . Problem Statement Decimation More structure Tensor approach Simulation Results Conclusions L. De Lathauwer 6
Problem Statement Overview If the signal poles are close (closely spaced peaks), the Vandermonde vectors are almost dependent. Therefore, in the presence of noise, � Introduction U Decimative method might yield a very poor estimate of the column space of S . Problem Statement Decimation More structure S 1 U 1 Tensor approach U 2 Simulation Results S 2 Conclusions 0 f − f s f s 2 2 L. De Lathauwer 6
b b b b b b b b b b b b b b b b b b b b Problem Statement Overview If the signal poles are close (closely spaced peaks), the Vandermonde vectors are almost dependent. Therefore, in the presence of noise, � Introduction U Decimative method might yield a very poor estimate of the column space of S . Problem Statement Decimation More structure S 1 U 1 Tensor approach U 2 Simulation Results S 2 Conclusions 0 f − f s f s x 2 2 Oversampling yields higher accuracy t L. De Lathauwer 6
b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b Problem Statement Overview If the signal poles are close (closely spaced peaks), the Vandermonde vectors are almost dependent. Therefore, in the presence of noise, � Introduction U Decimative method might yield a very poor estimate of the column space of S . Problem Statement Decimation More structure S 1 U 1 Tensor approach U 2 Simulation Results S 2 Conclusions 0 f − f s f s x 2 2 b b b b b b b b b b b Oversampling yields higher accuracy b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b t b b b b b L. De Lathauwer 6
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