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An operator perspective on signals and systems by Arthur Frazho, Wisuwat Bhosri

By Arthur Frazho, Wisuwat Bhosri

During this monograph, we mix operator concepts with nation house the way to remedy factorization, spectral estimation, and interpolation difficulties coming up on top of things and sign processing. We current either the speculation and algorithms with a few Matlab code to unravel those difficulties. A classical method of spectral factorization difficulties on top of things conception is predicated on Riccati equations bobbing up in linear quadratic keep watch over conception and Kalman ?ltering. One good thing about this process is that it with no trouble ends up in algorithms within the non-degenerate case. however, this method doesn't simply generalize to the nonrational case, and it's not consistently obvious the place the Riccati equations are coming from. Operator conception has built a few based how you can turn out the life of an answer to a few of those factorization and spectral estimation difficulties in a truly normal environment. besides the fact that, those thoughts are normally no longer used to enhance computational algorithms. during this monograph, we are going to use operator idea with kingdom area the way to derive computational how to clear up factorization, sp- tral estimation, and interpolation difficulties. it truly is emphasised that our strategy is geometric and the algorithms are got as a unique software of the speculation. we are going to current equipment for spectral factorization. One approach derives al- rithms in line with ?nite sections of a definite Toeplitz matrix. the opposite procedure makes use of operator thought to advance the Riccati factorization process. ultimately, we use isometric extension options to unravel a few interpolation difficulties.

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Inner-outer factorizations play a fundamental role in many optimization and interpolation problems arising in systems theory and signal processing. In Chapter 4 we will study state space realizations for rational inner and outer functions. Finally, recall that throughout this monograph, we assume that the spaces E and Y in H 2 (E, Y), L2 (E, Y), H ∞ (E, Y) and L∞ (E, Y) are all finite dimensional. Many of our results hold in the case when E and Y are separable Hilbert spaces. However, they are finite dimensional in our applications.

Because TΘ and TΨ are both isometries with the same range M, it follows that W = TΨ∗ TΘ is a unitary operator mapping 2+ (E) onto 2+ (D). (If V1 : V1 → K and V2 : V2 → K are two isometries with the same range M, then V1 : V1 → M and V2 : V2 → M can be viewed as unitary operators whose ranges are onto M. ) We claim that TΨ W = TΘ . Because M is the range of an isometry TΨ , this implies that PM = TΨ TΨ∗ . Using this, we obtain TΨ W = TΨ TΨ∗ TΘ = PM TΘ = TΘ . Hence TΨ W = TΘ . Using this along with the fact that both TΘ and TΨ intertwine the appropriate unilateral shifts, we obtain T Ψ SD W = SY T Ψ W = SY T Θ = T Θ SE = T Ψ W S E .

1) Here {Θk }∞ 0 is a sequence of operators mapping E into Y. The box around Θ0 represents the 0-0 component of the Laurent matrix. All the entries above the main diagonal are zero, and the diagonal entries of the Laurent matrix are the same. Now assume that g is a vector in c (E). Then Lg is well defined and the n-th component of Lg is given by n (Lg)n = Θn−j gj j=−∞ (⊕∞ −∞ gj ∈ c (E)). 6. Toeplitz Matrices and H ∞ Functions 37 The symbol for this Laurent matrix is the function with values in L(E, Y) formally defined by Θ(eıω ) = ∞ Θk e−ıωk .

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