System Theory

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

By Arthur Frazho, Wisuwat Bhosri

During this monograph, we mix operator strategies with nation house the right way to resolve factorization, spectral estimation, and interpolation difficulties bobbing up on top of things and sign processing. We current either the speculation and algorithms with a few Matlab code to resolve those difficulties. A classical method of spectral factorization difficulties on top of things idea is predicated on Riccati equations bobbing up in linear quadratic keep watch over thought and Kalman ?ltering. One good thing about this process is that it without problems results in algorithms within the non-degenerate case. however, this strategy doesn't simply generalize to the nonrational case, and it isn't constantly obvious the place the Riccati equations are coming from. Operator thought has constructed a few based how to turn out the lifestyles of an answer to a few of those factorization and spectral estimation difficulties in a truly normal atmosphere. despite the fact that, those innovations are regularly now not used to increase computational algorithms. during this monograph, we are going to use operator concept with kingdom house ways to derive computational how to remedy factorization, sp- tral estimation, and interpolation difficulties. it really is emphasised that our procedure is geometric and the algorithms are bought as a different software of the speculation. we'll current tools for spectral factorization. One technique derives al- rithms in keeping with ?nite sections of a definite Toeplitz matrix. the opposite process makes use of operator conception to improve the Riccati factorization strategy. eventually, we use isometric extension ideas to unravel a few interpolation difficulties.

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Example text

Finally, 0 z it is noted that TΘ defines an operator from 2+ (E) into 2+ (Y) if and only if Θ(z) is a function in H ∞ (E, Y). In this case, LΘ = TΘ = Θ ∞ . 4) with respect to the standard basis for 2+ (E) and 2+ (Y). In this case, T is denoted by TΘ where Θ is the symbol for T . Let SV denote the unilateral shift on 2+ (V). Let T be an operator mapping 2+ (E) into 2+ (Y). We claim that T is a lower triangular Toeplitz operator if and only if T intertwines SE with SY , that is, T SE = SY T . If T is a Toeplitz operator, then using the matrix representations for T , SE and SY , it is easy to verify that T intertwines SE with SY .

Moreover, H is an invariant subspace for the backward shift if and only if H = H(Θ) for some inner function Θ in H ∞ (E, Y). Furthermore, this inner function Θ is unique up to a constant unitary operator on the right. Let Θ be a function in H ∞ (E, Y) and Ψ a function in H ∞ (D, Y). Then we say that Ψ is a left divisor of Θ if there exists a function Φ in H ∞ (E, D) such that Θ(z) = Ψ(z)Φ(z) for all z in D+ . In this case, if both Θ and Ψ are inner functions, then Φ must also be an inner function.

Causal Laurent matrix if L is a Laurent .. .. 0 Θ0 0 0 Θ1 .. Θ0 .. ⎤ ⎥ ⎥ ··· ⎥ ⎥ ⎥ ··· ⎥. ⎥ .. ⎥ . ⎥ ⎦ .. 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)).

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