By Arthur Frazho, Wisuwat Bhosri
In this monograph, we mix operator concepts with nation house how to resolve factorization, spectral estimation, and interpolation difficulties bobbing up up to the mark and sign processing. We current either the idea and algorithms with a few Matlab code to unravel those difficulties. A classical method of spectral factorization difficulties up to speed thought is predicated on Riccati equations bobbing up in linear quadratic regulate idea and Kalman ?ltering. One good thing about this method is that it comfortably results in algorithms within the non-degenerate case. nonetheless, this technique doesn't simply generalize to the nonrational case, and it isn't constantly obvious the place the Riccati equations are coming from. Operator idea has built a few based ways to turn out the life of an answer to a few of those factorization and spectral estimation difficulties in a really basic atmosphere. in spite of the fact that, those suggestions are often no longer used to improve computational algorithms. during this monograph, we'll use operator concept with country area tips on how to derive computational easy methods to resolve factorization, sp- tral estimation, and interpolation difficulties. it really is emphasised that our process is geometric and the algorithms are bought as a distinct program of the speculation. we'll current tools for spectral factorization. One strategy derives al- rithms in response to ?nite sections of a definite Toeplitz matrix. the opposite process makes use of operator idea to boost the Riccati factorization process. ultimately, we use isometric extension ideas to resolve a few interpolation problems.
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Additional info for An Operator Perspective on Signals and Systems
Recall that Θ is an inner function if Θ is in H ∞ (E, Y) and Θ(eıω ) is almost everywhere an isometry mapping E into Y with respect to the Lebesgue measure. 1 (Beurling-Lax-Halmos). Let S be a unilateral shift on 2+ (Y). Then M is an invariant subspace for S if and only if M admits a representation of the form M = TΘ 2+ (E), where Θ is an inner function in H ∞ (E, Y). Moreover, this representation is unique up to a constant unitary operator on the right. To be precise, if M = TΨ 2+ (D) where Ψ is an inner function in H ∞ (D, Y), then Θ(z) = Ψ(z)Ω where Ω is a constant unitary operator mapping E onto D.
As before, let M be an invariant subspace for the unilateral shift SY on 2+ (Y). Let Φ be any isometry mapping a space E into 2+ (Y) such that the range of Φ equals M SY M. The proof of the Beurling-Lax-Halmos Theorem shows that Θ(z) = (FY+ Φ)(z) is an inner function in H ∞ (E, Y) satisfying M = TΘ 2+ (Y). 7). 1, we obtain the following H 2 version of the BeurlingLax-Halmos theorem. 3. Let S be a unilateral shift on H 2 (Y), then M is an invariant subspace for S if and only if M = ΘH 2 (E) where Θ is an inner function in H ∞ (E, Y).
Toeplitz and Laurent Operators Finally, Parserval’s theorem shows that ∞ (fk , gk )E , (f, g) = k=−∞ ∞ ∞ where f (eıω ) = −∞ e−ıωk fk , and g(eıω ) = −∞ e−ıωk gk are the Fourier series 2 expansion for f and g respectively. In particular, f 2 = ∞ −∞ fk . 2 The Fourier transform FE is the unitary operator mapping (E) onto L2 (E) deﬁned by FE ··· f−2 f−1 f0 f1 f2 ··· ∞ tr e−ıωk fk . 1) k=−∞ The box around f0 represents the zero component of a vector f in 2 (E). We say that Z is the bilateral shift on L2 (E) if Z is the unitary operator on L2 (E) deﬁned by (Zg)(eıω ) = e−ıω g(eıω ) (g ∈ L2 (E)).
An Operator Perspective on Signals and Systems by Arthur Frazho, Wisuwat Bhosri