By Arthur Frazho, Wisuwat Bhosri
In this monograph, we mix operator recommendations with kingdom area how you can resolve factorization, spectral estimation, and interpolation difficulties coming up up to speed 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 coming up in linear quadratic keep an eye on idea and Kalman ?ltering. One good thing about this technique is that it quite simply results in algorithms within the non-degenerate case. nonetheless, this strategy doesn't simply generalize to the nonrational case, and it's not constantly obvious the place the Riccati equations are coming from. Operator conception has built a few stylish tips on how to end up the life of an answer to a couple of those factorization and spectral estimation difficulties in a truly basic atmosphere. even if, those thoughts are usually now not used to enhance computational algorithms. during this monograph, we'll use operator idea with nation area how you can derive computational ways to resolve factorization, sp- tral estimation, and interpolation difficulties. it's emphasised that our process is geometric and the algorithms are received as a distinct program of the idea. we are going to current equipment for spectral factorization. One technique derives al- rithms according to ?nite sections of a undeniable Toeplitz matrix. the opposite technique makes use of operator thought to improve the Riccati factorization technique. eventually, we use isometric extension suggestions to resolve a few interpolation problems.
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Extra resources for An Operator Perspective on Signals and Systems
First we introduce the Fourier transform. Then we will study Toeplitz and Laurent operators. The Fourier transform will be used to turn Laurent operators into multiplication operators and visa versa. 1 The Fourier Transform In this section, we will present a short review of the Fourier transform. Throughout, L2 (E) is the Hilbert space formed by the set of all Lebesgue measurable, square integrable functions with values in E over the interval [0, 2π). In all of our applications concerning the space L2 (E) (or L2+ (E) or H 2 (E) introduced below), the subspace E is ﬁnite dimensional.
Due to this unitary identiﬁcation we drop the tilde notation on h and simply write h(eıω ) for the function Φh. In other words, if h is in H 2 (E) and we write h(eıω ), then we mean that h(eıω ) is the function in L2+ (E) given by h(eıω ) = (Φh)(eıω ). Finally, using this identiﬁcation we also view H 2 (E) as the subspace of L2 (E) corresponding to L2+ (E). Motivated by this identiﬁcation, we use H 2 (E) and L2+ (E) interchangeably. Due to the previous identiﬁcation between H 2 (E) and L2+ (E), we also view the Fourier transform FE+ as the unitary operator from 2+ (E) onto H 2 (E) deﬁned by ⎤ ⎤ ⎡ ⎡ f0 f0 ∞ ⎢ f1 ⎥ ⎢ f1 ⎥ ⎥ ⎥ ⎢ ⎢ (FE+ ⎢ f2 ⎥)(z) = z −k fk where ⎢ f2 ⎥ ∈ 2+ (E).
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 . On the other hand, if T intertwines SE with SY , then using Tj,k as the j-k entry of T , we obtain ⎡ ⎡ ⎤ ⎤ 0 0 0 ··· T01 T02 T03 · · · ⎢ T00 T01 T02 · · · ⎥ ⎢ T11 T12 T13 · · · ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ T10 T11 T12 · · · ⎥ = SY T = T SE = ⎢ T21 T32 T24 · · · ⎥ .
An Operator Perspective on Signals and Systems by Arthur Frazho, Wisuwat Bhosri