By Arthur Frazho, Wisuwat Bhosri
During this monograph, we mix operator ideas with nation house tips on how to clear up factorization, spectral estimation, and interpolation difficulties bobbing 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 up to the mark conception is predicated on Riccati equations coming up in linear quadratic keep an eye on conception and Kalman ?ltering. One benefit of this process is that it without problems ends up in algorithms within the non-degenerate case. nevertheless, this strategy doesn't simply generalize to the nonrational case, and it's not continually obvious the place the Riccati equations are coming from. Operator concept has built a few based how to end up the lifestyles of an answer to a couple of those factorization and spectral estimation difficulties in a truly basic atmosphere. although, those recommendations are normally now not used to enhance computational algorithms. during this monograph, we'll use operator idea with kingdom house tips on how to derive computational tips on how to remedy factorization, sp- tral estimation, and interpolation difficulties. it really is emphasised that our strategy is geometric and the algorithms are bought as a unique software of the speculation. we'll current tools for spectral factorization. One technique derives al- rithms in line with ?nite sections of a definite Toeplitz matrix. the opposite procedure makes use of operator idea to enhance the Riccati factorization technique. ultimately, we use isometric extension options to resolve a few interpolation difficulties.
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Extra resources for An operator perspective on signals and systems
3). In other words, T is a Toeplitz operator. Recall that the bilateral shift UE on 2 (E), respectively UY on 2 (Y) is the minimal unitary extension of SE , respectively SY . Moreover, L is a Laurent operator mapping 2 (E) into 2 (Y) if and only if L is in I(UE , UY ). Let T be an operator mapping 2+ (E) into 2+ (Y). 4, the operator T is Toeplitz if and only if there exists a Laurent operator L such that T = P+ L| 2+ (E) where P+ is the orthogonal projection onto 2+ (Y). In this case, the Laurent operator L is uniquely determined by T .
Let SE be the unilateral shift on 2+ (E) and SY be the unilateral shift on 2 2 2 + (Y). Let T be an operator mapping + (E) into + (Y). Then we claim that T ∗ is a Toeplitz operator if and only if SY T SE = T . 3) shows that SY∗ T SE = T . On the other hand, if SY∗ T SE = T , then the entries Tjk of T are determined by ⎡ ⎡ ⎤ ⎤ T00 T01 T02 · · · T11 T12 T13 · · · ⎢ T10 T11 T12 · · · ⎥ ⎢ T21 T22 T23 · · · ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ T20 T21 T22 · · · ⎥ = T = SY∗ T SE = ⎢ T31 T31 T33 · · · ⎥ . ⎣ ⎣ ⎦ ⎦ .. .. .. ..
In particular, we will show that any function in H 2 (E, Y) admits a unique inner-outer factorization. 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.