Extending Wavelet Filters: Infinite Dimensions, the Nonrational Case, and Indefinite Inner Product Spaces

Alpay, Daniel; Jørgensen, Palle E. T.; Lewkowicz, Izchak

doi:10.1007/978-0-8176-8379-5_5

Cited by 11 publications

(15 citation statements)

References 47 publications

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“…2 For example, in studying classical filters a "high-pass" could be viewed as an "all-pass" minus a "low-pass". 3 In control engineering circles a Schur stable functions in U is called "inner", see e.g. [ Books like [12], [14], [40], and the theses [27], [34] have made an effort to be at least "bi-lingual".…”

Section: Introductionmentioning

confidence: 99%

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Characterizations of rectangular (para)-unitary rational functions

Alpay¹,

Jørgensen²,

Lewkowicz³

2016

OpMath

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Abstract. We here present three characterizations of not necessarily causal, rational functions which are (co)-isometric on the unit circle:(i) through the realization matrix of Schur stable systems, (ii) the Blaschke-Potapov product, which is then employed to introduce an easy-to-use description of all these functions with dimensions and McMillan degree as parameters, (iii) through the (not necessarily reducible) Matrix Fraction Description (MFD). In cases (ii) and (iii) the poles of the rational functions involved may be anywhere in the complex plane, but the unit circle (including both zero and infinity). A special attention is devoted to exploring the gap between the square and rectangular cases.

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Section: Introductionmentioning

confidence: 99%

“…(ii) If for p ≥ m (m ≥ p) the matrix I m − (F (z)) * F (z) (I p − F (z)(F (z)) * ) is positive semi-definite, within the unit disk, 1 ≥ |z|, then F (z) is anti Schur stable 3 , i.e. its conjugate F # (z) is Schur stable…”

Section: Introductionmentioning

confidence: 99%

Characterizations of rectangular (para)-unitary rational functions

Alpay¹,

Jørgensen²,

Lewkowicz³

2016

OpMath

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“…In this paper we remove the L 2 (R, dx) requirement (which we assumed in [2,3]) from the wavelet setting. Now wavelet multiresolutions may be viewed as a special case of a probability space multiresolution.…”

Section: 2mentioning

confidence: 99%

W-Markov measures, transfer operators, wavelets and multiresolutions

Alpay,

Jorgensen,

Lewkowicz

2016

Preprint

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In a general setting we solve the following inverse problem: Given a positive operators R, acting on measurable functions on a fixed measure space (X, B X ), we construct an associated Markov chain. Specifically, starting with a choice of R (the transfer operator), and a probability measure µ 0 on (X, B X ), we then build an associated Markov chain T 0 , T 1 , T 2 , . . ., with these random variables (r.v) realized in a suitable probability space (Ω, F , P), and each r.v. taking values in X, and with T 0 having the probability µ 0 as law. We further show how spectral data for R, e.g., the presence of R-harmonic functions, propagate to the Markov chain. Conversely, in a general setting, we show that every Markov chain is determined by its transfer operator. In a range of examples we put this correspondence into practical terms: (i) iterated function systems (IFS), (ii) wavelet multiresolution constructions, and (iii) IFSs with random control. Our setting for IFSs is general as well: a fixed measure space (X, B X ) and a system of mappings τ i , each acting in (X, B X ), and each assigned a probability, say p i which may or may not be a function of x. For standard IFSs, the p i 's are constant, but for wavelet constructions, we have functions p i (x) reflecting the multi-band filters which make up the wavelet algorithm at hand. The sets τ i (X) partition X, but they may have overlap, or not. For IFSs with random control, we show how the setting of transfer operators translates into explicit Markov moves: Starting with a point x ∈ X, the Markov move to the next point is in two steps, combined yielding the move from T 0 = x to T 1 = y, and more generally from T n to T n+1 . The initial point x will first move to one of the sets τ i (X) with probability p i , and once there, it will choose a definite position y (within τ i (X)), now governed by a fixed law (a given probability distribution). For Markov chains, the law is the same in each move from T n to T n+1 .

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“…To this end, in Subsections 3.1, 3.2 and 4.1 respectively, we present three characterizations of Laurent polynomials in U. 3. minimal realization of (co)-isometric FIR 3.1.…”

Section: Doubling the Powersmentioning

confidence: 99%

Characterizations of families of rectangular, finite impulse response, para-unitary systems

Alpay

Jørgensen

Lewkowicz

2016

J. Appl. Math. Comput.

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We here study Finite Impulse Response (FIR) rectangular, not necessarily causal, systems which are (para)-unitary on the unit circle (=the class U ). First, we offer three characterizations of these systems. Then, introduce a description of all FIRs in U , as copies of a real polytope, parametrized by the dimensions and the McMillan degree of the FIRs.Finally, we present six simple ways (along with their combinations) to construct, from any FIR, a large family of FIRs, of various dimensions and McMillan degrees, so that whenever the original system is in U , so is the whole family.A key role is played by Hankel matrices.1 β=0,Hence, in Engineering terminology Φ(t) (and often F (z)) is referred to as Finite Impulse Response (i.e. the support of Φ(t) is finite).Moreover F (z) will be called causal whenever 1 ≥ q (i.e. Φ(t) ≡ 0 for all 0 > t) and strictly causal if 0 ≥ q. Similarly, (strictly) anti-causal when (q ≥ n + 1) q ≥ n.

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Extending Wavelet Filters: Infinite Dimensions, the Nonrational Case, and Indefinite Inner Product Spaces

Cited by 11 publications

References 47 publications

Characterizations of rectangular (para)-unitary rational functions

Characterizations of rectangular (para)-unitary rational functions

W-Markov measures, transfer operators, wavelets and multiresolutions

Characterizations of families of rectangular, finite impulse response, para-unitary systems

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