TIME DOMAIN CONSIDERATIONS IN OPTIMAL ESTIMATION FOR BANDLIMITED SIGNALS**This work was supported by NSF Grant #ENG-780903.

Parks, T.W.; Kolba, Dean P.

doi:10.1016/b978-0-12-256420-8.50014-9

Cited by 4 publications

(3 citation statements)

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“…The hounds for o} and 02' some of which are given in Table I, evidently do not meet; their role is only to fix roughly the range of accurate extrapolatability. Table I also shows that this range is not very accurately estimated by considerations of the Nyquist rate [8], which would suggest that (J -(a -1 -1)/2.…”

Section: Statement Of Resultsmentioning

confidence: 95%

“…Analogously to the definition of r(O, 1), let and applying Schwarz's inequality, we find On expanding F( w) in its Fourier series, we obtain the series converging in norm, and the Fourier transform of this series yields the sampling expansion (1). In (1) the series converges in norm by (6), hence also uniformly by (8). The construction shows that (sin 'IT( t -k )j'IT( t -k)} form a complete orthonormal set in 9/2 ('IT), that while However, and choosing D so that it intersects the reals in an interval barely larger than [ -1,0], the first integral approaches a finite negative value, while the second approaches zero.…”

Section: Statement Of Resultsmentioning

confidence: 99%

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Extrapolating a band-limited function from its samples taken in a finite interval

Landau¹

1986

IEEE Trans. Inform. Theory

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Section: Statement Of Resultsmentioning

confidence: 95%

Section: Statement Of Resultsmentioning

confidence: 99%

Extrapolating a band-limited function from its samples taken in a finite interval

Landau¹

1986

IEEE Trans. Inform. Theory

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“…l ) , the fundamental non-uniqueness of this exercise, induced by data incompleteness, is sometimes obscured in the discussion (with some welcome exceptions; e.g. Slepian 1978, Parks andKolba 1981). As reviewed below, for some x not equal to any x,, any prediction valuef(s) is compatible with membership in B(u) and the data constraints (1.1).…”

Section: Introductionmentioning

confidence: 99%

The Backus-Gilbert problem for sampled band-limited functions

Huestis¹

1992

Inverse Problems

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While the recovery of a time series f(x) from any finite sample set (f(xi); i=1. . .,N), is always non-unique, the smoothness implied by an assumption of band-limitedness allows certain definite quantitative inferences to be made about the solution set. Band-limitedness allows sample values to be expressed as a set of linear functionals of the unknown f(x), specifically integrals of f(x) weighted by known data kernel functions. The form of the data kernels depends upon the assumed spectral cutoff. The theory of Backus and Gilbert (1968) is applied to find weighted averages common to all solutions f(x) which satisfy these data constraints. The weighting function is a linear combination of data kernels which is designed to be as concentrated as possible at a target x=x0, and thereby gives an average as strongly dependent on solution values as near x0 as possible, while being relatively insensitive to values elsewhere. An apparent paradox arises when standard Backus-Gilbert theory is applied: for some values of x0, a smaller assumed spectral cutoff results in averaging functions which are much broader than the corresponding ones for a larger cutoff. This contradicts the intuitive notion that the increased smoothness implied by a smaller cutoff should result in increased knowledge of f(x), here reflected in narrower averaging functions. Band-limited solutions compatible with a smaller cutoff are a proper subset of those with a larger cutoff, and resolving power should not be lost when attention is restricted to these smoother solutions. The resolution of the paradox comes from the recognition that this argument fails because the data constraints admit a larger solution set than desired, including solutions violating the assumed spectral cutoff; Backus-Gilbert theory yields averages shared by all solutions, not just those which are properly band-limited.

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