Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty-III: The Dimension of the Space of Essentially Time- and Band-Limited Signals

Abstract

The purpose of this paper is to examine the mathematical truth in the engineering intuition that there are approximately 2WT independent signalsV'i of bandwidth W conceturated in an interval of length T. Roughly speaking, the result is true for the best choice of the V'i (prolate spheroidal wave functions), but not for sampling functions (of the form sin tit). Some typical conclusions are: Let f( t), of total energy 1, be band-limited to bandwidth W, and let +N 12 anV'n dt < CE(

Time-Frequency Localization and Sampling of Multiband Signals

“…Fourier uniqueness now implies that the functions ϕn are complete in In their works [29,34,24], Landau, Slepian, Pollak and Widom proved a number of statements of the "ΩT " stating, in essence, that the dimension of the space of essentially T -timelimited and Ω-bandlimited signals is essentially the time-frequency area ΩT . One version-Theorem 3 below-says that P Ω Q T has about ΩT eigenvalues close to one, and that the eigenvalues plunge rapidly from λ ≈ 1 to λ ≈ 0 over a transition band of width around log ΩT .…”

“…We begin reviewing the work of Landau, Slepian and Pollak [35,28,29,32] -the so-called Bell Labs theory -concerning "time-and bandlimiting" (see also [22,24,34,26]) in Section 2. The Paley-Wiener space PW consists of those functions in L 2 (R) whose Fourier transforms vanish outside the set [−1/2, 1/2].…”

Nufft-Accelerated Plane-Polar (Also Phaseless) Near-Field/Far-Field Transformation

“…In order to improve the antenna characterization results as compared to a standard transformation by exploiting the a priori information on the shape and size of the source [8,9,21], the aperture field E a is represented by the Prolate Spheroidal Wave Functions (PSWFs) as [9,[21][22][23]]…”

“…where Φ i [c w , w] is the i-th, 1D PSWF with "space-bandwidth product" c w [22,23], c x = au , c y = bv and u and v locate the spectral region of interest [21], as u ≤ β and v ≤ β. In Eq.…”

Pseudo prolate spheroidal functions

“…In a nowadays classical paper [18], whose purpose was Ω are real intervals, P T,Ω can be written explicitly as…”