Duration and bandwidth limiting : prolate functions, by Jeffrey A. Hogan

By Jeffrey A. Hogan

Preface.- bankruptcy 1: The Bell Labs Theory.- bankruptcy 2: Numerical facets of Time- and Bandlimiting.- bankruptcy three: Thomson's Multitaper approach and functions to Channel Modeling.- bankruptcy four: Time- and Bandlimiting of Multiband Signals.- bankruptcy five: Sampling of Bandlimited and Multiband Signals.- bankruptcy 6: Time-localized Sampling Approximations.- Appendix: Notation and Mathematical Prerequisites.- References.- Index

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21) with btrn ∈ RNtr and Atr the top left Ntr × Ntr submatrix of the matrix A above. 22) with Ao , Ae both (Ntr /2) × (Ntr /2) tridiagonal matrices (assuming Ntr even) given by (Ae )mk = a2m,2k and (Ao )mk = a2m+1,2k+1 . 22) (c) (c) yield approximations of the Legendre coefficients βnm of the prolates φ¯0 , φ¯1 , . . 18) and the associated eigenvalues χ0 , χ1 , . . , χN−1 . Plots of φ¯n for c = 5 and n = 0, 3, 10 using Boyd’s method can be found in Fig. 1. Boyd [44] reported that for all N and c, the worst approximated prolate is that of highest order (c) (c) φ¯N−1 so that if the truncation Ntr is chosen large enough so that φ¯N−1 is computed (c) with sufficient accuracy, then so too will φ¯n with 0 ≤ n ≤ N − 2.

The inverse Fourier transform, in turn, maps a function F(ω ) ∈ L2 (T) to its sequence of Fourier coefficients F[k] = 01 F(ω )e−2π ikω dω . While it is common to think of the forward transform, instead, as a mapping from functions to sequences, in the present context it is preferable to regard ω as a frequency variable and the integers as discrete time indices. The index-limiting operator QN,k0 truncates a sequence to N of its nonzero values centered at k0 , while the frequency-limiting operation BW multiplies a function in L2 (T) by the periodic extension of ½[−W,W ] where 0 ≤ W ≤ 1/2.

29) is real, symmetric, and Hilbert–Schmidt, that is, square-integrable over the region in which max{|ξ |, |ω |} ≤ W . The characteristic equation L Ψ = λΨ has as solutions real eigenfunctions Ψ0 , Ψ1 , . . whose restrictions to [−W,W ] are complete in L2 [−W,W ]. The eigenvalues λn of L are real and those that are nonzero have finite multiplicity. The eigenvalues and eigenvectors depend continuously on N. The eigenfunctions for λ = 0 can be extended to R by extending the characteristic equation to R.

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