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

By Jeffrey A. Hogan

Preface.- bankruptcy 1: The Bell Labs Theory.- bankruptcy 2: Numerical features of Time- and Bandlimiting.- bankruptcy three: Thomson's Multitaper process and purposes 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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Example text

N − 1. 34) must commute as their counterparts L and M do. Consequently, the Slepian vectors sn , n ∈ ZN , are also the orthogonal eigenvectors of S(N,W ) . Slepian employed a continuation argument along the same lines as in Slepian and Pollak [309] in order to prove that the ordering of the nonzero eigenvalues λn of L follows that of the nonzero eigenvalues of M . Specifically, we have already arranged the eigenvalues μn of M in decreasing order. We order λn in such a way that, if M σn = μn σn , then L σn = λn σn .

Let A = {amk }∞ m,k=0 be the doubly infinite tridiagonal matrix with nonzero elements ⎧ c2 m(m − 1) ⎪ ⎪ ⎪ ⎪ ⎪ (2m − 1) (2m − 3)(2m + 1) ⎪ ⎪ ⎪ 2 2 ⎪ ⎨m(m + 1) + c (2m + 2m − 1) am,k = (2m + 3)(2m − 1) ⎪ 2 (m + 2)(m + 1) ⎪ c ⎪ ⎪ ⎪ ⎪ ⎪ (2m + 3) (2m + 5)(2m + 1) ⎪ ⎪ ⎩ 0 if m ≥ 2, k = m − 2 if m = k ≥ 0 if m ≥ 0, k = m + 2 else. ¯ Then P P¯m = ∑∞ k=0 amk Pk . 2 Time- and Frequency-localization Operators χn φ¯n = ∞ 17 ∞ ∑ ∑ βnm amk P¯k . 19) k=0 m=0 For fixed n ≥ 0, let bn = (βn0 , βn1 , βn2 , . . )T .

The Legendre polynomials can be defined iteratively by means of the Gram–Schmidt process in order that the polynomials be orthogonal over [−1, 1] with 1 f (t) g(t) dt. It is standard to normalize Pn so that Pn (1) = 1. respect to f , g = −1 Among algebraic relationships satisfied by the Legendre functions, the most important for us will be Bonnet’s recurrence formula (n + 1) Pn+1(t) − (2n + 1)tPn(t) + nPn−1(t) = 0 . 12), namely d 2 dPn (t − 1) = n(n + 1)Pn(t), dt dt χn (0) = n(n + 1) . 13) P2 (t) = 3t 2 /2 − 1/2 and so on.

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