Duration and Bandwidth Limiting: Prolate Functions, by Jeffrey A. Hogan

By Jeffrey A. Hogan

Increasingly vital within the box of communications, the research of time and band proscribing is important for the modeling and research of multiband signs. This concise yet accomplished monograph is the 1st to be dedicated in particular to this subdiscipline, offering an intensive research of its idea and functions. via state-of-the-art numerical equipment, it develops the instruments for functions not just to communications engineering, but additionally to optical engineering, geosciences, planetary sciences, and biomedicine.

With wide insurance and a cautious stability among rigor and clarity, Duration and Bandwidth Limiting is a very unique and worthwhile source either for mathematicians drawn to the sector and for pro engineers with an curiosity in concept. whereas its major target market is practising scientists, the ebook can also function worthwhile supplemental interpreting fabric for mathematically-based graduate classes in communications and sign processing.

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40) n=0 This error converges to zero rapidly if f is essentially in the span of the first [2T ] PSWFs, but could converge very slowly otherwise. Papoulis also analyzed aliasing errors and discussed using the maximum entropy method as a means of approximating the PG iterates from (over)sampled values of f . Extrapolation can be carried out in practice for the case of discrete signals. Schlebusch and Splettst¨osser [284] proved that, by approximating band-limited signals by trigonometric polynomials with increasing degree and period length, the solution of the discrete extrapolation problem converges to that of the corresponding continuous extrapolation problem, answering in the affirmative a question raised by Sanz and Huang [283]; cf.

N − 1} then σn , regarded as an eigenfunction of M , belongs to one of the N largest eigenvalues μ0 , . . , μN−1 of M . To do so, denote by μnν the eigenvalue of M to which σn belongs. First, for each n = 0, . . , N − 1, σn is the product of the unimodular function eπ i(1−N)ω with the restriction to T of a polynomial of degree N − 1 in C. Thus, σn has at most N − 1 zeros in T, and so at most N − 1 zeros in (−W,W ). Thus, nν ≤ N − 1. 2, μnν ≤ N − 1, which was to be shown. Ordering Discrete Prolate Spheroidal Sequences We have shown that each of σ0 , .

X(N − 1)]T ∈ CN = 2 (ZN ), whose time–bandwidth product is N. Examples include the spike vectors x = ek satisfying ek ( j) = δ jk whose DFTs are the Fourier basis vectors ωk where ωk ( ) = e2π ik /N . , [152, Chap. 4]. Time and band limiting to such cosets gives rise to operators having some number of eigenvalues equal to one. For K fixed such that 2K + 1 ≤ N, define the Toeplitz matrix A = AK , Ak = ak− = sin (2K + 1)(k − )π /N , N sin ((k − )π /N) (k, = 0, . . , N − 1) . 37) Vectors in the image of A will be said to be K-band limited, since the DFT of any such vector will be zero at an index m such that m mod N > K.

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