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

By Jeffrey A. Hogan

Increasingly vital within the box of communications, the learn 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 a radical 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 assurance and a cautious stability among rigor and clarity, Duration and Bandwidth Limiting is a very unique and helpful source either for mathematicians drawn to the sphere and for pro engineers with an curiosity in concept. whereas its major audience is training scientists, the ebook can also function precious supplemental interpreting fabric for mathematically-based graduate classes in communications and sign processing.

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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.

When f is odd, its action is the same as that of (iF Q√a /2 )2 . 4 and using the known parities of the PSWFs gives the following. 5. If ψ is an eigenfunction of P√a Q√a /2 with eigenvalue λn = λn (a ), then ψ is an eigenfunction of F Q√a /2 with eigenvalue in λn (a ). √ The eigenfunctions of P√a Q√a /2 are dilates by a of those of PQa /2 . 4 also suggests that F QT is half of a timeand band-limiting operation. 11), (Da F Q)2 = ±P2a Q when acting, respectively, on real-valued even or odd functions.

3. An orthogonal matrix B = [b1 , . . , bN ] is the transpose of the eigenvector matrix of some matrix T ∈ UST if and only if (i) b1 ( ) = 0 for = 1, . . , N, (ii) b2 ( )/b1 ( ) = λ , = 1, . . , N, with λk = λ if k = , (iii) BT Λ B ∈ UST where Λ is the diagonal matrix with Λkk = λk . The theorem indicates that the pair b1 and b2 and the eigenvalues λk determine one another, so the remaining eigenvectors are determined once the first two are chosen in such a way that the λk are strictly decreasing.

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