Algebraic Number Theory and Code Design for Rayleigh Fading - download pdf or read online

By F. Oggier, E. Viterbo, Frederique Oggier

ISBN-10: 1933019077

ISBN-13: 9781933019079

Algebraic quantity idea is gaining an expanding effect in code layout for lots of diverse coding functions, comparable to unmarried antenna fading channels and extra lately, MIMO structures. prolonged paintings has been performed on unmarried antenna fading channels, and algebraic lattice codes were confirmed to be an efficient device. the overall framework has been constructed within the final ten years and many particular code structures in response to algebraic quantity thought are actually on hand. Algebraic quantity conception and Code layout for Rayleigh Fading Channels offers an outline of algebraic lattice code designs for Rayleigh fading channels, in addition to an instructional advent to algebraic quantity thought. the elemental proof of this mathematical box are illustrated via many examples and by means of machine algebra freeware so one can make it extra available to a wide viewers. This makes the ebook compatible to be used via scholars and researchers in either arithmetic and communications.

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Extra resources for Algebraic Number Theory and Code Design for Rayleigh Fading Channels (Foundations and Trends in Communications and Information Theory)

Sample text

9. [18] Algebraic lattices exhibit a diversity L = r1 + r2 . Proof. Let x = 0 be an arbitrary point of Λ: x = (σ1 (x), . . , σr1 (x), σr1 +1 (x), . . , σr1 +r1 (x)) with x ∈ OK . Since x = 0, we have x = 0 and the first r1 coefficients are non-zero. The minimum number of non-zero coefficients among the 2r2 that are left is r2 since the real and imaginary parts of any one of the complex embeddings may not be null together. We thus have a diversity L ≥ r1 + r2 . Applying the canonical embedding to x = 1 gives exactly r1 + r2 non-zero coefficients (σj (1) = 1 for any j), which concludes the proof.

N are linearly independent, so they define a full rank algebraic lattice Λ = Λ(OK ) = σ(OK ). 5) that the lattice Λ = σ(OK ) can be expressed by means of its generator matrix M . Λ = {x = λM ∈ Rn | λ ∈ Zn } The lattice generator matrix M is given explicitly by ⎛ ⎜ ⎝ σ1 (ω1 ) . . σr1 (ω1 ) σ1 (ω2 ) . . σr1 (ω2 ) σ1 (ωn ) . . σr1 (ωn ) σr1 +1 (ω1 ) σr1 +1 (ω2 ) σr1 +1 (ω1 ) . . σr1 +r2 (ω1 ) σr1 +r2 (ω1 ) σr1 +1 (ω2 ) . . σr1 +r2 (ω2 ) σr1 +r2 (ω2 ) .. σr1 +1 (ωn ) σr1 +1 (ωn ) . . 1) where the vectors vi are the rows of M .

6. Let L/K be a field extension, and let α ∈ L. If there exists a non-zero irreducible monic (with highest coefficient 1) polynomial p ∈ K[X] such that p(α) = 0, we say that α is algebraic over K. Such a polynomial is called the minimal polynomial of α over K. We denote it by pα . In our example, the polynomial X 2 − 2 is the minimal polynomial of √ 2 over Q. 7. If all the elements of K are algebraic, we say that K is an algebraic extension of Q. √ √ Consider the field Q( 2) = {a+b 2, a, b ∈ Q}. It is simple to see that √ any α ∈ Q( 2) is a root of the polynomial pα (X) =√X 2 − 2aX + a2 − 2b2 with rational coefficients.

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Algebraic Number Theory and Code Design for Rayleigh Fading Channels (Foundations and Trends in Communications and Information Theory) by F. Oggier, E. Viterbo, Frederique Oggier


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