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Download e-book for kindle: Algebraic Methods for Signal Processing and Communications by Richard E. Blahut, C.S. Burrus

By Richard E. Blahut, C.S. Burrus

ISBN-10: 1461228263

ISBN-13: 9781461228264

ISBN-10: 146127687X

ISBN-13: 9781461276876

Algorithms for computation are a vital a part of either electronic sign professional­ cessing and decoders for error-control codes and the vital algorithms of the 2 matters percentage many similarities. every one topic makes large use of the discrete Fourier rework, of convolutions, and of algorithms for the inversion of Toeplitz structures of equations. electronic sign processing is now a longtime topic in its personal correct; it now not has to be considered as a digitized model of analog sign procedure­ ing. Algebraic buildings have gotten extra vital to its improvement. the various recommendations of electronic sign processing are legitimate in any algebraic box, even if typically at the very least a part of the matter will evidently lie both within the genuine box or the advanced box simply because that's the place the information originate. In different circumstances the alternative of box for computations will be as much as the set of rules dressmaker, who often chooses the genuine box or the complicated box due to familiarity with it or since it is acceptable for the actual program. nonetheless, it's applicable to catalog the numerous algebraic fields in a fashion that's obtainable to scholars of electronic sign processing, in hopes of stimulating new functions to engineering tasks.

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R/16)4 near the origin. in which the coefficients are restricted to be small (three-bit) integers, provides surprisingly good approximations near the origin of the complex plane. 1 shows the set embedded in the complex plane; each point is represented by a dot. 2 shows a similar set of points from Z[ej21r/16]6 embedded in the complex plane. 28 2. 2. Z[ei21r/16J6 near the origin. 3 Sequences and Spectra Traditionally, digital signal processing deals with vectors whose components are elements of the complex number system or the real number system.

We shall call this vector the Legendre vector. Such a vector has only components equal to 0 or ±1, so it can be regarded as a vector over any field F. We shall see in the next theorem that it is an eigenvector of the Fourier transform operator of blocklength p in F (provided a Fourier transform of blocklength p exists in F). 1 In any field F, not of chamcteristic p, the Legendre vector of prime blocklength p is an eigenvector of the Fourier tmnsform opemtor of blocklength p and the corresponding eigenvalue is the Gaussian sum ().

2. All of the properties hold in any field F; all of the properties except the last few are well known. 1. The translation property is the dual of the modulation property. 38 3. Sequences and Spectra 1. V + JLV' +---+ >"V + JLV' 2. Inverse Vi n-l =~ L i W-ikVk k=O where n = 1 + 1 + ... + 1 = 0, ... ,n-1 (~ terms) 3. Modulation 4. Translation 5. Convolution Theorem n-l ei =L i f((i-£»9£ = 0, ... ,n-1 £=0 if and only if k 6. v(x) n-l = L i=O = 0, ... ,n-1 Vixi has a zero at wk if and only if Vk = ° 7.

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Algebraic Methods for Signal Processing and Communications Coding by Richard E. Blahut, C.S. Burrus

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