By Richard E. Blahut, C.S. Burrus
Algorithms for computation are a crucial a part of either electronic sign seasoned cessing and decoders for error-control codes and the important algorithms of the 2 matters proportion many similarities. each one topic makes huge use of the discrete Fourier remodel, of convolutions, and of algorithms for the inversion of Toeplitz platforms of equations. electronic sign processing is now a longtime topic in its personal correct; it not has to be seen as a digitized model of analog sign approach ing. Algebraic constructions have gotten extra vital to its improvement. a number of the suggestions of electronic sign processing are legitimate in any algebraic box, even though often at the least a part of the matter will clearly lie both within the actual box or the advanced box simply because that's the place the information originate. In different instances the alternative of box for computations should be as much as the set of rules fashion designer, who often chooses the genuine box or the complicated box as a result of familiarity with it or since it is acceptable for the actual program. nonetheless, it really is acceptable to catalog the various algebraic fields in a fashion that's available to scholars of electronic sign processing, in hopes of stimulating new functions to engineering tasks.
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Extra resources for Algebraic Methods for Signal Processing and Communications Coding
Then, by the conjugacy constraint, if v is in GF(p), But the left side can be expanded into a real part and an imaginary part as follows: v: = = (VRk + jVlk)P Vk k + jPVt,. 5. Decimation of Shift Register Sequences 45 because VRk and Vlk are each in GF(p) (where all elements satisfy (3P = (3) and jP = - j if p = -1 (mod 4). Therefore in GF(p2), with p = -1 (mod 4) the conjugacy constraint is V'; = y((pk)), which resembles the complex case. The final case that we consider is the case. of a cyclotomic extension of the rationals.
We can represent the elements of GF((2 16 + 1)2) in the form a + jb, where a and b are elements of GF(2 16 + 1), and j2 = -j - 1. Because j does not act the same as A does in the complex field, we cannot embed complex integers into GF((216 + 1)2) but we can embed real integers into the real part of GF((216 + 1)2). Therefore Fourier transforms in GF((216 + 1)2) can be used to convolve sequences ofreal integers in the same way that Fourier transforms in the complex field can be used to convolve sequences of real integers.
Hence c - c' has weight at least 2t + 1 if C - C' has 2t consecutive components equal to zero. Hence c and c' differ in at least 2t + 1 places if C and C' have the same 2t consecutive components equal to zero. A Reed-Solomon code is the set of all vectors with a fixed set of 2t consecutive components equal to zero; any two such vectors must differ in at least 2t + 1 places.