By Richard E. Blahut, C.S. Burrus
Algorithms for computation are a critical a part of either electronic sign professional cessing and decoders for error-control codes and the primary algorithms of the 2 matters percentage many similarities. every one topic makes wide 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 not should be considered as a digitized model of analog sign method ing. Algebraic buildings have gotten extra vital to its improvement. a few of the concepts of electronic sign processing are legitimate in any algebraic box, even if as a rule no less than a part of the matter will evidently lie both within the actual box or the complicated box simply because that's the place the information originate. In different situations the alternative of box for computations should be as much as the set of rules clothier, who often chooses the genuine box or the advanced box due to familiarity with it or since it is appropriate for the actual program. nonetheless, it truly is acceptable to catalog the numerous algebraic fields in a manner that's available to scholars of electronic sign processing, in hopes of stimulating new purposes to engineering tasks.
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Additional info for Algebraic Methods for Signal Processing and Communications Coding
Unless the additive order is explicitly mentioned, the term "order" in a finite field refers to multiplicative order. The finite field GF(q) has q - 1 elements in its multiplicative group. Hence the order of every nonzero element of GF(q) divides q - 1. 5 Algebraic Integers Let w be a primitive nth root of unity in the complex field with n ;::: 3. It is a zero of the cyclotomic polynomial IT GCD(k,n)=l where the product is over all k less than n and relatively prime to n. The degree of the polynomial n(x) is equal to tj>(n).
But j is relatively prime to p so the sum in the first term reduces to xU)O. Therefore Dj p-l p-l 0=1 k=1 = X( -1)coo [l- XU)O-lO) + ~X( -1) ~wij LwtkXGk )X(k)Ck' p The term X(k) in the second term can be replaced by -1 because Ck whenever X(k) =F -1. Thus Dj p-l p-l 0=1 k=l =0 = X(-I) [coo (l- xU)) - ~ ~wij LwtkX(~k )Ck]. p The first term is zero if j is a nonzero square, so we ignore this term henceforth by restricting attention only to those j that are nonzero squares. Next observe that the indices are elements ofthe field GF(p), so the nonzero indices can be written as powers of a primitive element 7r.
The following theorem shows that, for m 2:: 2, = 0 in any field F. Thus em p"'-l I: X(i)w i =0 i=O leading to a large collection of identities, including trigonometric identities, which are related to the cyclotomic polynomials. In particular the theorem implies that the polynomial p"'-l g(x) = I: X(i)x i i=O is divisible by the cyclotomic polynomial