By Richard E. Blahut, C.S. Burrus

Algorithms for computation are a significant a part of either electronic sign seasoned cessing and decoders for error-control codes and the imperative algorithms of the 2 matters percentage many similarities. each one topic makes huge use of the discrete Fourier remodel, 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 approach ing. Algebraic constructions have gotten extra vital to its improvement. some of the innovations of electronic sign processing are legitimate in any algebraic box, even supposing normally at the very least a part of the matter will evidently lie both within the actual box or the advanced box simply because that's the place the knowledge originate. In different circumstances the alternative of box for computations can be as much as the set of rules dressmaker, who often chooses the true box or the advanced box as a result of familiarity with it or since it is appropriate for the actual program. nonetheless, it's applicable to catalog the various 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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Moreover, for some a and e, the equation ab = e may have a solution for b even though a has no inverse. The set of nonnegative integers smaller than q under modulo q arithmetic is called the ring of integers modulo q and is denoted by Zq. It is a field if and only if q is a prime. Thus Zp Zp'" GF(p) , =F GF(pm). ) algebraic structure that we shall consider. A group has only a single operation defined on it. 4. Rings and Groups 25 A formal definition of a group is as follows. A 9rouP G is a set together with an operation, denoted *, on pairs of elements satisfying four properties: 1) 2) (Closure) c = a * b is in the group.

2 Let GCD(b,n) decimation = 1 and Bb = 1 (mod n). The cyclic 40 3. Sequences and Spectra has transform with components V£ = V(Bk» , where all indices are interpreted modulo n. Proof The corollary to the Euclidean algorithm implies that Bb+Nn = 1 for some integers B and N. Hence the required B exists. Therefore, V~ n-l = LwikV: i=O n-l " = ~ w(Bb+Nn)ik v «bi» i=O n-l = "~w biBk V«bi». i=O Because i' = ((bi)) is just a permutation, it does not affect the sum. Thus n-l V£ L wi' Bk vi , i'=O V(Bk» , as was to be proved.

0 A Universal Eigenvector Does there exist a vector whose Fourier transform is proportional to itself? When the blocklength is a prime p, the vector whose components are Legendre symbols has this property. The ith component, for i = 0, ... ,p - 1, is defined as Xi = X(i). 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).