By Richard E. Blahut, C.S. Burrus

Algorithms for computation are a important a part of either electronic sign professional­ cessing and decoders for error-control codes and the important algorithms of the 2 matters proportion many similarities. each one topic makes vast 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 not has to be considered as a digitized model of analog sign technique­ ing. Algebraic buildings have gotten extra vital to its improvement. the various thoughts of electronic sign processing are legitimate in any algebraic box, even though mostly no less than a part of the matter will clearly lie both within the genuine box or the complicated box simply because that's the place the information originate. In different instances the alternative of box for computations might be as much as the set of rules clothier, who often chooses the genuine box or the advanced box as a result of familiarity with it or since it is appropriate for the actual program. nonetheless, it really is acceptable to catalog the numerous algebraic fields in a fashion that's obtainable to scholars of electronic sign processing, in hopes of stimulating new purposes to engineering initiatives.

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T. 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).

The complex spectrum V is the Fourier transform of a real-valued vector if and only if V: = Vn-k. This follows by writing n-l V':_k = (Lwi(n-k)Vir i=O = n-l LwikV;. i=O The asserted conjugacy constraint holds if and only if Vi = vi for all i. The conjugacy constraint in fields of characteristic 2 is that the GF(2m)ary spectrum is the Fourier transform of a vector over GF(2) if and only if 44 3. Sequences and Spectra This follows by writing n-l (LWikVi) 2 i=O n-l LW 2ik V; i=O because all other terms of the expansion contain a 2 and so are zero in fields of characteristic 2.

Every element of a finite group has a well-defined order. The order of any element w of a finite group divides the number of elements in the group. To prove this, form the array W 92 *w 93 *w w2 92 *w 2 93 *w 2 w3 92 *w 3 93 *w 3 9m*w 9m *w 2 9m*W 3 92 93 wn - 1 * * wn - 1 wn = 1 92 *w n 93 *w n 9m * wn - 1 9m*W n wn - 1 as follows. Write the powers of w in the first row. In forming the ith row choose any 9i that does not appear in any previous row and place 9i * wj in the ith row and jth column.

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