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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 critical algorithms of the 2 topics percentage many similarities. each one topic makes huge 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 has to be seen as a digitized model of analog sign procedure­ ing. Algebraic constructions have gotten extra vital to its improvement. some of the suggestions of electronic sign processing are legitimate in any algebraic box, even though more often than not at the very least a part of the matter will obviously 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 clothier, who frequently chooses the genuine box or the advanced box due to familiarity with it or since it is acceptable for the actual program. nonetheless, it truly is applicable to catalog the various algebraic fields in a manner that's obtainable to scholars of electronic sign processing, in hopes of stimulating new functions to engineering tasks.

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Example text

I=O The components ofv and V are elements of GF(5) and all arithmetic is the arithmetic of GF(5). If 36 3. Sequences and Spectra then the Fourier transform of v is 4. GF(31). In GF(31), the element 2 has order 5. Therefore we have the five-point Fourier transform in GF(31). 4 Vk " ·k Vi, = '~2z k =0, ... ,4. i=O For example, ~ [ 1 [; j 1~ 1~ 1~][ i1 [J1 + 1). An element w of order n exists if and only if n divides 2m + 1 - 1. Hence a Fourier transform of blocklength n exists if and only if n = 2r for some r ~ m.

Proof Let v be a vector of weight d - 1 or less. The cyclic complexity of V is equal to the weight of v. Therefore d-l Vk = - LAjV((k-m· j=l If V has d - 1 consecutive components equal to zero, the recursion implies that V is everywhere zero and that v must be the all-zero vector. 0 Two other bounds generalize the HCR bound. The next theorem says that the consecutive d -1 spectral zeros of the HCR bound can be replaced by a pattern of s uniformly spaced subblocks each of d - s consecutive spectral zeros.

In other fields there need not exist an element of order n for every n. In particular, in the rational field or the real field we can only choose w = -1 and so can only have a Fourier transform of blocklength 2 in those fields. That Fourier transform is too small to be of much use. This is one reason why computational problems that start out in the real or rational field often are embedded into the complex field for processing. The inverse Fourier transform is given next; it appears much the same as the Fourier transform except that w- 1 replaces w, and that n- 1 appears multiplying the sum.

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