By Richard E. Blahut, C.S. Burrus
Algorithms for computation are a important 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 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 now not has to be seen as a digitized model of analog sign procedure ing. Algebraic buildings have gotten extra very important to its improvement. a few of the suggestions of electronic sign processing are legitimate in any algebraic box, even supposing commonly 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 could be as much as the set of rules dressmaker, 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 software. nonetheless, it really 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
Although division is not defined in general, for some a the equation ab = 1 may have a solution b which is called the inverse of a. 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.
By definition, C'(x) has only odd-indexed coefficients nonzero. Therefore g(x)g'(x)C'(x) has all even-indexed coefficients equal to zero. Hence D~ = 0 if s is even, so Vj = 0 if j is a nonzero square. This completes the proof of part i. Proof of ii). p-l X(-l)coo + LX(-l/i)Cl/i i=l p-l X(-l)coo + LX(i)Ci. i=l 56 3. Sequences and Spectra Replacing Ci by the expression of the inverse Fourier transform gives p-1 p-1p-1 LX(i)Ci = ~ LLX(i)[w-ikCk + Col· i=l p i=l k=l Because X( i) equals 1 for (p - 1)/2 values of i and equals -1 for (p - 1) /2 values of i the term in Co is zero.
Then the Gleason-Prange permutation of c satisfies these same two conditions. Proof The proof consists of verifying in turn that each of the two conditions holds. i) Dk = 0 if k is a nonzero square Ldi. () p-l ii) doo = - P i=O Proof of i). Let C be the Fourier transform of c and D be the Fourier transform of d. By definition Ok is zero if k is a nonzero square. To get D from C, first compute the inverse Fourier transform of C to get C; next permute components of c to get d; then take the direct Fourier transform of d to get D.
Algebraic Methods for Signal Processing and Communications Coding by Richard E. Blahut, C.S. Burrus