Algebraic Methods for Signal Processing and Communications by Richard E. Blahut, C.S. Burrus

By Richard E. Blahut, C.S. Burrus

Algorithms for computation are a primary a part of either electronic sign seasoned­ cessing and decoders for error-control codes and the valuable algorithms of the 2 topics percentage many similarities. every one topic makes large 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 should be seen as a digitized model of analog sign approach­ 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 if commonly at the least a part of the matter will obviously lie both within the genuine box or the advanced box simply because that's the place the knowledge originate. In different situations the alternative of box for computations will be as much as the set of rules dressmaker, who frequently chooses the genuine box or the advanced box due to familiarity with it or since it is appropriate for the actual software. nonetheless, it's applicable to catalog the numerous algebraic fields in a manner that's available to scholars of electronic sign processing, in hopes of stimulating new functions to engineering tasks.

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A Universal Eigenvector 47 Proof Let w be an element of order p in the field F. The Fourier transform of x, with components denoted Xk, is p-l X k = "" w XiW ik . i=O Using the fact that XiXj = Xij this becomes p-l Xk = Xk w " " Xik W''k i=O p-l = Xk LXiWi . i=O The last line follows because k and p are relatively prime, so ik modulo p ranges over the same values as i modulo p. Therefore Xk = (}Xk, where (} is the Gaussian sum. jP or FP (as interpreted in the field F), as stated in the following theorem.

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 If GCD(b, n) i= 1, then the cyclic decimation v~ = V«bi» has period smaller than n - the period is n' = nIGCD(b, n).

Thus = 0, ... , s - 1 and V~ may be zero in other components as well, depending on the choice of at. The set of such vectors V(t) is linearly independent because the set of their transforms (Viwit) is linearly independent, so we may choose the d - s arbitrary coefficients at to obtain the d - s - 1 additional zeros: Cjb = ° for j = s, ... ,d - 2. Now C' has spectral zeros at k = jb for j = 0, ... , d - 2. Finally, because GCD(b, n) = 1, the components of the spectrum of C' can be permuted by cyclic decimation, beginning with component Cb and taking every bth component of C' cyclically.

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