By Richard E. Blahut, C.S. Burrus
Algorithms for computation are a critical a part of either electronic sign professional cessing and decoders for error-control codes and the significant algorithms of the 2 topics percentage many similarities. every one topic makes vast 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 method ing. Algebraic constructions have gotten extra very important to its improvement. a number of the thoughts of electronic sign processing are legitimate in any algebraic box, even if generally a minimum of a part of the matter will clearly lie both within the actual box or the advanced box simply because that's the place the information originate. In different circumstances the alternative of box for computations could be as much as the set of rules dressmaker, who frequently chooses the true box or the advanced box as a result of familiarity with it or since it is appropriate for the actual software. 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 purposes to engineering tasks.
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Additional info for Algebraic Methods for Signal Processing and Communications Coding
N' - 1, k" = 0, ... ,n" - 1. n' -1 n"-l , 1 '"'" '"'" -bik' W -n' bik" Vk'k", vi=- ~ ~ W n k'=O k"=O h Vik'k" = Vik'+n'k", Because b = b'n" and wn ,,, n were in w can be replaced by 1. Then , = -n'1 L n'-l Vi -n" b' ik' W k'=O [1nil L ] n"-1 - = 1, the second term Vk'k" . k"=O Next replace k' by B'k' (mod n'). This is a permutation of the indices (0,1, ... ,n' - 1) so it does not affect the sum. Then , = -1 L [1 L n' -1 vi Because b' B' n' n'" k'=O VB'k' kIf . k"=O = 1 (mod n'), this becomes , = -n'1 L n' -1 vi k'=O where 'Y ] n"-l -n"b'B'ik' W [1-n" L n"-l 'Y -ik' = wn ".
Ek = FkGk, Proof This holds because n-l L f(Ci-l)) l=O ei n-l = ~L 1 n-l [n L k=O w-klGk] n-l n-l w-ikGk [L wCi-l)k f((i-l))] k=O l=O =~L w-ikGkFk. 1 CYCLIC DECIMATION If b and n are relatively prime, then i -+ bi (mod n) defines a permutation of the components of a vector v. The permutation v~ = V((bi)) is called cyclic decimation because every bth component, cyclically, is chosen. 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.
Properties of the Fourier Transform The modulation property is verified as follows: The transform of Viwil is 39 v~ = n-l = V£ LWikv~ i=O n-l LWi(kH)Vi = i=O VkH' The convolution theorem is an important and familiar link between the Fourier transform and the cyclic convolution. We shall prove the convolution theorem here to make it quite clear that we are not restricted to any particular field when using it. The cyclic convolution of two vectors f and g, denoted f * g, is defined as n-l ei = L f((i-mg l , i = 0, ...