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 topics proportion 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 considered as a digitized model of analog sign strategy ing. Algebraic buildings have gotten extra vital to its improvement. a number of the suggestions of electronic sign processing are legitimate in any algebraic box, even if often not less than a part of the matter will clearly lie both within the actual box or the complicated 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 fashion designer, who frequently chooses the true box or the complicated box as a result of familiarity with it or since it is appropriate for the actual program. nonetheless, it's acceptable to catalog the various 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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Additional resources for Algebraic Methods for Signal Processing and Communications Coding
S - 1. The linear combination V' = ° L d-l-s atV(t) t=o is zero in each component where every for j V(l) equals zero. 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, ...
N -1, where w is a number of order n in the field F. Thus wn = 1 and no smaller power of w equals 1. The Fourier transform looks quite familiar, but the operations of addition and multiplication that are indicated are operations in the field F. ) We shall see that, in any field, all of the familiar 34 3. Sequences and Spectra properties of the Fourier transform are valid because as we look into the proof of these properties we find that only the abstract structure of a field is used; properties of addition and multiplication specific to a particular field are not used.
For example, w = (e- j21r / n )3 works if n is not a multiple of 3. 3. GF(5). In GF(5) the element 2 has order 4 (because 24 = 1 modulo 5). Therefore we have the four-point Fourier transform in GF(5): Vk = 3 ' " ok Vi, ~2' k=O, ... ,3. 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).