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 valuable algorithms of the 2 matters proportion many similarities. each one topic makes huge use of the discrete Fourier rework, of convolutions, and of algorithms for the inversion of Toeplitz platforms of equations. electronic sign processing is now a longtime topic in its personal correct; it now not has to be considered as a digitized model of analog sign method ing. Algebraic constructions have gotten extra vital to its improvement. the various options of electronic sign processing are legitimate in any algebraic box, even if normally 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 knowledge originate. In different situations the alternative of box for computations will be as much as the set of rules fashion designer, 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's applicable to catalog the numerous algebraic fields in a fashion 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
We shall define the discrete Fourier transform in an arbitrary field. Because we only consider the case where the index set is discrete (and finite), we shall refer to the discrete Fourier transform simply as the Fourier transform. 1 Weight and Complexity Let F be a field and let v = (vo, ... , Vn-l) be a vector of blocklength n of numbers from the field F. We shall be interested in studying the structure and properties of such vectors. The two properties that we define in this section are the weight and the complexity.
4 Rings and Groups A ring is an algebraic system that is weaker than a field in that division is not necessarily defined for all nonzero elements. A group is weaker still in that only a single operation is defined. 24 2. Mathematical Fundamentals A formal definition of a ring is as follows. A ring R is a set that has two arithmetic operations defined on it: addition and multiplication, such that the following properties are satisfied. 1) (Addition Axiom) The set R is closed under addition, and addition is associative and commutative a (a+ b) + c, b+a.
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, using the definition of the Gaussian sum, we have p-1 Ldi i=O But Ck because 1 p-1 p k=l = X(-l)coo + -X(-l) LCkX(k)O. = 0 whenever X(k) -# -1. Therefore Coo = -(O/p)Co. Therefore p-1 Ldi = -X(-l)Oeo = -X(-l)Odoo • i=O Consequently, because 02 X ( -1) as was to be proved.
Algebraic Methods for Signal Processing and Communications Coding by Richard E. Blahut, C.S. Burrus