By Richard E. Blahut, C.S. Burrus
Algorithms for computation are a principal a part of either electronic sign professional cessing and decoders for error-control codes and the critical algorithms of the 2 topics percentage many similarities. each one topic makes huge use of the discrete Fourier remodel, 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 not should be seen as a digitized model of analog sign technique ing. Algebraic constructions have gotten extra very important to its improvement. some of the thoughts of electronic sign processing are legitimate in any algebraic box, even if commonly a minimum of a part of the matter will evidently lie both within the genuine box or the advanced box simply because that's the place the information originate. In different circumstances the alternative of box for computations will be as much as the set of rules clothier, who often chooses the genuine box or the advanced box due to 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 fashion that's obtainable to scholars of electronic sign processing, in hopes of stimulating new purposes to engineering tasks.
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Extra resources for Algebraic Methods for Signal Processing and Communications Coding
In a finite group, w has order n if n is the least positive integer such that w n = e, where w n = w * w * ... * w with n copies of w on the right. Every element of a finite group has a well-defined order. The order of any element w of a finite group divides the number of elements in the group. To prove this, form the array W 92 *w 93 *w w2 92 *w 2 93 *w 2 w3 92 *w 3 93 *w 3 9m*w 9m *w 2 9m*W 3 92 93 wn - 1 * * wn - 1 wn = 1 92 *w n 93 *w n 9m * wn - 1 9m*W n wn - 1 as follows. Write the powers of w in the first row.
The ith component, for i = 0, ... ,p - 1, is defined as Xi = X(i). We shall call this vector the Legendre vector. Such a vector has only components equal to 0 or ±1, so it can be regarded as a vector over any field F. We shall see in the next theorem that it is an eigenvector of the Fourier transform operator of blocklength p in F (provided a Fourier transform of blocklength p exists in F). 1 In any field F, not of chamcteristic p, the Legendre vector of prime blocklength p is an eigenvector of the Fourier tmnsform opemtor of blocklength p and the corresponding eigenvalue is the Gaussian sum ().
2) (Multiplication Axiom) The set R is closed under multiplication, and multiplication is associative a(bc) = (ab)e. 3) (Joint Axiom) The distributive laws (a + b)e e(a + b) = ae+ be, ca+eb hold for all numbers a, b, and e in the set R. A ring need not have an identity under multiplication but if it does that element is denoted 1, and called one, and the ring is called a ring with identity. The multiplication operation in a ring need not be commutative but if it is, the ring is called a commutative ring.
Algebraic Methods for Signal Processing and Communications Coding by Richard E. Blahut, C.S. Burrus