By Richard E. Blahut, C.S. Burrus

Algorithms for computation are a critical a part of either electronic sign seasoned cessing and decoders for error-control codes and the significant algorithms of the 2 matters proportion many similarities. each one topic makes broad 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 seen as a digitized model of analog sign strategy ing. Algebraic buildings have gotten extra very important to its improvement. the various thoughts of electronic sign processing are legitimate in any algebraic box, even if commonly no less than a part of the matter will certainly lie both within the actual box or the complicated box simply because that's the place the knowledge originate. In different instances the alternative of box for computations will be as much as the set of rules fashion designer, who frequently chooses the genuine box or the advanced box due to familiarity with it or since it is acceptable for the actual software. nonetheless, it's acceptable to catalog the numerous algebraic fields in a fashion that's available to scholars of electronic sign processing, in hopes of stimulating new purposes to engineering projects.

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**Extra resources for Algebraic Methods for Signal Processing and Communications Coding (Signal Processing and Digital Filtering)**

**Example text**

Let B'b' = 1 (mod n'). The cyclic decimation of V by b, denoted V' has an inverse Fourier transform v', with components v~. 3 the vector v' is equal to the cyclic decimation by B' of v, a folded version of v. J ~ Vi'+n'i" i"=O and can be nonzero only if Vi'+n'i" is nonzero for some i". 6 = "(b'it for t = 1, ... , t. 0 A Universal Eigenvector Does there exist a vector whose Fourier transform is proportional to itself? When the blocklength is a prime p, the vector whose components are Legendre symbols has this property.

8. J w''k C;. Ok i=O The Gleason-Prange permutation of c is defined as the vector where di do doo = = X( -l/i)c_l/i' X( -l)coo , Co, i = 1, ... ,p -1, and x( i) is the Legendre symbol. 1 (Gleason-Prange) Let p be a prime. Suppose that over the field F, the vector satisfies i) if k is a nonzero square, L c;, () p-l ii) Coo = -- P i=O where () is the Gaussian sum. Then the Gleason-Prange permutation of c satisfies these same two conditions. Proof The proof consists of verifying in turn that each of the two conditions holds.

217 - 37 1; that is, the 9. GF((2m + 1)2). This field allows a Fourier transform of any blocklength that divides (2m + 1)2 -1 = 2m+1(2m- 1+ 1). We can represent the elements of GF((2 16 + 1)2) in the form a + jb, where a and b are elements of GF(2 16 + 1), and j2 = -j - 1. Because j does not act the same as A does in the complex field, we cannot embed complex integers into GF((216 + 1)2) but we can embed real integers into the real part of GF((216 + 1)2). Therefore Fourier transforms in GF((216 + 1)2) can be used to convolve sequences ofreal integers in the same way that Fourier transforms in the complex field can be used to convolve sequences of real integers.