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Fast Convolution and Filtering
Published in Vijay K. Madisetti, The Digital Signal Processing Handbook, 2017
Ivan W. Selesnick, C. Sidney Burrus
The use of polynomials in representing elements of a digital sequence and in representing the convolution operation has led to the development of a family of algorithms based on the fast polynomial transform [4,16,21]. These algorithms are especially useful for two-dimensional convolution. The CRT for polynomials, which is central to Winograd’s short convolution algorithm, is also conveniently described in polynomial notation. An interesting approach combines the use of the polynomial-based methods with the number theoretic approach to convolution (NTTs), wherein the elements of a sequence are taken to lie in a finite field [9,15]. In [15] the CRT is extended to the case of a ring of polynomials with coefficients from a finite ring of integers. It removes the limitations on both word length and sequence length of NNTs and serves as a link between the two methods (CRT and NNT). The new result so obtained, which specializes to both the NNTs and the CRT for polynomials, has been called the AICE-CRT (the American-Indian-Chinese extension of the CRT). A complex version has also been derived.
Technology, Applications, and Computation
Published in Vassil Dimitrov, Graham Jullien, Roberto Muscedere, Multiple-Base Number System, 2017
Vassil Dimitrov, Graham Jullien, Roberto Muscedere
where z is a complex variable. If the sequence h0, h1, … , hN–1 represents the impulse response of a discrete system, then H(z) is referred to as the transfer function of the system. The z-transform, as with the Fourier transform and the Laplace transform in the continuous domain, has the bidirectional multiplication ⇔ convolution property. The DFT has the same property, but with sample indices over a finite ring, as discussed earlier. Thus, the output from a system transfer function, H(z), in response to an input sequence, X(z), is given by the multiplication of the transfer function with the input sequence transform, since this is the z-transform of the convolution of the input sequence with the impulse response of the system; see Equation (1.14).
Signed ring families and signed posets
Published in Optimization Methods and Software, 2021
Kazutoshi Ando, Satoru Fujishige
The one-to-one correspondence between finite distributive lattices and finite partially ordered sets (posets) is a well-known theorem of G. Birkhoff (see [5,6]). This implies a nice representation of any distributive lattice by its corresponding poset, where the size of the former (distributive lattice) is often exponential in the size of the underlying set of the latter (poset). A lot of engineering and economic applications bring us distributive lattices as a ring family of sets which is closed with respect to the two binary operations of set union and intersection. Typically we have such a ring family of sets as a family of minimizers of a submodular set function (see [11]). When it comes to a ring family of sets, the underlying set is partitioned into subsets (or components) and we have a poset structure on the partition. This decomposition with a poset structure on the set of components plays important and crucial roles in many practical problems related, for example, to the decomposition of a directed graph into strongly connected components, the Dulmage-Mendelsohn decomposition of a bipartite graph [29], etc. This is a set-theoretical variant of the original Birkhoff theorem, to be called the Birkhoff-Iri decomposition, revealing the correspondence between finite ring families and finite posets on partitions of the underlying sets, which was intensively pursued by Masao Iri around 1978, especially concerned with the problem of what is called the principal partition of discrete systems [13,22–25,28].
Fuzzy Linear Codes
Published in Fuzzy Information and Engineering, 2018
S. Atamewoue Tsafack, S. Ndjeya, L. Strüngmann, C. Lele
Since is a finite ring, also is finite. Let , then . Let be the generator polynomial of . Note that is the Hensel lift of order k of some polynomial which divides . The cyclic code is called the lift code of the cyclic code . For more information about Hensel lifting see [5].