共查询到20条相似文献,搜索用时 31 毫秒
1.
Frames have become standard tools in signal processing due to their robustness against transmission errors and their resilience to noise. Equiangular tight frames (ETFs) are particularly useful and have been shown to be optimal for transmission under a certain number of erasures. Unfortunately, ETFs do not exist in many cases and are hard to construct when they do exist. However, it is known that an ETF of d + 1 vectors in a d dimensional space always exists. This article gives an explicit construction of ETFs of d + 1 vectors in a d dimensional space. This construction works for both real and complex cases and is simpler than existing methods. The absence of ETFs of arbitrary sizes in a given space leads to generalizations of ETFs. One way to do this to consider tight frames where the set of (acute) angles between pairs of vectors has k distinct values. This article presents a construction of tight frames such that for a given value of k, the angles between pairs of vectors take at most k distinct values. These tight frames can be related to regular graphs and association schemes. 相似文献
2.
提出了广义差集的概念,并且给出了广义差集的一些初等性质.从应用的角度讲,广义差集就是使得其±1特征序列的自相关函数是(最多)三值的一种组合结构.因此,广义差集不仅仅是在概念(理论)上的推广,它还具有深层次的应用背景.事实上,给出了一些广义差集,它不是可分差集,也不是相对差集.同时也给出了一类广义差集存在的一些必要条件,使得这些广义差集对应的±1特征序列成为几乎完美序列.并举例说明本文中的方法是有效的. 相似文献
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4.
Matthew Fickus John Jasper Dustin G. Mixon Jesse D. Peterson Cody E. Watson 《Applied and Computational Harmonic Analysis》2018,44(2):476-496
An equiangular tight frame (ETF) is a set of unit vectors whose coherence achieves the Welch bound, and so is as incoherent as possible. Though they arise in many applications, only a few methods for constructing them are known. Motivated by the connection between real ETFs and graph theory, we introduce the notion of ETFs that are symmetric about their centroid. We then discuss how well-known constructions, such as harmonic ETFs and Steiner ETFs, can have centroidal symmetry. Finally, we establish a new equivalence between centroid-symmetric real ETFs and certain types of strongly regular graphs (SRGs). Together, these results give the first proof of the existence of certain SRGs, as well as the disproofs of the existence of others. 相似文献
5.
Darryn Bryant Judith Egan Barbara Maenhaut Ian M. Wanless 《Designs, Codes and Cryptography》2009,50(1):93-105
In this paper, we present two constructions of divisible difference sets based on skew Hadamard difference sets. A special
class of Hadamard difference sets, which can be derived from a skew Hadamard difference set and a Paley type regular partial
difference set respectively in two groups of orders v
1 and v
2 with |v
1 − v
2| = 2, is contained in these constructions. Some result on inequivalence of skew Hadamard difference sets is also given in
the paper. As a consequence of Delsarte’s theorem, the dual set of skew Hadamard difference set is also a skew Hadamard difference
set in an abelian group. We show that there are seven pairwisely inequivalent skew Hadamard difference sets in the elementary
abelian group of order 35 or 37, and also at least four pairwisely inequivalent skew Hadamard difference sets in the elementary abelian group of order 39. Furthermore, the skew Hadamard difference sets deduced by Ree-Tits slice symplectic spreads are the dual sets of each other
when q ≤ 311.
相似文献
6.
Karlheinz Grö chenig Amos Ron 《Proceedings of the American Mathematical Society》1998,126(4):1101-1107
Based on Ron and Shen's new method for constructing tight wave-let frames, we show that one can construct, for any dilation matrix, and in any spatial dimension, tight wavelet frames generated by compactly supported functions with arbitrarily high smoothness.
7.
In this paper we demonstrate that there are distinct differences between real and complex equiangular tight frames (ETFs) with regards to erasures. For example, we prove that there exist arbitrarily large non-trivial complex equiangular tight frames which are optimal against three erasures, and that such frames come from a unique class of complex ETFs. In addition, we extend certain results in Bodmann and Paulsen (2005) [2] to complex vector spaces as well as show that other results regarding real ETFs are not valid for complex ETFs. 相似文献
8.
External difference families (EDFs) are a type of new combinatorial designs originated from cryptography. In this paper, some
earlier ideas of recursive and cyclotomic constructions of combinatorial designs are extended, and a number of classes of
EDFs and disjoint difference families are presented. A link between a subclass of EDFs and a special type of (almost) difference
sets is set up. 相似文献
9.
This article introduces a new approach to studying difference sets via their additive properties. We introduce the concept
of special subsets, which are interesting combinatorial objects in their own right, but also provide a mechanism for measuring
additive regularity. Skew Hadamard difference sets are given special attention, and the structure of their special subsets
leads to several results on multipliers, including a categorisation of the full multiplier group of an abelian skew Hadamard
difference set. We also count the number of ways to write elements as a product of any number of elements of a skew Hadamard
difference set.
相似文献
10.
The aim of this paper is to investigate the symmetry properties of tight frames,
with a view to constructing tight frames of orthogonal polynomials
in several variables which share the symmetries of the weight function,
and other similar applications.
This is achieved by using representation theory to
give methods for constructing tight frames as orbits of groups of unitary
transformations acting on a given finite-dimensional Hilbert space. Along the way, we show
that a tight frame is determined by its Gram matrix and discuss how the symmetries of a
tight frame are related to its Gram matrix. We also give a complete classification of
those tight frames which arise as orbits of an abelian group of symmetries. 相似文献
11.
Finite tight frames are widely used for many applications. An important problem is to construct finite frames with prescribed norm for each vector in the tight frame. In this paper we provide a fast and simple algorithm for such a purpose. Our algorithm employs the Householder transformations. For a finite tight frame consisting of m vectors in ?n or ?n only O(nm) operations are needed. In addition, we also study the following question: Given a set of vectors in ?n or ?n, how many additional vectors, possibly with constraints, does one need to add in order to obtain a tight frame? 相似文献
12.
Equal-Norm Tight Frames with Erasures 总被引:2,自引:0,他引:2
Equal-norm tight frames have been shown to be useful for robust data transmission. The losses in the network are modeled as erasures of transmitted frame coefficients. We give the first systematic study of the general class of equal-norm tight frames and their properties. We search for efficient constructions of such frames. We show that the only equal-norm tight frames with the group structure and one or two generators are the generalized harmonic frames. Finally, we give a complete classification of frames in terms of their robustness to erasures. 相似文献
13.
该文研究对应于3带尺度函数的小波紧框架,这个小波紧框架是由V_1中的l个函数ψ^1, ψ^2, ψ^n 构成.给出这l个函数构成小波紧框架的充分条件.由此给出由3 带尺度函数构造出一个小波紧框架的显式公式.特别的,如果给定尺度函数的符号是有理函数,则可以构造出符号为有理函数的小波紧框架.最后还给出类似于小波的小波紧框架的分解与重构算法. 相似文献
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15.
16.
Afonso S. Bandeira Matthew Fickus Dustin G. Mixon Percy Wong 《Journal of Fourier Analysis and Applications》2013,19(6):1123-1149
The restricted isometry property (RIP) is a well-known matrix condition that provides state-of-the-art reconstruction guarantees for compressed sensing. While random matrices are known to satisfy this property with high probability, deterministic constructions have found less success. In this paper, we consider various techniques for demonstrating RIP deterministically, some popular and some novel, and we evaluate their performance. In evaluating some techniques, we apply random matrix theory and inadvertently find a simple alternative proof that certain random matrices are RIP. Later, we propose a particular class of matrices as candidates for being RIP, namely, equiangular tight frames (ETFs). Using the known correspondence between real ETFs and strongly regular graphs, we investigate certain combinatorial implications of a real ETF being RIP. Specifically, we give probabilistic intuition for a new bound on the clique number of Paley graphs of prime order, and we conjecture that the corresponding ETFs are RIP in a manner similar to random matrices. 相似文献
17.
Due to its good potential for digital signal processing, discrete Gabor analysis has interested some mathematicians. This
paper addresses Gabor systems on discrete periodic sets, which can model signals to appear periodically but intermittently.
Complete Gabor systems and Gabor frames on discrete periodic sets are characterized; a sufficient and necessary condition
on what periodic sets admit complete Gabor systems is obtained; this condition is also proved to be sufficient and necessary
for the existence of sets E such that the Gabor systems generated by χ
E
are tight frames on these periodic sets; our proof is constructive, and all tight frames of the above form with a special
frame bound can be obtained by our method; periodic sets admitting Gabor Riesz bases are characterized; some examples are
also provided to illustrate the general theory.
This work was supported by National Natural Science Foundation of China (Grant No. 10671008), Beijing Natural Science Foundation
(Grant No. 1092001), PHR (IHLB) and the project sponsored by SRF for ROCS, SEM of China 相似文献
18.
Oversampling and preservation of tightness in affine frames 总被引:1,自引:0,他引:1
The problem of how an oversampling of translations affects the bounds of an affine frame has been proposed by Chui and Shi. In particular, they proved that tightness is preserved if the oversampling factor is coprime with the dilation factor. In this paper we study, in the dyadic dilation case, oversampling of translation by factors which do not satisfy the above condition, and prove that tightness is preserved only in the case of affine frames generated by wavelets having frequency support with very particular properties.
19.
G. Garrigó s E. Herná ndez H. Sikic F. Soria 《Proceedings of the American Mathematical Society》2006,134(11):3211-3221
In a previous paper, the authors introduced new ideas to treat the problem of connectivity of Parseval frames. With these ideas it was shown that a large set of Parseval frames is arcwise connected. In this article we exhibit a larger class of Parseval frames for which the arcwise connectivity is true. This larger class fails to include all Parseval frames.
20.
FAN CuiLing LEI JianGuo & SHAN XiuLing School of Sciences Nantong University Nantong China College of Mathematics Information Science Hebei Normal University Shijiazhuang 《中国科学 数学(英文版)》2011,(1)
Difference systems of sets (DSSs) are combinatorial configurations which were introduced in 1971 by Levenstein for the construction of codes for synchronization. In this paper, we present two kinds of constructions of difference systems of sets by using disjoint difference families and a special type of difference sets, respectively. As a consequence, new infinite classes of optimal DSSs are obtained. 相似文献