Asymptotically locally optimal weight vector design for a tighter correlation lower bound of quasi-complementary sequence sets

A quasi-complementary sequence set (QCSS) refers to a set of two-dimensional matrices with low nontrivial aperiodic auto- and cross-correlation sums. For multicarrier code-division multiple-access applications, the availability of large QCSSs with low correlation sums is desirable. The generalized L...

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Main Authors: Liu, Zilong, Guan, Yong Liang, Mow, Wai Ho
Other Authors: School of Electrical and Electronic Engineering
Format: Article
Language:English
Published: 2019
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Online Access:https://hdl.handle.net/10356/105153
http://hdl.handle.net/10220/49529
http://dx.doi.org/10.1109/TSP.2017.2684740
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Institution: Nanyang Technological University
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spelling sg-ntu-dr.10356-1051532019-12-06T21:46:38Z Asymptotically locally optimal weight vector design for a tighter correlation lower bound of quasi-complementary sequence sets Liu, Zilong Guan, Yong Liang Mow, Wai Ho School of Electrical and Electronic Engineering Correlation Multicarrier Code Division Multiple Access Engineering::Electrical and electronic engineering A quasi-complementary sequence set (QCSS) refers to a set of two-dimensional matrices with low nontrivial aperiodic auto- and cross-correlation sums. For multicarrier code-division multiple-access applications, the availability of large QCSSs with low correlation sums is desirable. The generalized Levenshtein bound (GLB) is a lower bound on the maximum aperiodic correlation sum of QCSSs. The bounding expression of GLB is a fractional quadratic function of a weight vector w and is expressed in terms of three additional parameters associated with QCSS: the set size K, the number of channels M, and the sequence length N. It is known that a tighter GLB (compared to the Welch bound) is possible only if the condition M ≥ 2 and K ≥ K̅ + 1, where K̅ is a certain function of M and N, is satisfied. A challenging research problem is to determine if there exists a weight vector that gives rise to a tighter GLB for all (not just some) K ≥ K̅ + 1 and M ≥ 2, especially for large N, i.e., the condition is asymptotically both necessary and sufficient. To achieve this, we analytically optimize the GLB which is (in general) nonconvex as the numerator term is an indefinite quadratic function of the weight vector. Our key idea is to apply the frequency domain decomposition of the circulant matrix (in the numerator term) to convert the nonconvex problem into a convex one. Following this optimization approach, we derive a new weight vector meeting the aforementioned objective and prove that it is a local minimizer of the GLB under certain conditions. NRF (Natl Research Foundation, S’pore) Accepted version 2019-08-05T05:32:03Z 2019-12-06T21:46:38Z 2019-08-05T05:32:03Z 2019-12-06T21:46:38Z 2017 Journal Article Liu, Z., Guan, Y. L., & Mow, W. H. (2017). Asymptotically locally optimal weight vector design for a tighter correlation lower bound of quasi-complementary sequence sets. IEEE Transactions on Signal Processing, 65(12), 3107-3119. doi:10.1109/TSP.2017.2684740 1053-587X https://hdl.handle.net/10356/105153 http://hdl.handle.net/10220/49529 http://dx.doi.org/10.1109/TSP.2017.2684740 en IEEE Transactions on Signal Processing © 2017 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. The published version is available at: https://doi.org/10.1109/TSP.2017.2684740 13 p. application/pdf
institution Nanyang Technological University
building NTU Library
country Singapore
collection DR-NTU
language English
topic Correlation
Multicarrier Code Division Multiple Access
Engineering::Electrical and electronic engineering
spellingShingle Correlation
Multicarrier Code Division Multiple Access
Engineering::Electrical and electronic engineering
Liu, Zilong
Guan, Yong Liang
Mow, Wai Ho
Asymptotically locally optimal weight vector design for a tighter correlation lower bound of quasi-complementary sequence sets
description A quasi-complementary sequence set (QCSS) refers to a set of two-dimensional matrices with low nontrivial aperiodic auto- and cross-correlation sums. For multicarrier code-division multiple-access applications, the availability of large QCSSs with low correlation sums is desirable. The generalized Levenshtein bound (GLB) is a lower bound on the maximum aperiodic correlation sum of QCSSs. The bounding expression of GLB is a fractional quadratic function of a weight vector w and is expressed in terms of three additional parameters associated with QCSS: the set size K, the number of channels M, and the sequence length N. It is known that a tighter GLB (compared to the Welch bound) is possible only if the condition M ≥ 2 and K ≥ K̅ + 1, where K̅ is a certain function of M and N, is satisfied. A challenging research problem is to determine if there exists a weight vector that gives rise to a tighter GLB for all (not just some) K ≥ K̅ + 1 and M ≥ 2, especially for large N, i.e., the condition is asymptotically both necessary and sufficient. To achieve this, we analytically optimize the GLB which is (in general) nonconvex as the numerator term is an indefinite quadratic function of the weight vector. Our key idea is to apply the frequency domain decomposition of the circulant matrix (in the numerator term) to convert the nonconvex problem into a convex one. Following this optimization approach, we derive a new weight vector meeting the aforementioned objective and prove that it is a local minimizer of the GLB under certain conditions.
author2 School of Electrical and Electronic Engineering
author_facet School of Electrical and Electronic Engineering
Liu, Zilong
Guan, Yong Liang
Mow, Wai Ho
format Article
author Liu, Zilong
Guan, Yong Liang
Mow, Wai Ho
author_sort Liu, Zilong
title Asymptotically locally optimal weight vector design for a tighter correlation lower bound of quasi-complementary sequence sets
title_short Asymptotically locally optimal weight vector design for a tighter correlation lower bound of quasi-complementary sequence sets
title_full Asymptotically locally optimal weight vector design for a tighter correlation lower bound of quasi-complementary sequence sets
title_fullStr Asymptotically locally optimal weight vector design for a tighter correlation lower bound of quasi-complementary sequence sets
title_full_unstemmed Asymptotically locally optimal weight vector design for a tighter correlation lower bound of quasi-complementary sequence sets
title_sort asymptotically locally optimal weight vector design for a tighter correlation lower bound of quasi-complementary sequence sets
publishDate 2019
url https://hdl.handle.net/10356/105153
http://hdl.handle.net/10220/49529
http://dx.doi.org/10.1109/TSP.2017.2684740
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