Design of sequences with good correlation properties

This thesis is dedicated to exploring sequences with good correlation properties. Periodic sequences with desirable correlation properties have numerous applications in communications. Ideally, one would like to have a set of sequences whose out-of-phase auto-correlation magnitudes and cross-correlation magnitudes are very small, preferably zero. However, theoretical bounds show that the maximum magnitudes of auto-correlation and cross-correlation of a sequence set are mutually constrained, i.e., if a set of sequences possesses good auto-correlation properties, then the cross-correlation properties are not good and vice versa. The design of sequence sets that achieve those theoretical bounds is therefore of great interest. In addition, instead of pursuing the least possible correlation values within an entire period, it is also interesting to investigate families of sequences with ideal correlation in a smaller zone around the origin. Such sequences are referred to as sequences with zero correlation zone or ZCZ sequences, which have been extensively studied due to their applications in 4G LTE and 5G NR systems, as well as quasi-synchronous code-division multiple-access communication systems. Paper I and a part of Paper II aim to construct sequence sets with low correlation within a whole period. Paper I presents a construction of sequence sets that meets the Sarwate bound. The construction builds a connection between generalised Frank sequences and combinatorial objects, circular Florentine arrays. The size of the sequence sets is determined by the existence of circular Florentine arrays of some order. Paper II further connects circular Florentine arrays to a unified construction of perfect polyphase sequences, which include generalised Frank sequences as a special case. The size of a sequence set that meets the Sarwate bound, depends on a divisor of the period of the employed sequences, as well as the existence of circular Florentine arrays. Paper III-VI and a part of Paper II are devoted to ZCZ sequences. Papers II and III propose infinite families of optimal ZCZ sequence sets with respect to some bound, which are used to eliminate interference within a single cell in a cellular network. Papers V, VI and a part of Paper II focus on constructions of multiple optimal ZCZ sequence sets with favorable inter-set cross-correlation, which can be used in multi-user communication environments to minimize inter-cell interference. In particular, Paper~II employs circular Florentine arrays and improves the number of the optimal ZCZ sequence sets with optimal inter-set cross-correlation property in some cases.Doktorgradsavhandlin

Filter Bank Fusion Frames

In this paper we characterize and construct novel oversampled filter banks implementing fusion frames. A fusion frame is a sequence of orthogonal projection operators whose sum can be inverted in a numerically stable way. When properly designed, fusion frames can provide redundant encodings of signals which are optimally robust against certain types of noise and erasures. However, up to this point, few implementable constructions of such frames were known; we show how to construct them using oversampled filter banks. In this work, we first provide polyphase domain characterizations of filter bank fusion frames. We then use these characterizations to construct filter bank fusion frame versions of discrete wavelet and Gabor transforms, emphasizing those specific finite impulse response filters whose frequency responses are well-behaved.Comment: keywords: filter banks, frames, tight, fusion, erasures, polyphas

A study of correlation of sequences.

by Wai Ho Mow.Thesis (Ph.D.)--Chinese University of Hong Kong, 1993.Includes bibliographical references (leaves 116-124).Chapter 1 --- Introduction --- p.1Chapter 1.1 --- Spread Spectrum Technique --- p.2Chapter 1.1.1 --- Pulse Compression Radars --- p.3Chapter 1.1.2 --- Spread Spectrum Multiple Access Systems --- p.6Chapter 1.2 --- Definitions and Notations --- p.8Chapter 1.3 --- Organization of this Thesis --- p.12Chapter 2 --- Lower Bounds on Correlation of Sequences --- p.15Chapter 2.1 --- Welch's Lower Bounds and Sarwate's Generalization --- p.16Chapter 2.2 --- A New Construction and Bounds on Odd Correlation --- p.23Chapter 2.3 --- Known Sequence Sets Touching the Correlation Bounds --- p.26Chapter 2.4 --- Remarks on Other Bounds --- p.27Chapter 3 --- Perfect Polyphase Sequences: A Unified Approach --- p.29Chapter 3.1 --- Generalized Bent Functions and Perfect Polyphase Sequences --- p.30Chapter 3.2 --- The General Construction of Chung and Kumar --- p.32Chapter 3.3 --- Classification of Known Constructions ...........； --- p.34Chapter 3.4 --- A Unified Construction --- p.39Chapter 3.5 --- Desired Properties of Sequences --- p.41Chapter 3.6 --- Proof of the Main Theorem --- p.45Chapter 3.7 --- Counting the Number of Perfect Polyphase Sequences --- p.49Chapter 3.8 --- Results of Exhaustive Searches --- p.53Chapter 3.9 --- A New Conjecture and Its Implications --- p.55Chapter 3.10 --- Sets of Perfect Polyphase Sequences --- p.58Chapter 4 --- Aperiodic Autocorrelation of Generalized P3/P4 Codes --- p.61Chapter 4.1 --- Some Famous Polyphase Pulse Compression Codes --- p.62Chapter 4.2 --- Generalized P3/P4 Codes --- p.65Chapter 4.3 --- Asymptotic Peak-to-Side-Peak Ratio --- p.66Chapter 4.4 --- Lower Bounds on Peak-to-Side-Peak Ratio --- p.67Chapter 4.5 --- Even-Odd Transformation and Phase Alphabet --- p.70Chapter 5 --- Upper Bounds on Partial Exponential Sums --- p.77Chapter 5.1 --- Gauss-like Exponential Sums --- p.77Chapter 5.1.1 --- Background --- p.79Chapter 5.1.2 --- Symmetry of gL(m) and hL(m) --- p.80Chapter 5.1.3 --- Characterization on the First Quarter of gL(m) --- p.83Chapter 5.1.4 --- Characterization on the First Quarter of hL(m) --- p.90Chapter 5.1.5 --- Bounds on the Diameters of GL(m) and HL(m) --- p.94Chapter 5.2 --- More General Exponential Sums --- p.98Chapter 5.2.1 --- A Result of van der Corput --- p.99Chapter 6 --- McEliece's Open Problem on Minimax Aperiodic Correlation --- p.102Chapter 6.1 --- Statement of the Problem --- p.102Chapter 6.2 --- A Set of Two Sequences --- p.105Chapter 6.3 --- A Set of K Sequences --- p.110Chapter 7 --- Conclusion --- p.113Bibliography --- p.12

A Direct Construction of Prime-Power-Length Zero-Correlation Zone Sequences for QS-CDMA System

In recent years, zero-correlation zone (ZCZ) sequences are being studied due to their significant applications in quasi-synchronous code division multiple access (QS-CDMA) systems and other wireless communication domains. However, the lengths of most existing ZCZ sequences are limited, and their parameters are not flexible, which are leading to practical limitations in their use in QS-CDMA and other communication systems. The current study proposes a direct construction of ZCZ sequences of prime-power length with flexible parameters by using multivariable functions. In the proposed construction, we first present a multivariable function to generate a vector with specific properties; this is further used to generate another class of multivariable functions to generate the desired $(p^t,(p-1)p^n,p^{n+t+1})$-ZCZ sequence set, where $p$ is a prime number, $t,n$ are positive integers, and $t\leq n$. The constructed ZCZ sequence set is optimal for the binary case and asymptotically optimal for the non-binary case by the \emph{Tang-Fan-Matsufuji} bound. Moreover, a relation between the second-order cosets of first-order generalized Reed-Muller code and the proposed ZCZ sequences is also established. The proposed construction of ZCZ sequences is compared with existing constructions, and it is observed that the parameters of this ZCZ sequence set are a generalization of that of in some existing works. Finally, the performance of the proposed ZCZ-based QS-CDMA system is compared with the Walsh-Hadamard and Gold code-based QS-CDMA system