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

Filterbank optimization with convex objectives and the optimality of principal component forms

This paper proposes a general framework for the optimization of orthonormal filterbanks (FBs) for given input statistics. This includes as special cases, many previous results on FB optimization for compression. It also solves problems that have not been considered thus far. FB optimization for coding gain maximization (for compression applications) has been well studied before. The optimum FB has been known to satisfy the principal component property, i.e., it minimizes the mean-square error caused by reconstruction after dropping the P weakest (lowest variance) subbands for any P. We point out a much stronger connection between this property and the optimality of the FB. The main result is that a principal component FB (PCFB) is optimum whenever the minimization objective is a concave function of the subband variances produced by the FB. This result has its grounding in majorization and convex function theory and, in particular, explains the optimality of PCFBs for compression. We use the result to show various other optimality properties of PCFBs, especially for noise-suppression applications. Suppose the FB input is a signal corrupted by additive white noise, the desired output is the pure signal, and the subbands of the FB are processed to minimize the output noise. If each subband processor is a zeroth-order Wiener filter for its input, we can show that the expected mean square value of the output noise is a concave function of the subband signal variances. Hence, a PCFB is optimum in the sense of minimizing this mean square error. The above-mentioned concavity of the error and, hence, PCFB optimality, continues to hold even with certain other subband processors such as subband hard thresholds and constant multipliers, although these are not of serious practical interest. We prove that certain extensions of this PCFB optimality result to cases where the input noise is colored, and the FB optimization is over a larger class that includes biorthogonal FBs. We also show that PCFBs do not exist for the classes of DFT and cosine-modulated FBs

Theory of optimal orthonormal subband coders

The theory of the orthogonal transform coder and methods for its optimal design have been known for a long time. We derive a set of necessary and sufficient conditions for the coding-gain optimality of an orthonormal subband coder for given input statistics. We also show how these conditions can be satisfied by the construction of a sequence of optimal compaction filters one at a time. Several theoretical properties of optimal compaction filters and optimal subband coders are then derived, especially pertaining to behavior as the number of subbands increases. Significant theoretical differences between optimum subband coders, transform coders, and predictive coders are summarized. Finally, conditions are presented under which optimal orthonormal subband coders yield as much coding gain as biorthogonal ones for a fixed number of subbands

Sequence Design for Cognitive CDMA Communications under Arbitrary Spectrum Hole Constraint

To support interference-free quasi-synchronous code-division multiple-access (QS-CDMA) communication with low spectral density profile in a cognitive radio (CR) network, it is desirable to design a set of CDMA spreading sequences with zero-correlation zone (ZCZ) property. However, traditional ZCZ sequences (which assume the availability of the entire spectral band) cannot be used because their orthogonality will be destroyed by the spectrum hole constraint in a CR channel. To date, analytical construction of ZCZ CR sequences remains open. Taking advantage of the Kronecker sequence property, a novel family of sequences (called "quasi-ZCZ" CR sequences) which displays zero cross-correlation and near-zero auto-correlation zone property under arbitrary spectrum hole constraint is presented in this paper. Furthermore, a novel algorithm is proposed to jointly optimize the peak-to-average power ratio (PAPR) and the periodic auto-correlations of the proposed quasi-ZCZ CR sequences. Simulations show that they give rise to single-user bit-error-rate performance in CR-CDMA systems which outperform traditional non-contiguous multicarrier CDMA and transform domain communication systems; they also lead to CR-CDMA systems which are more resilient than non-contiguous OFDM systems to spectrum sensing mismatch, due to the wideband spreading.Comment: 13 pages,10 figures,Accepted by IEEE Journal on Selected Areas in Communications (JSAC)--Special Issue:Cognitive Radio Nov, 201

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

Generalized discrete Fourier transform with non-linear phase : theory and design

Constant modulus transforms like discrete Fourier transform (DFT), Walsh transform, and Gold codes have been successfully used over several decades in various engineering applications, including discrete multi-tone (DMT), orthogonal frequency division multiplexing (OFDM) and code division multiple access (CDMA) communications systems. Among these popular transforms, DFT is a linear phase transform and widely used in multicarrier communications due to its performance and fast algorithms. In this thesis, a theoretical framework for Generalized DFT (GDFT) with nonlinear phase exploiting the phase space is developed. It is shown that GDFT offers sizable correlation improvements over DFT, Walsh, and Gold codes. Brute force search algorithm is employed to obtain orthogonal GDFT code sets with improved correlations. Design examples and simulation results on several channel types presented in the thesis show that the proposed GDFT codes, with better auto and cross-correlation properties than DFT, lead to better bit-error-rate performance in all multi-carrier and multi-user communications scenarios investigated. It is also highlighted how known constant modulus code families such as Walsh, Walsh-like and other codes are special solutions of the GDFT framework. In addition to theoretical framework, practical design methods with computationally efficient implementations of GDFT as enhancements to DFT are presented in the thesis. The main advantage of the proposed method is its ability to design a wide selection of constant modulus orthogonal code sets based on the desired performance metrics mimicking the engineering .specs of interest. Orthogonal Frequency Division Multiplexing (OFDM) is a leading candidate to be adopted for high speed 4G wireless communications standards due to its high spectral efficiency, strong resistance to multipath fading and ease of implementation with Fast Fourier Transform (FFT) algorithms. However, the main disadvantage of an OFDM based communications technique is of its high PAPR at the RF stage of a transmitter. PAPR dominates the power (battery) efficiency of the radio transceiver. Among the PAPR reduction methods proposed in the literature, Selected Mapping (SLM) method has been successfully used in OFDM communications. In this thesis, an SLM method employing GDFT with closed form phase functions rather than fixed DFT for PAPR reduction is introduced. The performance improvements of GDFT based SLM PAPR reduction for various OFDM communications scenarios including the WiMAX standard based system are evaluated by simulations. Moreover, an efficient implementation of GDFT based SLM method reducing computational cost of multiple transform operations is forwarded. Performance simulation results show that power efficiency of non-linear RF amplifier in an OFDM system employing proposed method significantly improved

Signals on Networks: Random Asynchronous and Multirate Processing, and Uncertainty Principles

The processing of signals defined on graphs has been of interest for many years, and finds applications in a diverse set of fields such as sensor networks, social and economic networks, and biological networks. In graph signal processing applications, signals are not defined as functions on a uniform time-domain grid but they are defined as vectors indexed by the vertices of a graph, where the underlying graph is assumed to model the irregular signal domain. Although analysis of such networked models is not new (it can be traced back to the consensus problem studied more than four decades ago), such models are studied recently from the view-point of signal processing, in which the analysis is based on the "graph operator" whose eigenvectors serve as a Fourier basis for the graph of interest. With the help of graph Fourier basis, a number of topics from classical signal processing (such as sampling, reconstruction, filtering, etc.) are extended to the case of graphs. The main contribution of this thesis is to provide new directions in the field of graph signal processing and provide further extensions of topics in classical signal processing. The first part of this thesis focuses on a random and asynchronous variant of "graph shift," i.e., localized communication between neighboring nodes. Since the dynamical behavior of randomized asynchronous updates is very different from standard graph shift (i.e., state-space models), this part of the thesis focuses on the convergence and stability behavior of such random asynchronous recursions. Although non-random variants of asynchronous state recursions (possibly with non-linear updates) are well-studied problems with early results dating back to the late 60's, this thesis considers the convergence (and stability) in the statistical mean-squared sense and presents the precise conditions for the stability by drawing parallels with switching systems. It is also shown that systems exhibit unexpected behavior under randomized asynchronicity: an unstable system (in the synchronous world) may be stabilized simply by the use of randomized asynchronicity. Moreover, randomized asynchronicity may result in a lower total computational complexity in certain parameter settings. The thesis presents applications of the random asynchronous model in the context of graph signal processing including an autonomous clustering of network of agents, and a node-asynchronous communication protocol that implements a given rational filter on the graph. The second part of the thesis focuses on extensions of the following topics in classical signal processing to the case of graph: multirate processing and filter banks, discrete uncertainty principles, and energy compaction filters for optimal filter design. The thesis also considers an application to the heat diffusion over networks. Multirate systems and filter banks find many applications in signal processing theory and implementations. Despite the possibility of extending 2-channel filter banks to bipartite graphs, this thesis shows that this relation cannot be generalized to M-channel systems on M-partite graphs. As a result, the extension of classical multirate theory to graphs is nontrivial, and such extensions cannot be obtained without certain mathematical restrictions on the graph. The thesis provides the necessary conditions on the graph such that fundamental building blocks of multirate processing remain valid in the graph domain. In particular, it is shown that when the underlying graph satisfies a condition called M-block cyclic property, classical multirate theory can be extended to the graphs. The uncertainty principle is an essential mathematical concept in science and engineering, and uncertainty principles generally state that a signal cannot have an arbitrarily "short" description in the original basis and in the Fourier basis simultaneously. Based on the fact that graph signal processing proposes two different bases (i.e., vertex and the graph Fourier domains) to represent graph signals, this thesis shows that the total number of nonzero elements of a graph signal and its representation in the graph Fourier domain is lower bounded by a quantity depending on the underlying graph. The thesis also presents the necessary and sufficient condition for the existence of 2-sparse and 3-sparse eigenvectors of a connected graph. When such eigenvectors exist, the uncertainty bound is very low, tight, and independent of the global structure of the graph. The thesis also considers the classical spectral concentration problem. In the context of polynomial graph filters, the problem reduces to the polynomial concentration problem studied more generally by Slepian in the 70's. The thesis studies the asymptotic behavior of the optimal solution in the case of narrow bandwidth. Different examples of graphs are also compared in order to show that the maximum energy compaction and the optimal filter depends heavily on the graph spectrum. In the last part, the thesis considers the estimation of the starting time of a heat diffusion process from its noisy measurements when there is a single point source located on a known vertex of a graph with unknown starting time. In particular, the Cramér-Rao lower bound for the estimation problem is derived, and it is shown that for graphs with higher connectivity the problem has a larger lower bound making the estimation problem more difficult.</p