Code design and analysis for multiple access communications

This thesis explores various coding aspects of multiple access communications, mainly for spread spectrum multiaccess(SSMA) communications and collaborative coding multiaccess(CCMA) communications. Both the SSMA and CCMA techniques permit efficient simultaneous transmission by several users sharing a common channel, without subdivision in time or frequency. The general principle behind these two multiaccess schemes is that one can find sets of signals (codes) which can be combined together to form a composite signal; on reception, the individual signals in the set can each be recovered from the composite signal. For the CCMA scheme, the isolation between users is based on the code structure; for the SSMA scheme, on the other hand, the isolation between users is based on the autocorrelation functions(ACFs) and crosscorrelation functions (CCFs) of the code sequences. It is clear that, in either case, the code design is the key to the system design.For the CCMA system with a multiaccess binary adder channel, a class of superimposed codes is analyzed. It is proved that every constant weight code of weight w and maximal correlation λ corresponds to a subclass of disjunctive codes of order T 3, the out-of-phase ACFs and CCFs of the codes are constant and equal to √L. In addition, all codes of the same length are mutually orthogonal.2. Maximal length sequences (m-sequences) over Gaussian integers, suitable for use with QAM modulation, are considered. Two sub-classes of m-sequences with quasi-perfect periodic autocorrelations are obtained. The CCFs between the decimated m-sequences are studied. By applying a simple operation, it is shown that some m-sequences over rational and Gaussian integers can be transformed into perfect sequences with impulsive ACFs.3. Frank codes and Chu codes have perfect periodic ACFs and optimum periodic CCFs. In addition, it is shown that they also have very favourable nonperiodic ACFs; some new results concerning the behaviour of the nonperiodic ACFs are derived. Further, it is proved that the sets of combinedFrank/Chu codes, which contain a larger number of codes than either of the two constituent sets, also have very good periodic CCFs. Based on Frank codes and Chu codes, two interesting classes of real-valued codes with good correlation properties are defined. It is shown that these codes have periodic complementary properties and good periodic and nonperiodic ACF/CCFs.Finally, a hybrid CCMA/SSMA coding scheme is proposed. This new hybrid coding scheme provides a very flexible and powerful multiple accessing capability and allows simple and efficient decoding. Given an SSMA system with K users and a CCMA system with N users, where at most T users are active at any time, then the hybrid system will have K . N users with at most T.K users active at any time. The hybrid CCMA/SSMA coding scheme is superior to the individual CCMA system or SSMA system in terms of information rate, number of users, decoding complexity and external interference rejection capability

Code design and analysis for multiple access communications

This thesis explores various coding aspects of multiple access communications, mainly for spread spectrum multiaccess(SSMA) communications and collaborative coding multiaccess(CCMA) communications. Both the SSMA and CCMA techniques permit efficient simultaneous transmission by several users sharing a common channel, without subdivision in time or frequency. The general principle behind these two multiaccess schemes is that one can find sets of signals (codes) which can be combined together to form a composite signal; on reception, the individual signals in the set can each be recovered from the composite signal. For the CCMA scheme, the isolation between users is based on the code structure; for the SSMA scheme, on the other hand, the isolation between users is based on the autocorrelation functions(ACFs) and crosscorrelation functions (CCFs) of the code sequences. It is clear that, in either case, the code design is the key to the system design.For the CCMA system with a multiaccess binary adder channel, a class of superimposed codes is analyzed. It is proved that every constant weight code of weight w and maximal correlation λ corresponds to a subclass of disjunctive codes of order T 3, the out-of-phase ACFs and CCFs of the codes are constant and equal to √L. In addition, all codes of the same length are mutually orthogonal.2. Maximal length sequences (m-sequences) over Gaussian integers, suitable for use with QAM modulation, are considered. Two sub-classes of m-sequences with quasi-perfect periodic autocorrelations are obtained. The CCFs between the decimated m-sequences are studied. By applying a simple operation, it is shown that some m-sequences over rational and Gaussian integers can be transformed into perfect sequences with impulsive ACFs.3. Frank codes and Chu codes have perfect periodic ACFs and optimum periodic CCFs. In addition, it is shown that they also have very favourable nonperiodic ACFs; some new results concerning the behaviour of the nonperiodic ACFs are derived. Further, it is proved that the sets of combinedFrank/Chu codes, which contain a larger number of codes than either of the two constituent sets, also have very good periodic CCFs. Based on Frank codes and Chu codes, two interesting classes of real-valued codes with good correlation properties are defined. It is shown that these codes have periodic complementary properties and good periodic and nonperiodic ACF/CCFs.Finally, a hybrid CCMA/SSMA coding scheme is proposed. This new hybrid coding scheme provides a very flexible and powerful multiple accessing capability and allows simple and efficient decoding. Given an SSMA system with K users and a CCMA system with N users, where at most T users are active at any time, then the hybrid system will have K . N users with at most T.K users active at any time. The hybrid CCMA/SSMA coding scheme is superior to the individual CCMA system or SSMA system in terms of information rate, number of users, decoding complexity and external interference rejection capability

Recursion Polynomials of Unfolded Sequences

Watermarking digital media is one of the important chal- lenges for information hiding. Not only the watermark must be resistant to noise and against attempts of modification, legitimate users should not be aware that it is embedded in the media. One of the techniques for watermarking is using an special variant of spread-spectrum tech- nique, called frequency hopping. It requires ensembles of periodic binary sequences with low off-peak autocorrelation and cross-correlation. Un- fortunately, they are quite rare and difficult to find. The small Kasami, Kamaletdinov, and Extended Rational Cycle constructions are versatile, because they can also be converted into Costas-like arrays for frequency hopping. We study the implementation of such ensembles using linear feedback shift registers. This permits an efficient generation of sequences and arrays in real time in FPGAs. Such an implementation requires minimal memory usage and permits dynamic updating of sequences or arrays. The aim of our work was to broaden current knowledge of sets of se- quences with low correlation studying their implementation using linear feedback shift registers. A remarkable feature of these families is their similarities in terms of implementation and it may open new way to characterize sequences with low correlation, making it easier to gener- ate them. It also validates some conjectures made by Moreno and Tirkel about arrays constructed using the method of composition.Supported by Consejería de Universidades e Investigación, Medio Ambiente y Política Social, Gobierno de Cantabria (ref. VP34

Construction of pp-ary Sequence Families of Period (pn−1)/2(p^n-1)/2 and Cross-Correlation of pp-ary m-Sequences and Their Decimated Sequences

학위논문 (박사)-- 서울대학교 대학원 : 전기·컴퓨터공학부, 2015. 2. 노종선.This dissertation includes three main contributions: a construction of a new family of $p$-ary sequences of period $\frac{p^n-1}{2}$ with low correlation, a derivation of the cross-correlation values of decimated $p$-ary m-sequences and their decimations, and an upper bound on the cross-correlation values of ternary m-sequences and their decimations. First, for an odd prime $p = 3 \mod 4$ and an odd integer $n$, a new family of $p$-ary sequences of period $N = \frac{p^n-1}{2}$ with low correlation is proposed. The family is constructed by shifts and additions of two decimated m-sequences with the decimation factors 2 and $d = N-p^{n-1}$. The upper bound on the maximum value of the magnitude of the correlation of the family is shown to be $2\sqrt{N+1/2} = \sqrt{2p^n}$ by using the generalized Kloosterman sums. The family size is four times the period of sequences, $2(p^n-1)$. Second, based on the work by Helleseth \cite{Helleseth1}, the cross-correlation values between two decimated m-sequences by 2 and $4p^{n/2}-2$ are derived, where $p$ is an odd prime and $n = 2m$ is an integer. The cross-correlation is at most 4-valued and their values are $\{\frac{-1\pm p^{n/2}}{2}, \frac{-1+3p^{n/2}}{2}, \frac{-1+5p^{n/2}}{2}\}$. As a result, for $p^m \neq 2 \mod 3$, a new sequence family with the maximum correlation value $\frac{5}{\sqrt{2}} \sqrt{N}$ and the family size $4N$ is obtained, where $N = \frac{p^n-1}{2}$ is the period of sequences in the family. Lastly, the upper bound on the cross-correlation values of ternary m-sequences and their decimations by $d = \frac{3^{4k+2}-3^{2k+1}+2}{4}+3^{2k+1}$ is investigated, where $k$ is an integer and the period of m-sequences is $N = 3^{4k+2}-1$. The magnitude of the cross-correlation is upper bounded by $\frac{1}{2} \cdot 3^{2k+3}+1 = 4.5 \sqrt{N+1}+1$. To show this, the quadratic form technique and Bluher's results \cite{Bluher} are employed. While many previous results using quadratic form technique consider two quadratic forms, four quadratic forms are involved in this case. It is proved that quadratic forms have only even ranks and at most one of four quadratic forms has the lowest rank $4k-2$.Abstract i Contents iii List of Tables vi List of Figures vii 1. Introduction 1 1.1. Background 1 1.2. Overview of Dissertation 9 2. Sequences with Low Correlation 11 2.1. Trace Functions and Sequences 11 2.2. Sequences with Low Autocorrelation 13 2.3. Sequence Families with Low Correlation 17 3. A New Family of p-ary Sequences of Period (p^n−1)/2 with Low Correlation 21 3.1. Introduction 22 3.2. Characters 24 3.3. Gaussian Sums and Kloosterman Sums 26 3.4. Notations 28 3.5. Definition of Sequence Family 29 3.6. Correlation Bound 30 3.7. Size of Sequence Family 35 3.8. An Example 38 3.9. Related Work 40 3.10. Conclusion 41 4. On the Cross-Correlation between Two Decimated p-ary m-Sequences by 2 and 4p^{n/2}−2 44 4.1. Introduction 44 4.2. Decimated Sequences of Period (p^n−1)/2 49 4.3. Correlation Bound 53 4.4. Examples 59 4.5. A New Sequence Family of Period (p^n−1)/2 60 4.6. Discussions 61 4.7. Conclusion 67 5. On the Cross-Correlation of Ternary m-Sequences of Period 3^{4k+2} − 1 with Decimation (3^{4k+2}−3^{2k+1}+2)/4 + 3^{2k+1} 69 5.1. Introduction 69 5.2. Quadratic Forms and Linearized Polynomials 71 5.3. Number of Solutions of x^{p^s+1} − cx + c 78 5.4. Notations 79 5.5. Quadratic Form Expression of the Cross-Correlation Function 80 5.6. Ranks of Quadratic Forms 83 5.7. Upper Bound on the Cross-Correlation Function 89 5.8. Examples 93 5.9. Related Works 94 5.10. Conclusion 94 6. Conclusions 96 Bibliography 98 초록 109Docto

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

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

두 p진 데시메이션 수열 간의 상호상관도

학위논문 (박사)-- 서울대학교 대학원 : 전기·컴퓨터공학부, 2017. 2. 노종선.In this dissertation, the cross-correlation between two differently decimated sequences of a $p$-ary m-sequence is considered. Two main contributions are as follows. First, for an odd prime $p$, $n=2m$, and a $p$-ary m-sequence of period $p^n -1$, the cross-correlation between two decimated sequences by $2$ and $d$ are investigated. Two cases of $d$, $d=\frac{(p^m +1)^2}{2}$ with $p^m \equiv 1 \pmod4$ and $d=\frac{(p^m +1)^2}{p^e +1}$ with odd $m/e$ are considered. The value distribution of the cross-correlation function for each case is completely deterimined. Also, by using these decimated sequences, two new families of $p$-ary sequences of period $\frac{p^n -1}{2}$ with good correlation property are constructed. Second, an upper bound on the magnitude of the cross-correlation function between two decimated sequences of a $p$-ary m-sequence is derived. The two decimation factors are $2$ and $2(p^m +1)$, where $p$ is an odd prime, $n=2m$, and $p^m \equiv 1 \pmod4$. In fact, these two sequences corresponds to the sequences used for the construction of $p$-ary Kasami sequences decimated by $2$. The upper bound is given as $\frac{3}{2}p^m + \frac{1}{2}$. Also, using this result, an upper bound of the cross-correlation magnitude between a $p$-ary m-sequence and its decimated sequence with the decimation factor $d=\frac{(p^m +1)^2}{2}$ is derived.1 Introduction 1 1.1 Background 1 1.2 Overview of This Dissertation 7 2 Preliminaries 9 2.1 Finite Fields 9 2.2 Trace Functions and Sequences 11 2.3 Cross-Correlation Between Two Sequences 13 2.4 Characters and Weils Bound 15 2.5 Trace-Orthogonal Basis 16 2.6 Known Exponential Sums 17 2.7 Cross-Correlation of $p$-ary Kasami Sequence Family 18 2.8 Previous Results on the Cross-Correlation for Decimations with $\gcd(p^n -1, d)=\frac{p^{n/2}+1}{2}$ 20 2.9 Cross-Correlation Between Two Decimated Sequences by $2$ and $d=4$ or $\frac{p^n +1}{2}$ 23 3 New $p$-ary Sequence Families of Period $\frac{p^n -1}{2}$ with Good Correlation Property Using Two Decimated Sequences 26 3.1 Cross-Correlation for the Case of $d=\frac{(p^m +1)^2}{2}$ 27 3.2 Cross-Correlation for the Case of $d=\frac{(p^m +1)^2}{p^e +1}$ 37 3.3 Construction of New Sequence Families 43 4 Upper Bound on the Cross-Correlation Between Two Decimated $p$-ary Sequences 52 4.1 Cross-Correlation Between $s(2t+i)$ and $s(2(p^m +1)t +j)$ 53 4.2 Cross-Correlation Between $s(t)$ and $s(\frac{(p^m +1)^2}{2} t)$ 66 5 Conclusions 69 Bibliography 72 Abstract (In Korean) 80Docto

Chip and Signature Interleaving in DS CDMA Systems

Siirretty Doriast