634 research outputs found
λ pμ§ λ°μλ©μ΄μ μμ΄ κ°μ μνΈμκ΄λ
νμλ
Όλ¬Έ (λ°μ¬)-- μμΈλνκ΅ λνμ : μ κΈ°Β·μ»΄ν¨ν°κ³΅νλΆ, 2017. 2. λ
Έμ’
μ .In this dissertation, the cross-correlation between two differently decimated sequences of a -ary m-sequence is considered. Two main contributions are as follows.
First, for an odd prime , , and a -ary m-sequence of period , the cross-correlation between two decimated sequences by and are investigated. Two cases of , with and with odd 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 -ary sequences of period with good correlation property are constructed.
Second, an upper bound on the magnitude of the cross-correlation function between two decimated sequences of a -ary m-sequence is derived. The two decimation factors are and , where is an odd prime, , and . In fact, these two sequences corresponds to the sequences used for the construction of -ary Kasami sequences decimated by . The upper bound is given as .
Also, using this result, an upper bound of the cross-correlation magnitude between a -ary m-sequence and its decimated sequence with the decimation factor 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 -ary Kasami Sequence Family 18
2.8 Previous Results on the Cross-Correlation for Decimations with 20
2.9 Cross-Correlation Between Two Decimated Sequences by and or 23
3 New -ary Sequence Families of Period with Good Correlation Property Using Two Decimated Sequences 26
3.1 Cross-Correlation for the Case of 27
3.2 Cross-Correlation for the Case of 37
3.3 Construction of New Sequence Families 43
4 Upper Bound on the Cross-Correlation Between Two Decimated -ary Sequences 52
4.1 Cross-Correlation Between and 53
4.2 Cross-Correlation Between and 66
5 Conclusions 69
Bibliography 72
Abstract (In Korean) 80Docto
Construction of -ary Sequence Families of Period and Cross-Correlation of -ary m-Sequences and Their Decimated Sequences
νμλ
Όλ¬Έ (λ°μ¬)-- μμΈλνκ΅ λνμ : μ κΈ°Β·μ»΄ν¨ν°κ³΅νλΆ, 2015. 2. λ
Έμ’
μ .This dissertation includes three main contributions: a construction of a new family of -ary sequences of period with low correlation, a derivation of the cross-correlation values of decimated -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 and an odd integer , a new family of -ary sequences of period with low correlation is proposed. The family is constructed by shifts and additions of two decimated m-sequences with the decimation factors 2 and . The upper bound on the maximum value of the magnitude of the correlation of the family is shown to be by using the generalized Kloosterman sums. The family size is four times the period of sequences, .
Second, based on the work by Helleseth \cite{Helleseth1}, the cross-correlation values between two decimated m-sequences by 2 and are derived, where is an odd prime and is an integer. The cross-correlation is at most 4-valued and their values are . As a result, for , a new sequence family with the maximum correlation value and the family size is obtained, where 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 is investigated, where is an integer and the period of m-sequences is . The magnitude of the cross-correlation is upper bounded by . 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 .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^nβ1)/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}β2 44
4.1. Introduction 44
4.2. Decimated Sequences of Period (p^nβ1)/2 49
4.3. Correlation Bound 53
4.4. Examples 59
4.5. A New Sequence Family of Period (p^nβ1)/2 60
4.6. Discussions 61
4.7. Conclusion 67
5. On the Cross-Correlation of Ternary m-Sequences of Period 3^{4k+2} β 1 with Decimation (3^{4k+2}β3^{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} β 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
μ΄λ‘ 109Docto
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 Ξ» 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 βL. 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 Ξ» 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 βL. 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
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