Applications of Complete Complementary Codes and Propositions for ...

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ScienceDirect Procedia Computer Science 83 (2016) 592 – 599

The 7th International Conference on Ambient Systems, Networks and Technologies (ANT 2016)

Applications of Complete Complementary Codes and Propositions for Future Research Areas of These Codes Monika Dávidekováa,b*, Michal Greguš ml. c, Peter Farkaša, Martin Rákusa a

Institute of Telecommunication, Faculty of Electrical Engineering and Inforamtion Technology, Ilkovičova 3, Bratislava , 812 19, Slovakia b Vysoká škola manažmentu, Panónska cesta 17, Bratislava, 851 01, Slovakia c Department of Information Systems, Comenius University, Odbojárov 10, Bratislava, 820 05, Slovakia

Abstract Complete complementary codes represent special group of codes with unique properties, which were not detected for any other codes. These codes found a wide application in several science areas with the broadest application possibilities in telecommunications. This review paper analyses the applications of these codes proposed so far and aims to propose not sufficiently explored and new areas for further research endeavors. © 2016 2016The TheAuthors. Authors.Published Published Elsevier © by by Elsevier B.V.B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Conference Program Chairs. Peer-review under responsibility of the Conference Program Chairs Keywords: Telecommunications; complete complementary codes (CCC); mobile wireless networks; coding theory;

1. Introduction The theory of Complete Complementary Codes (CCC) represents a vivid area of research. These codes attracts interest of many researchers thanks to their unique properties, which are not detected for another code groups so far. These codes possesses ideal cross- and auto-correlation properties that allow concurrent interference free transmission of multiple users and recognition of particular user by the use of matched filter on receiving end in Code Division Multiplex Access (CDMA) systems. These codes have found broad application in several technical areas, however, the broadest application has been proposed in the field of telecommunications.

* Corresponding author. Tel.: +4212 68279 402. E-mail address: [email protected]

1877-0509 © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Conference Program Chairs doi:10.1016/j.procs.2016.04.279

Monika Dávideková et al. / Procedia Computer Science 83 (2016) 592 – 599

In this paper, several application areas of CCCs are summarized. Next, the development of the theory of complete complementary codes is briefly described with a discussion of the impact of each milestone followed by the summarization of the most important construction milestones in field of CDMA systems. Finally, the conclusion discusses the research implications and proposes future areas of research for these codes. 2. Application Fields Extensive ongoing research of complementary and complete complementary codes contributed to the wide application of theoretical knowledge gained from research and studies in different disciplines. Today it is possible to find the application of these codes in different areas of high-end technologies and diverse disciplines: - physics - Ising spin model1-2; - mathematics: combinatorics - orthogonal model1-2, Hadamard matrices3-4; and the widest application was found in information and communication technology (ICT) including telecommunication digital systems and networks. Complete complementary codes found their application in following digital processing: - model surface acoustic waves5, - navigation system LORAN C6, - radar7-13, - optical transmission14, - measuring characteristics of the channel measurement and analysis of optical routes (OTDR)15-17, - image processing and transmission18-21, - watermarking22-28, - fingerprinting29, - steganography30-31, - video transmission and processing32-33, - eHealth34, - data hiding method to an evacuation guiding system35, - disaster prevention broadcasting36, The application of these codes in modern telecommunication systems and architecture: - signal processing37, - synchronization38-39, - cell interference elimination40, - multiterminal source41, - network spread coding42-43, - body area network44, - modern communication wireless architectures such as spread spectrum systems (SSS), orthogonal frequency division multiplexing (OFDM) and CDMA and others. The ideal autocorrelation property allows the use of a matched filter on the receiving end for optimal and rapid recognition of the respective set. The ideal characteristic of mutual correlation function allows simultaneous transmission of multiple users using different sets of the same CCC without mutual interference (ICI)45-46 . Full utilization efficiency of spectrum in the Direct Spread Spectrum (DS-CDMA) is possible assuming ideal correlation properties. In the DS-CDMA system using the CCC each user is allocated to at least one unique set, comprising different sequences. Each sequence has to be transferred through separate channel, selected by the matched filter and sequences are re-combined into one set at the receiving end. Individual channels in the case one-dimensional completely complementary codes (1D-CCC) can be distributed in time47 and the frequency48 space. Channels can be separated in frequency or with the time. The second case was also analyzed by SM Tseng and MR Bell49, while in the second case the numerical results can be found in works of C Hsiao-Hwa et al.50 and T Kojima et al.51. The use of CCCs for CDMA systems is currently also studied in Multicarrier Systems (MC-CDMA) and in systems with multiple input and multiple output (MIMO). In view of the extensive use of the knowledge of the theory of the CCC, a continuous research and continued studies in this field may be assumed.

593

594


3. Milestones in the Development of the Theory of Complete Complementary Codes First traces of the theory of complete complementary codes can be tracked back to the late forties of the twentieth century when the term "complementary sequences" (CS) first appeared in the work of MJE Golay52. Later, in 1961, he analyzed pairs of binary sequences in his further work53, where the term “complementary sequences” was defined. Researchers (MJE Golay53, R Turyn54) in their works studied pairs of binary sequences and discovered couples with unique correlation properties. Their aperiodic autocorrelation functions (ACF) are equal to zero for all shifts except the zero shift (fig. 1). This allows a very fast recognition of the transmission of a particular user with a simple use of a matched filter on the receiving end. Since then, these sequences attracted attention of several researchers55-57. An example of a complementary pair with ideal auto-correlation properties is depicted in fig. 1. As it can be seen, the aperiodic ACF is not ideal if it is counted per sequence. However, it is possible to achieve overall ideal autocorrelation if complementary sequences are collected in one set as shown in fig. 1.

1.sequence +

Shift +

-1 0 1

ACF

2.sequence

+

+ +

+

+ +

1 2 1

+

+

ACF

Overall ACF

-

+

+

-1 2 -1

-

0 4 0

Fig. 1.aperiodic auto-correlation function (ACF) of complete complementary sequences.

Investigation of complementary pairs and their relation to other types of sequences complemented the idea of binary complementary pairs into binary complementary sequence sets (CSS). Thus a set of sequences with ideal autocorrelation properties, in other words, the value of aperiodic autocorrelation function reaches zero for all possible shifts except the zero shift. In a set, two and more sequences are present. Fig. 1 shows a CSS consisting of two sequences for demonstrational purposes. A CCC consisting of more than two sequences allows a concurrent transmission of more than just two users with the very fast recognition of the transmission of a particular user with a simple use of a matched filter on the receiving end. Later, researchers (R Sivaswamy58, RL Frank59) dedicated their works to multi-phase complementary sequences and extended the theory of CSS codes in the complex area. This allowed an even higher amount of concurrent transmissions with the use of complex sequences maintaining the advantageous attribute of the very fast recognition of the transmission of a particular user with a simple use of a matched filter on the receiving end.

1.sequence

User 1 User 2 Shift -1 0 1

+ +

+ +

XCF

2.sequence

-

+ +

+

+

1 0 -1

+

+

XCF

Overall ACF

+

+

-

-1 0 1

0 0 0

Fig. 2.aperiodic cross-correlation function (XCF) of complete complementary sequences.

A significant contribution to the theory was the expansion of the concept of complementarity to full complementarity, in other words, the introduction of codes with ideal overall aperiodic cross-correlation properties (fig. 2) in addition to their ideal overall aperiodic auto-correlation properties which was introduced by N Suehiro et

595


al.60 who also introduced the first published generating algorithm of Complete Complementary Codes (CCC) called one-dimensional CCC (1D-CCC) today and where the length of the sequences L in a sequence set is given by E2 (E is the number of sequences in one set). This allows an interference free transmission of concurrently communicating users based on ideal overall cross-correlation properties. It also allows to transmit a superposition of these signals with a pure chip shift and allows a much faster communication transmission as was conventional before (fig. 3). In traditional transmissions, it is necessary to wait until the whole spread sequence (all chips) has been transmitted to begin with the transmission of the spread sequence of the next user (fig. 3 a). CCCs allows to begin with the transmission of the next user after first chip of the spread sequence of the previous user has been transmitted (fig. 3 b). This reduces the transmission time extremely (fig. 3). The transmissions will not interfere thanks to the overall cross-correlation properties of CCCs. The auto-correlation properties of these codes allow the very fast recognition of the transmission of a particular user with a simple use of a matched filter on the receiving end. The introduction of a generating algorithm allowed to construct these codes instead of searching for possible sequences by brute force that represented a huge burden. The use of these codes in proposed MC-CDMA architecture61 was closely analyzed in several simulations and further studies62-64. The terminology used by C Hsiao-Hwa et al.61 will be used further: signature denotes a set of sequences, element is a sequence in a signature and the collection of all signatures is the CCC.

Fig. 3. (a) traditional transmission; (b) transmission with applied CCCs. Table 1. Overview of the parameters of individual CCCs generated by processed construction algorithms Source

Maximal number of signatures

Number of elements in each signature

Shortest element length

198860

ܰൌ‫ܧ‬

E=N

‫ ܮ‬ൌ ‫ܧ‬ଶ

63

ܰൌ‫ܧ‬

E=N

‫ ܮ‬ൌ ܰ‫ܧ‬

64

ܰൌ‫ܧ‬

E=N

‫ ܮ‬ൌ ܰ‫ܧ‬Ȁܲ

200567

ܰൌ‫ܧ‬

E=N

ʹ‫ܧ‬

68

ܰൌ‫ܧ‬

E=N

2004

2004

2011 2003

69

2013

70

ܰൌ

ଶ ‫ܧ‬ଵ஽

ܰଶ஽ ൌ

ଶ ‫ܧ‬ଵ஽

‫ܧ‬ଶ஽ ൌ ܰଶ஽ ൌ

ଶ ‫ܧ‬ଵ஽

‫ܧ‬ଶ஽ ൌ ܰଶ஽ ൌ

ଶ ‫ܧ‬ଵ஽

‫ܧ‬ ൌ

ଶ ܰଵ஽

ଶ ‫ܮ‬ଶ஽ ǣ ‫ܧ‬ଵ஽

ൌ

ଶ ܰଵ஽

ଶ ൈ ‫ܧ‬ଵ஽

‫ܮ‬ଶ஽ ǣ ‫ܧ‬ଵ஽ ൈ ‫ܧ‬ଵ஽

where ‫ܮ‬ǡ ܰǡ ‫ܧ‬ǡ ܲ ‫ א‬Ժା are arbitrary numbers that are power of two 2 fulfilling ‫ ܧ‬൒ ܰǡ ܲ and ܲ is the common divisor of N and E, indexes ଵ஽ and ଶ஽ describe the dependency among parameters of 2D- and 1D-CCCs, where the parameters of 2D-CCCs directly depend on the parameters of underlying input 1D-CCCs used for their the generating of 2D-CCCs.

The length limitation (L=E2 introduced by N Suehiro and M Hatori60) as well as the limiting maximal number of signatures N represented the major limitation of the spectral efficiency of proposed CDMA system with applied CCCs. To overcome these drawbacks further research intensified in last 16 years and lead to the achievement of great results in recent years: - new complex CCCs65, - several methods of generating 1D-CCC with various element lengths66-68, - introduction of two-, three- and multi-dimensional CCCs69-70, 71, 72,

596


- introduction of quasi orthogonal CCCs (QOCCC)73-75, - introduction of zero cross-correlation zone codes (ZCCZ)76-77, - several proposals of CDMA architectures and performance evaluations78-83, - verification of simultaneous use of two different code families84-89, - new analysis of Reed-Mueller Codes in connection to CCCs90-91, - several others. From all these research findings, some proposed generating algorithms represent milestones in the development of these codes, namely the generating algorithms that construct shorter element length. The parameters of these constructions are listed in the tab.1 with a reference. 4. Future Research These codes allows an interference free communication for multiple devices thanks to the ideal overall crosscorrelation properties. The recognition of the sender is very fast by using a simple matched filter on receiving end and the overall ideal auto-correlation property. Thanks to these two unique attributes, these codes have found and will find a broad application in several areas of technology. Their application in the eHealth systems in the body area networks 34 and 44 is very promising as well as the cell division of mobile network. The use in wireless home networks seems also as a future field of these codes. In all these areas, the amount of communicating devices is limited and relatively small if compared with the number of user transmission through one sender in a cell. Therefore, the use of the ideal codes seems very promising. However, in the telecommunication systems and network architectures, the number of concurrently communicating users is huge if compared to a home WLAN or a body area network. In these core mobile networks the use of quasi orthogonal CCCs, zero correlation zone codes and other non-ideal codes derived from CCCs seems more promising. These non-ideal variants give a trade-off between the maximal number of concurrently communicating users and the ideality of the correlation properties. The non-ideal variants allow to increase the number of communicating users for a slight loss of some ideality of the overall correlation properties. However, this loss is limited and can be avoided by using appropriate protocols that will not allow forbidden shifts during the transmission. In exchange, the maximal number of users increases enormously and allow to use the great unique advantages of these codes. The area of the possibility of 1D- and 2D- mixed transmissions seems to be insufficiently researched and seems to be promising for combined network architectures. Another area of research seem to be the use of these codes in network coding, image processing and video processing that becomes a great topic for further research trends. Due to the unique properties of these codes it can be assumed that research in these areas will continue to run continuous development and search for new forms and structures as well as the search for new forms of application of these codes. Conclusion This paper contains an overview of the applications of complete complementary codes that were published so far and explains why these codes are attracting researchers´ attention at the same time. It briefly describes the historical development of the theory of these codes and summarizes the attributes of the generating algorithms that represent milestones of the development of the theory. Possibilities of future research are proposed that seem to be becoming trends in near future. Acknowledgements This work was supported by the Department of Information Systems of the Faculty of Management, Comenius University in Bratislava, Slovakia and partially supported by the Scientific Grant Agency of Ministry of Education of Slovak Republic and the Slovak Academy of Sciences under contract VEGA 1/0518/13.


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