### Nº 10 2013 > Academia — Innovating for society

# Chaos theory as the answer to limited spectrum?

## Ken Umeno, Professor at the Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Japan and Minghui Kao, Chairman, ChaosWare, Inc., Japan

In the near future, there will be an uncountable number of sensors and terminal devices, while the number of users demanding wireless connection will increase exponentially. So far, frequency spectrum bandwidth has been limited because of the physical nature of spectrum. All the technologies that use spectrum are confronted by this physical boundary. How can we connect an unlimited number of devices using the limited spectrum bands? This is the ultimate question on spectrum usage. It has been a central theme of communications technology and will be one of the most crucial issues in the coming decades.

**A paradigm shift from periodicity to chaos?**

Today, spread spectrum communication is mainly third generation/Universal Mobile Telecommunication System (3G/UMTS). The ITU Radiocommunication Sector (ITU–R) standards for the evolution of communications technology notably indicate the recent development of high-end digital signal processing technology as well as diverse applications using different frequency spectrum bandwidths. Problems such as power control, energy efficiency and fading effects are currently more worrying than capacity.

The central research target of the Physical Statistics Laboratory at Kyoto University is to solve all these problems by taking a radical approach to information and communication technologies (ICT) using a relatively new area of mathematics and physics, namely chaos theory. This implies a paradigm shift from the periodicity regime that underpins the frequency spectrum to a chaos regime that will give rise to what we may call the chaos spectrum. The idea is to use the chaotic nature of telecommunication signals to develop a new unified communications technology beyond 3G code division multiple access (CDMA) and 4G orthogonal frequency division multiplexing (OFDM).

The very concept of spectrum comes from the frequency of periodic signals that can be arbitrarily used for communications. According to Fourier analysis (the hidden mathematics behind this mechanism), an arbitrary physical signal can be given by sum of sine waves which are orthogonal to each other. We thus sometimes refer to frequency spectrum as Fourier spectrum. Frequency spectrum is the basis of OFDM technology, which is widely used for 4G (IMT-Advanced) and wireless local area networks.

In contrast, chaos is an aperiodic phenomenon which has the property of unpredictable randomness. The existence of chaos in nature was discovered in the 1960s. A more recent discovery — of importance for ICT — is that chaotic signals can be used as the arbitrary signals that represent communications. An arbitrary physical signal can be given by the sum of a series of chaotic signals which are orthogonal to each other. In other words, the nature of the signal for the transmission of information can be equally well represented in the chaos spectrum as in the Fourier (frequency) spectrum.

The chaos spectrum has never been used for telecommunications but the crucial point here is that it is mathematically proven to be infinite. The figure below shows examples of chaos codes for communications.

Fundamental research on the role of chaos in ICT led to the presentation of the first and second laws of informatics to an international conference in 2012. These research results were published in 2013.

The first law of informatics states that secure information is always preserved. Chaotic signals can carry a measurable amount of information, and an identical amount of information can be retrieved from the chaotic signals. In other words, if certain information is converted into chaotic signals for the purposes of transmission, then exactly the same information is received when the chaotic signals are decoded. Chaotic signals in informatics are thus equivalent to a counterpart of reproducible thermal noise in thermodynamics.

The second law of informatics states that information sharing is irreversible. Alice and Bob can share secure information via chaotic modulation. According to the first law of informatics, zero information cannot be converted into the secure shared information that Alice and Bob have. So, what is the cost of sharing such secure information? If the cost is measured in terms of the generation of the chaotic signal, then the following inequality holds: the chaos generation costs for information sharing are greater than or equal to the cost of sharing secure information. Based on this fundamental inequality, it is clear that once Alice and Bob share secure information, they can never be in a situation in the future when they will not share that information. This is an irreversibility effect arising from the nature of information. As is well known, once published, information can never be secure.

**Technological advantages of chaos spectrum**

It is possible to suppress interference using a simple filter at the base station, implemented by chaos theory. This is an improvement over frequency spectrum techniques, which require an intervention by the end user.

Fading is caused by multichannel communications and is unavoidable. In fading channels, however, chaotic code signals are superior to conventional OFDM code signals because the receiver can use independent component analysis to separate the signals in chaotic mixtures.

Thanks to the invention of primitive root codes, we can construct an infinite number of orthogonal chaos codes which can be assigned to be an infinite number of addresses of things. If the Internet of Things means that several hundred billion things will be connected wirelessly, then that kind of super multiplexing technology with a potentially infinite number of orthogonal spreading codes will be needed.

In our view, chaos technology offers a promising future for 5G mobile communications.

**About the authors**

**Ken Umeno** received his BSc degree in electronic communication from Waseda University, Japan, in 1990. He received his MSc and PhD degrees in physics from the University of Tokyo, Japan, in 1992 and 1995, respectively. From 1998 until he joined Kyoto University in 2012, he worked for Japan’s Ministry of Posts and Telecommunications in its Communications Research Laboratory (currently National Institute of Information and Communications Technology). From 2004 to 2012, he was CEO and President of ChaosWare, Inc. He received the LSI IP Award in 2003 and the Telecom-System Award in 2008. He holds 46 registered Japanese patents, 23 registered United States patents and more than 5 international patents in the fields of telecommunications, security, and financial engineering. His research interests include ergodic theory, statistical computing, coding theory, chaos theory, information security, and social systems.

**Minghui Kao** is co-founder and Chairman of ChaosWare, Inc., Japan.

Professor at the Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Japan and Minghui Kao, Chairman, ChaosWare, Inc., Japan