On an Improved Chaos Shift Keying Communication Scheme Timothy J. Wren & Tai C. Yang

Introduction Why are we interested in chaotic communication schemes? Secure communications Spread Spectrum Noise Rejection Advantages of proposed scheme Increased Data Transmission Rates Improved Noise Rejection

Presentation Overview Look at existing schemes Expand on one method Quadrature Chaos Shift Keying (QCSK) Extended the general notion to Orthogonal Chaos Shift Keying (OCSK) Show simulation results Update on further work being undertaken

Existing Schemes Pecora and Carroll synchronization type methods Signal Masking Parameter Variation Chaotic Attractor Synchronization Symmetric Chaos Shift Keying Non Reference Correlation Methods Correlation Delay Shift Keying Reference signal methods Differential Chaos Shift Keying Quadrature Chaos Shift Keying

QCSK- Chaotic Process Consider a signal generated by a chaotic process and modified so that is has zero mean; that is

QCSK- Fourier Expansion If the signal admits to a Fourier expansion then Define the average power of this signal as where or

QCSK- Sinusoid Properties Properties of Sinusoidal signals

QCSK- Hilbert Transform To derive an orthogonal signal to then Apply a Hilbert Transform with a phase shift of and

QCSK- Constellations Consider two possible maximally separated constellations (a)(b)

QCSK- Encoding Maps Encoding maps for the two constellations Symbol0123 (a) (b)

QCSK- Real Complex Mapping Each symbol can be represented in the complex plane as On the real time axis this can be represented as This is the signal sequence for each symbol in the message

QCSK- Correlation Integrals The encoded values can be recovered by using the two correlation integrals

Orthogonal Chaos Shift Keying

OCSK- n Dimensional Space Consider an n dimensional space Any point can be represented by an n dimensional vector A linear sum of orthonormal basis vectors

OCSK- m Dimensional Subspace Now consider a subset of size m of the basis vectors that describes an m dimensional subspace of the n dimensional space Further consider a vector set describing a hypersurface within the m dimensional subspace

OCSK- Real Function Mapping The selected subspace vectors can now be mapped onto the real time axis so that each basis vector represents a set of discrete time values of a real function Orthogonal encoding of our message can now be represented as This is the message sequence for each symbol in our message

OCSK- Vector Notation This can be represented in vector notation as where and

OCSK- Correlation Integrals The symbols can be recovered in the receiver using the m correlation integrals where

OCSK- Decoding In vector notation the correlation integrals become and therefore

OCSK- Orthogonal Problem This is a nice idea It has further advantages in noise rejection, security and data transmission rates that QCSK has shown us are available We need a way of generating a signal set with more than two orthogonal signal sequences But how?

OCSK- Singular Value Decomposition Consider a chaotic signal sampled at regular intervals and the values placed into a series of m vectors of length n These are then arranged into an nxm matrix X Now consider the Singular Value Decomposition of this matrix where

OCSK- Singular Value Decomposition The matrix X T X is symmetric and if the chaotic process is sufficiently varying so that the columns of X are independent then are the eigenvalues of X T X

OCSK- Orthogonal Signal Set Now the columns of U are orthogonal since So each column vector component can be considered as one signal sequence of a set of orthogonal signal sequences Each signal sequence has zero mean Average power of each sequence is 1/n These sequences can now be encoded and transmitted

OCSK- Encoding Scheme Consider the nxm matrix X and the orthonormal matrix U Generate an encoded signal sequence from a combination of the columns of U by using an encoding vector for each symbol Each symbol sequence is n long Symmetric solution is to transmit m sequences for each signal matrix X

OCSK- Symmetric Solution So it is possible to transmitdifferent symbols by Samples << 1 concatenating m samples shifting each sequence left by m samples

OCSK- Encoding Parameter Inversion Consider the nxm matrix U generated from matrix X The eigenvalues of X T X are unique but the eigenvector matrix V can have inverted eigenvectors If the symbol encoding map is symmetric inverted encoded parameters are undetectable and decoding will be incorrect

OCSK- Non-complementary Encoding Symbol0123 (a) Centre of hypersphere offset by

OCSK- Decoding Method Consider the i th received encoded signal sequence given by whereis unit variance Gaussian white noise so variance of noise is The estimate of the received signal is indicates a derived variable ^ indicates an estimated one

OCSK- Decoding Method So with respect to the coding vector estimateminimize

OCSK- Decoding Method Solving these equations gives For all m sequences the solution is and where

OCSK- System Architecture

OCSK- Simulation Results Transmitter Generated Chaotic Reference Signal Set Received Chaotic Reference with Channel Noise

OCSK- Simulation Results Transmitter Generated Orthogonal Signal Set Receiver Generated Orthogonal Signal Set Showing Signal Inversion

OCSK- Overall Results

OCSK- Simulation Results Transmitted 16 Bit Signal Received Time Delayed 16 Bit Signal Transmission Rate of m/2nT

Conclusions In this paper we have proposed a new form of multilevel chaotic communication scheme based on the DCSK schemes Have shown a method of deriving orthogonal signal sequences using the singular valued decomposition of vectors of signals in n space Have demonstrated advantages over QCSK in terms of extensibility, encoding and decoding Have shown improvements in noise rejection and data transmission rates

Review Looked at existing schemes Expanded an idea of QCSK Extended the general notion to OCSK Shown simulation results

Completed Doctoral Research Reversal Problem Solution SVD Algorithm Characterization BER calculations Dimensional Efficiency

Post Doctoral Research Hyperchaotic Signal Generation Transmission Efficiencies Real Time Implementation