On an Improved Chaos Shift Keying On an Improved Chaos Shift Keying Communication SchemeCommunication Scheme

Timothy J. Wren

&

Tai C. Yang

IntroductionIntroduction

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 OverviewPresentation 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 SchemesExisting 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

Quadrature Chaos Shift KeyingQuadrature Chaos Shift Keying

QCSK-QCSK-Chaotic ProcessChaotic Process

],0[)( Tttx

Consider a signal generated by a chaotic process

T

dttxT

0

0)(1

and modified so that is has zero mean; that is

QCSK-QCSK-Fourier ExpansionFourier Expansion

If the signal admits to a Fourier expansion then

Define the average power of this signal as

x(t) fm sin(mt m )m1

T

x dttxT

P0

2 )(1

where T/2

00 f

or

1

2

2

1

mmx fP

QCSK-QCSK-Sinusoid PropertiesSinusoid Properties

Properties of Sinusoidal signals

T

nnmm dttnftmfT 0

)sin()sin(1

)cos(2

1 2 mf

0

nm

nm

QCSK-QCSK-Hilbert TransformHilbert Transform

To derive an orthogonal signal to

2/

then

Apply a Hilbert Transform with a phase shift of

)(tx

)2

sin()(1

m

mm tmfty

T

dttytxT

yx0

0)()(1

yx PP and

QCSK-QCSK-ConstellationsConstellations

Consider two possible maximally separated constellations

(a) (b)

QCSK-QCSK-Encoding MapsEncoding Maps

Encoding maps for the two constellations

ac

bc2/1

2/1

2/1

2/1

2/1

2/1 2/1

2/1

Symbol 0 1 2 3

(a)1 0 -1 0

0 1 0 -1

(b)

QCSK-QCSK-Real Complex MappingReal Complex Mapping

Each symbol can be represented in the complex plane as

jc

On the real time axis this can be represented as

)()()( tytxts

This is the signal sequence for each symbol in the message

QCSK-QCSK-Correlation IntegralsCorrelation Integrals

The encoded values can be recovered by using the two correlation integrals

T

x

dttxtsTP

0

)()(1

T

y

dttytsTP

0

)()(1

Orthogonal Chaos Shift KeyingOrthogonal Chaos Shift Keying

OCSK-OCSK-nn Dimensional Space Dimensional Space

Consider an n dimensional space

• Any point can be represented by an n dimensional vector

• A linear sum of orthonormal basis vectors

nniiiip ...332211

OCSK-OCSK-mm Dimensional Subspace 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

mmaaaa iiiic ...332211

OCSK-OCSK-Real Function MappingReal 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

)(tui ],1[ mi

Orthogonal encoding of our message can now be represented as

)(...)()()()( 332211 tuatuatuatuats mm

This is the message sequence for each symbol in our message

OCSK-OCSK-Vector NotationVector Notation

This can be represented in vector notation as

where

cu )()( tts T

uT (t) [u1(t),u2 (t),u3 (t), ,um (t)]

],...,,[ 21 mT aaac

and

OCSK-OCSK-Correlation IntegralsCorrelation IntegralsThe symbols can be recovered in the receiver using the m correlation integrals

where

T

ii

i dttutsTP

a0

)()(1

T

ii dttuT

P0

2 )(1

],1[ mi

OCSK-OCSK-DecodingDecoding

In vector notation the correlation integrals become

and therefore

T

TT

dtttdttst00

)()()()( cuuu

TTT dttstdttt

0

1

0

)()()()( uuuc

OCSK-OCSK-Orthogonal ProblemOrthogonal 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-OCSK-Singular Value DecompositionSingular Value Decomposition

Consider a chaotic signal sampled at regular intervals and the values placed into a series of m vectors of length n

ix mii ,

These are then arranged into an nxm matrix X

Now consider the Singular Value Decomposition of this matrix

TUWVX

where

mTTT IVVVVUU

OCSK-OCSK-Singular Value DecompositionSingular Value DecompositionThe matrix XTX is symmetric and if the chaotic process is sufficiently varying so that the columns of X are independent

then

1XVWU

idiag W ],1[ mi

i are the eigenvalues of XTX

],,,[ 21 muuuU

OCSK-OCSK-Orthogonal Signal SetOrthogonal Signal Set

Now the columns of U are orthogonal since

mT IUU

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-OCSK-Encoding SchemeEncoding 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 eachsignal matrix X

OCSK-OCSK-Symmetric SolutionSymmetric Solution

So it is possible to transmit

2m mdifferent symbols by

mm

m

m

(m 1)

Samples Samples < < Samples

1

concatenating m samples shifting each sequence left by m samples

OCSK-OCSK-Encoding Parameter InversionEncoding Parameter Inversion

Consider the nxm matrix U generated from matrix X

1XVWU

The eigenvalues of XTX are unique but the eigenvectormatrix V can have inverted eigenvectors

If the symbol encoding map is symmetric inverted encoded parameters are undetectable and decoding will be incorrect

OCSK-OCSK-Non-complementary EncodingNon-complementary Encoding

c3/2

3/22

Symbol 0 1 2 3

(a)

3/2 3/2

3/23/223/22

3/22

Centre of hypersphere offset by m3/1

OCSK-OCSK-Decoding MethodDecoding Method

Consider the ith received encoded signal sequence given by

iii εUPcs

where is unit variance Gaussian white noise soiε

0ε iE variance of noise is 2

ii cPUs ˆˆ

The estimate of the received signal is

indicates a derived variable ^ indicates an estimated one

OCSK-OCSK-Decoding MethodDecoding Method

So

with respect to the coding vector estimateminimize

ei s i ˆ s i

iT

ii ee

i

ˆ c i

T

i

iTi 0

c

ee

ˆ2

OCSK-OCSK-Decoding MethodDecoding Method

Solving these equations gives

PUc

e

i

i

ˆ

iTTTT

i sUPPUUPc1

ˆ

For all m sequences the solution is

SUPPUUPC TTTT 1ˆ

ˆ C ˆ c 1, ˆ c 2 ,..., ˆ c m msssS ,...,, 21and

where

OCSK-OCSK-System ArchitectureSystem Architecture

OCSK-OCSK-Simulation ResultsSimulation Results

Transmitter Generated Chaotic

Reference Signal Set

Received Chaotic Reference with Channel Noise

OCSK-OCSK-Simulation ResultsSimulation Results

Transmitter Generated Orthogonal Signal Set

Receiver GeneratedOrthogonal Signal Set

Showing Signal Inversion

OCSK-OCSK-Overall ResultsOverall Results

OCSK-OCSK-Simulation ResultsSimulation Results

Transmitted 16 Bit Signal

Received Time Delayed

16 Bit Signal

Transmission Rate of

m/2nT

ConclusionsConclusions 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

ReviewReview

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

Completed Doctoral Research Completed Doctoral Research

Reversal Problem Solution SVD Algorithm Characterization BER calculations Dimensional Efficiency

Post Doctoral Research Post Doctoral Research

Hyperchaotic Signal Generation Transmission Efficiencies Real Time Implementation