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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