Chaotic Communication .

1. Chaotic Communication
Ph.D. Course_Broadband Communications
Lect#10
hikmat.abdullah@coie-nahrain.edu.iq

2.  Chaos is aperiodic time-asymptotic behavior in a
deterministic system which exhibits sensitive
dependence on initial conditions
 All systems can be basically divided into three types:
Deterministic, Stochastic or Chaotic

3.  The secret key includes the initial condition 𝒙𝟎 and
the control parameter p within the interval (0; 4]
 The attractor branches into two, then four, then eight
and so on, until chaos emerges for 3.56996 <= p
 If the initial condition is changed, the sequence
always converge to the same cycle period, but with
a different rate.
𝒙𝒏+1 = 𝒑𝒙𝒏 1 − 𝒙𝒏

4. Auto-correlation characteristics Cross-correlation characteristics
Power spectral density characteristics
Possible Chaos oscillator

5. • Easy to generate
• Low power
• Broadband spectrum
• Low multipath interference
• Noise-like appearance
• Good auto and cross correlation properties
• Self-synchronization property
• Chaotic signal can be used for providing security at physical level
• In recent experiments, digital messages were successfully sent at Gbps speeds
over 115 km of commercial optical fiber system using chaotic communication
with a BER of one in ten million. The BER was said to be limited by the
equipment rather than the technique itself

6. FIXED POINT
LIMIT CYCLE
Chaotic attractor is also known as a strange attractor,
a type of attractor (i.e., an attracting set of states) in a
complex dynamical system's phase space that shows
sensitivity to initial conditions
An attractor is informally described as strange if it has
non-integer dimension or if the dynamics on it are chaotic.
The term was coined by David Ruelle and Floris Takens

7. An attractor that is represented by a
particular point in phase space,
sometimes called an equilibrium
point. As a point it corresponds to a
very limited range of possible
behaviors of the system.
A limit cycle is a periodic orbit of
the system that is isolated.

9. - Based on Differential Equations
- Examples of Chaotic Flows are
Lorenz system
Rӧssler system
Chen’s system
Chua system
Lü system

10. Lorenz system

12. Sensitivity to Initial Conditions

13. /Rӧssler/system
Time series and strange attracter of Rӧssler
chaotic flow are shown below when the values
𝑞1 = 0.2, 𝑞2 =0.2 and 𝑞3 = 5.7

14. - Based on Difference Equations
- Examples of Chaotic Maps are
Logistic map
Henon map
Tent map
Baker map
Arnold cat map
Standard map

16. Sensitivity to parameter perturbation

17. Tent Map

18. Arnold Cat Map
A=3

22. If more than one Lyapunov exponents are positive then the system is called
Hyperchaotic

23. 0-1 Test
- One of the powerful testing of chaotic systems is 0-1 test which is presented
by Gottwald and Melbourne .
- The input numbers are the generated key from the system in time domain
and the output from the system is a number between 0 to1.
- The 0-1 test is exceeded the use of Lyapunov exponent in two points which
are:
1) There is no need for phase space reconstruction of the chaotic system. So,
it is applied directly on the generated key from the system.
2) It could be applied on the generated key even the system is continuous,
discrete, exponential data, maps, integer or fractional order system

24. This algorithm of the 0-1 test could be represented as follows:

26.  First generation
1.Additive chaos masking
2.Chaotic shift keying
 Second generation
1.Chaotic parameter modulation
 Third generation
1.Chaotic cryptosystem
 Forth generation
1.impulsive synchronization based cscs

27.  The additive
chaos masking
scheme

28. The chaotic shift keying schemes
Chaos On/Off Keying COOK

29. Chaotic Shift Keying CSK

30. C
B
Binary data
source
(0 & 5) Volts
Rate=100 kbps
Unipolar to
bipolar
converter
Chaotic
generator1
Chaotic
generator2
CSK
modulator
Transmitted
signal
HW
(a) What are the output data at point C, if the input data are 1001
(b) Generate 8 chaotic samples from the two chaotic generators shown assuming the first
generator is logistic map with p=3.8 and the second is cubic map with A=3. Assume
x0=0.2 for both generators.
(c) Find the output of CSK modulator using the generated chaotic samples in (b) and input
data at point C. Assume for each input data bit, two chaotic samples are produced.
(d) What is the chaotic carrier frequency?
(e) Draw a block diagram for the receiver.

31.  Differential Chaotic shift keying (DCSK) using correlation to demodulate
was proposed to solve the problem of synchronization.
 To enhanced the noise performance of DCSK, FM-DCSK scheme was
proposed where frequency modulation is utilized to achieve constant
energy per bit for the chaotic carrier.
Differential Chaotic Shift Keying DCSK

32. T T T T T T
T/2 T/2 T/2 T/2 T/2 T/2 T/2 T/2 T/2 T/2 T/2 T/2
R R R R R R
c(t) x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x
d(t)
+1 -1 +1 +1 -1 +1
DCSK x x x x x x x x x -x-x-x x x x x x x x x x x x x x x x -x-x-x x x x x x x
DCSK-
T/2
x x x x x x x x x -x-x-x x x x x x x x x x x x x x x x -x-x-x x x x
det. d(t) +1 -1 +1 +1 -1 +1

34.  Quadrature Chaos shift keying (QCSK) which can transmit 2 bits in a sample
function was designed to improve the speed of chaos shift keying. The FM-QCSK
is proposed to enhance the noise performance.
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35. HW
Consider the transmitter shown in Fig.1
i. Find the average power at point A.
ii. Show the first 12 generated chaotic samples at point B. Show the output
samples at point C assuming the generated data at point A are 101. Draw a
block diagram for the demodulator identifying the threshold value of the
detector.

36. The chaotic parameter modulation.

37. Initial Condition Modulation ICM

38. chaotic cryptosystem

39. Logistic Map
Permutation Matrix P
for an 8-bit grey scale image A (M×N matrix) is represented as a linear array
of size 1×MN.
each value in P (represents the pixel location of image A) is calculated for an
initial value x0 and parameter μ using : P(i) = int(108X(i)) mod MN +1
Example

40. Computer Experiment
the secret key is
{0.12345678, 3.99995}
Histogram
performance

41. Sensitivity Analysis
the plain image is encrypted
by using the secret keys
{0.12345678, 3.99995}
the secret keys
{0.12345678, 3.99995},
{0.12345678001, 3.99995} and
{0.12345677999, 3.99995}
are used to decrypt the cipher image

42. Chaos based
Stegonography
system

43.  The impulse synchronization

44. Conditional Lyapunov Exponents and the
Pecora-Carroll Chaotic Synchronization
- The necessary and sufficient condition, for master-slave synchronization
to occur, is that the non-driven slave subsystem must be asymptotically
stable
- Lyapunov’s direct method is one of the most powerful tools in the
nonlinear system stability analysis. However,
- In order for the master-slave system to synchronize, all the CLEs of the
non-driven slave subsystem must be negative

45. Pecora-Carroll master-slave system
divided into subsystems

46. Example: Synchronization of the simplest piecewise linear chaotic flow
It exhibits chaotic behaviour with the
parameter value A = 0.6 . Its dynamics are
shown below

48. Let the difference among the non-driving master subsystem and the non-
driven slave subsystem be denoted by ‘*’. When x drives this difference is
given by
Differentiating both sides of equation 3.2.2 it should be noted that
3.2.2
Then:
The conditional Lyapunov exponents are defined as the real parts of the
eigenvalues of the matrix B

49. Let the eigenvalues of matrix B of equation 3.2.5 be denoted by λ1 and λ2. Then
the two CLEs are determined by taking the real parts of the eigenvalues of the
matrix B:
Therefore, as both CLEs are negative, theoretically the master-slave
system of must synchronize. The numerical simulation, confirming the
theoretical result of the equation is shown in next Figures

50. Synchronization of the master-slave simplest piecewise linear chaotic
signals, with the x signal driving.

51. Chaos for Error Correcting Coding
a denotes a position of the top of the skew tent map

52. To transmit 1-bit information, N chaotic signals are generated
The received signals block is given by
The receiver calculates the shortest
distance (D) between the Rx signals and
the chaotic map in the Nd-dimensional
space using Nd successive Rx signals
We decided the symbol
is decoded as “1”
We decided the symbol
is decoded as “0”

54. K = 16, 32, 64.
a = 0.05.
skew tent map
the chaotic sequence length
N = 4 and 8
3 dB improvement over
traditional method!
The short sequences is
more suited for error
correcting codes

55. Chaos for Multiple Access Communications

56. Gram Schmidt Ortho-normalization