Secure Chaotic Spread Spectrum Systems Jin Yu Jin Yu WISELAB, ECE - - PowerPoint PPT Presentation

Secure Chaotic Spread Spectrum Systems

Jin Yu Jin Yu

WISELAB, ECE Department WISELAB, ECE Department Stevens Institute of Technology Stevens Institute of Technology Hoboken, NJ 07030 Hoboken, NJ 07030

Outline

Introduction Chaotic SS signals Security/ LPI performance

Intercept receivers

Binary correlating detection “Mismatch” problem Particle-filtering based approach Dual-antenna approach

Numerical results

Conclusions

Introduction

LPI/ LPD-Secure/ covert communications Spread-spectrum systems

Direct sequences

PN binary sequences Chaotic sequences

Frequency hopping Time hopping (UWB)

Interceptors

likelihood-ratio test Energy detector

Chaotic Signals

Generate chaotic spreading sequences

Discrete Chaotic Map

Exponential Map, Triangular Map…

.

For Example: logistic map Bipolar signaling PDF of { an}

) 1 ( 1 n x n x n x − = + α 0 ≤ xn ≤ 1, 0 ≤ α ≤ 4

1 2 − = n x n a

1 1 , 2 1 1 ) ( ≤ ≤ − − = n a n a n a f π

Properties of Chaotic Sequences

Non-binary and non-periodic Random-like behaviors Good auto- and cross-correlation Large number of available spreading sequences for

multiple-access applications

20 40 60 80 100 0.2 0.4 0.6 0.8 1

Chaotic sequence (logistic map: α = 4, x0 = 0.2) n xn

• 1.2
• 0.8
• 0.4

0.4 0.8 1.2 0.5 1 1.5 2 2.5

pdf of an an f(an)

Received Signals

System Model

⎪ ⎩ ⎪ ⎨ ⎧ ≤ ≤ + + = T t H t n H t n t t a P t r , ) ( 1 , ) ( ) cos( ) ( 2 ) ( φ ω

∑ ∞ −∞ = − − = n c T c nT t p n a t a ) ( ) ( τ

where

The chip epoch τTc is modeled by r.v. τ, uniformly distributed in [0, 1).

Binary Correlating Method

Likelihood ratio test (Optimum

Intercept Receivers)

Synchronous coherent Synchronous noncoherent Asynchronous coherent Asynchronous nonherent

Gaussian approximation ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = Λ

∏ ∑ ∫

− = − = + 1 1 ) 1 ( , , 1

) cos( ) ( ) ( 2 2 exp E )) ( (

N n Q q T n nT q

c c

dt t t b t r N P t r ω κ

φ ε φ ε

Synchronous Coherent Case

Using Gaussian approximation, we obtain

Antenna

Low Noise Amplifier

∫

T

1

• 1

Matched Filter EXP

) cos( 0 φ ω + t

1/N0

(a) Binary Synchronous Coherent Detector

) 2 2 4 1 2 ) ( 1 ( c D c C C c N FA P Q Q D P γ γ γ + + − − =

) 1 , ) 2 2 ( 5 . ( 2 ) ( 2 ) 1 , 5 . )( (

λ λ

k c D c C c T N N k C c c T N N m δ γ γ σ δ γ + + = + = ] [ 2 ) 2 ( ] [ ] 2 [ a E a Var D a E a E C = =

Synchronous Noncoherent Case

The mean and variance of λ is

) 1 , ) 2 5 . 2 ( 1 ( 2 ) ( 2 ) 1 , 1 )( ( k c D c C c T N N k C c c T N N m δ γ γ σ δ γ

λ λ

+ + = + = ) 2 5 . 2 1 ) ( 1 ( c D c C C c N FA P Q Q D P γ γ γ + + − − = ⇒

Antenna

Low Noise Amplifier

Ln( ) 1

• 1

Matched Filter

) sin(

0t

ω ) cos(

0t

ω

( )2 ( )2 ( )1/2 I0( ) COMB FILTER EXP P(0, Tc)

Asynchronous Cases

Assume chip epoch is U[0, Tc)

Coherent case Noncoherent case

∫

+ + − − − = ⇒ + + − − − = 1 ) 4 1 ) 2 2 2 1 ( 2 ) ( 1 ( ) 4 1 ) 2 2 1 ( 2 ) ( 1 ( ) /

2 (

τ γ γ τ τ γ γ τ τ τ λ d c C c C N FA P Q Q P c C c C N P Q Q P

D FA D

∫

+ + − − − = ⇒ + + − − − = 1 ) 2 1 ) 2 1 ( ) ( 1 ( ) 2 1 ) 1 ( ) ( 1 ( ) /

2 (

τ γ γ τ τ γ γ τ τ τ λ d c C c C N FA P Q Q P c C c C N P Q Q P

D FA D

Performance Comparison

• 30
• 25
• 20
• 15
• 10
• 5

10

• 2

10

• 1

10 SNR(chip) detection Engergy Coherent-binary Noncoherent-binary Coherent-chaotic Noncoherent-chaotic PFA=0.01 and N =1000

Chaotic vs. Binary PN (Sync)

Particle-Filtering Based Detector

Uncertainties in Chaotic Signals

Amplitude uncertainty (mismatch

problems)

For all detection scenarios with chaotic

signals

Phase uncertainty

Noncoherent detections

Delay uncertainty

Asynchronous detections

Particle-Filtering Based Detector

Design particle sets

approximate the unknown random

variables

select the most likely particle

statistically

combat the impact due to uncertainties

Reduce computational complexity

Updated particles for each iteration Fixed particles for each iteration

Particle-Filtering Based Detector

LRT function with particle filtering

Notice: probability density functions p(•) is used to select the particles ai (j) and φi(p) which are mostly close to the actual amplitude and phase.

Coherent detection Noncoherent detection

∏ ∏ ∑

= = =

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = Λ = Λ

N n n I N a N i L j i i I I I

p L j a p t r H p H p t r

a

1 , 1 1 , 1

) ( )) ( | ( )) ( ( ) | ( ) | ( )) ( ( r r r r

∏ ∏ ∑ ∑

= = = =

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = Λ = Λ

N n n Q n I N p N a N i L j L p i i i Q i I Q I Q I

p L L p j a p t r H p H p t r

a p

1 , , 1 1 1 , , 1

) , ( )) ( ), ( | , ( )) ( ( ) | , ( ) | , ( )) ( ( r r r r r r r r φ

∫

+

C C

T j jT ) 1 (

Antenna

Low Noise Amplifier

) cos( 0 φ ω + t

INITIALIZATION: Particles aj, j = 0, P0 = 1, P1 = 1 CALCULATE: p(rj|aj , H1), p(r j|H0) P1 = P1p(rj|H1), P0 = P0p(rj|H0) RESAMPLING (aj +1) j = j+1 j < N ? YES NO P1/P0 DECISION

• 30
• 25
• 20
• 15
• 10
• 5

10

• 2

10

• 1

10

SNR γc (chip) Detection probability

Binary seq.: Binary Detection Logistic seq.: Binary Detection [6] Triangular seq.: Binary Detection [6] Logistic seq.: Particle Filter Triangular seq.: Particle Filter

Particle-Filtering Based Detector

Synchronous coherent receivers with PFA = 0.01, La = 50, and N = 1000

Asynchronous Detection: Multiple sampling

Obtain multiple observations by multiple

sampling at τn (combat delay uncertainty)

⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ =

d d d

N N I N I N I N I I I I

r r r r r r

, 1 , 1 , 1 , 1 2 , 1 1 ,

M L O M M L R ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ =

d d d

N N Q N Q N Q N Q Q Q Q

r r r r r r

, 1 , 1 , 1 , 1 2 , 1 1 ,

M L O M M L R ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ =

d d d

N N I N I N I N I I I I

n n n n n n

, 1 , 1 , 1 , 1 2 , 1 1 ,

M L O M M L N ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ =

d d d

N N Q N Q N Q N Q Q Q Q

n n n n n n

, 1 , 1 , 1 , 1 2 , 1 1 ,

M L O M M L N

Parallel Detection Algorithm

LRT function Notice: The row having the minimum delay is automatically selected by probability density functions p(•) to detect the presence of radio signals. ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ = = Λ

d n I n I n

N n H p H p t r , , 2 , 1 , ) | ( ) | ( max )) ( (

1

K r r ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ = = Λ

d n Q n I n Q n I n

N n H p H p t r , , 2 , 1 , ) | , ( ) | , ( max )) ( (

1

K r r r r

Numerical Results

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• 25
• 20
• 15
• 10
• 5

10

• 2

10

• 1

10

SNR γc (chip) Detection probability

ASYN-CO: PF, La = 20, Nd = 2 ASYN-CO: PF, La = 10, Nd = 2 ASYN-CO: PF, La = 20, Nd = 1 ASYN-CO: PF, La = 10, Nd = 1 ASYN-NC: PF, La = 10, Lp = 10, Nd = 2 ASYN-NC: PF, La = 10, Lp = 10, Nd = 1 ASYN-NC: Chao. Seq - Bin. Detection

Asynchronous detectors with various Nd, PFA = 0.01, N = 1000, and parallel detection algorithm.

Dual-Antenna Approach: Synchronous coherent case

Signal Model

Antenna

Decision

Low Noise Amplifier

r1(t) r2(t)

d

Incident Wave

) cos( φ ω + t

) cos( 0 ψ φ ω + + t

θ

∫

T

∫

T

/ cos 2 / cos λ θ π ψ θ d c d = = ∆

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − = ⎪ ⎩ ⎪ ⎨ ⎧ + + + = + + =

− c c FA D

N P Q Q P t n t t a P t r t n t t a P t r γ γ ψ φ ω φ ω 4 1 2 ) ( ) ( ) cos( ) ( 2 ) ( ) ( ) cos( ) ( 2 ) (

1 2 2 1 1

Detection Probability

Dual-Antenna Approach: Synchronous noncoherent case

Antenna

Decision

Low Noise Amplifier

r1(t) r2(t)

d

Incident Wave

) cos(

0t

ω ) sin(

0t

ω

θ

∫

T

∫

T

∫

T

∫

T

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + Ω − =

− c c FA D

N P Q Q P γ γ θ

θ

2 1 ) ( 2 ) (

1 |

Dual-Antenna Approach: Asynchronous case

Antenna Low Noise Amplifier

r1(t) r2(t)

d

Incident Wave

θ

∫

T

Decision

) / cos 2 cos( ) ( 2 1 ) ( ) (

1 |

λ θ π θ γ γ θ

θ

d N P Q Q P

c c FA D

= Ω ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + Ω − =

−

Numerical Results

LPI performance of chaotic DS SS signals with N = 1000 and PFA = 0.01 for synchronous coherent detectors.

Numerical Results

LPI performance of chaotic DS SS signals with various d, N = 1000, and chip SNR = -15 dB. MC: Mutual coupling.

Conclusions

The mismatch between chaotic sequences

and binary detection results in the LPI performance improvement;

Particle-filtering based approach can be

used to combat the uncertainties and then improve the detection performance;

Dual antenna approach can also suppress

the uncertainties; however, it is subject to mutual coupling.

Thank You!

Any Questions?