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 = α − ( 1 ) 0 ≤ x n ≤ 1, 0 ≤ α ≤ 4 x x x + 1 n n n � Bipolar signaling = − 2 1 a x n n � PDF of { a n } 1 = − ≤ ≤ ( ) , 1 1 f a a n n 2 π − 1 a n

Properties of Chaotic Sequences Chaotic sequence (logistic map: α = 4, x 0 = 0.2) pdf of a n 2.5 1 2 0.8 1.5 0.6 f(a n ) x n 1 0.4 0.5 0.2 0 0 0 20 40 60 80 100 -1.2 -0.8 -0.4 0 0.4 0.8 1.2 n a n � Non-binary and non-periodic � Random-like behaviors � Good auto- and cross-correlation � Large number of available spreading sequences for multiple-access applications

System Model � Received Signals ⎧ ω + φ + ⎪ 2 ( ) cos( ) ( ) , P a t t n t H 0 1 = ≤ ≤ ( ) 0 ⎨ r t t T ( ) , ⎪ n t H ⎩ 0 where ∞ = ∑ − − τ ( ) ( ) a t a p t nT T n c c = −∞ n The chip epoch τ T c 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 ⎧ ⎫ ⎡ ⎤ − − 1 1 ⎛ ⎞ Q N ⎪ + ⎪ 2 2 ( 1 ) ∏ ∑ n T P ∫ ⎜ ⎟ ⎢ ⎥ Λ = κ c ω ( ( )) E exp ( ) ( ) cos( ) ⎨ ⎬ r t r t b t t dt ε φ ⎜ ε φ ⎟ 1 , , 0 q ⎢ ⎥ N ⎪ ⎪ ⎝ ⎠ nT ⎣ ⎦ 0 ⎩ = = ⎭ c 0 0 n q

Synchronous Coherent Case � Using Gaussian approximation, we obtain − 1 = + γ δ ( )( 0 . 5 ) − γ ( ) 2 m N N T C Q P N C 0 c c , 1 λ c k FA = ( ) P Q D 2 + γ + γ 2 2 2 1 4 2 C D σ = + γ + γ δ c c ( ) ( 0 . 5 ( 2 ) ) N N T C D 0 c c c , 1 λ k (a) Binary Synchronous Coherent Detector 2 [ ] E a = C ω + φ cos( 0 ) t [ ] E a Antenna 2 1 ( ) Var a = D 2 Matched Filter [ ] E a Low Noise -1 Amplifier 1/N 0 T EXP ∫ 0

Synchronous Noncoherent Case Antenna ( ) 2 1 ω cos( ) Matched Filter ( ) 1/2 0 t Low Noise -1 Amplifier ( ) 2 I 0 ( ) ω sin( ) 0 t COMB Ln( ) EXP P(0, T c ) FILTER The mean and variance of λ is − 1 − γ ( ) Q P N C = + γ δ ( )( 1 ) c m N N T C FA ⇒ = ( ) 0 , 1 λ c c P Q k D 2 + γ + γ 1 2 0 . 5 C D 2 2 2 c c σ = + γ + γ δ ( ) ( 1 ( 2 0 . 5 ) ) N N T C D 0 , 1 λ c c c k

Asynchronous Cases � Assume chip epoch is U[0, T c ) � Coherent case − 1 2 − − τ + τ γ ( ) 2 ( 1 2 2 ) Q P N C c λ τ = FA / ) ( ) ( P Q D + γ 1 4 C c − 1 2 1 − − τ + τ γ ( ) 2 ( 1 2 2 ) Q P N C FA c ⇒ = τ ∫ ( ) P Q d D + γ 1 4 C 0 c � Noncoherent case − 1 2 − − τ + τ γ ( ) ( 1 ) Q P N C c λ τ = FA / ) ( ) P ( Q D + γ 1 2 C c − 1 2 1 − − τ + τ γ ( ) ( 1 ) Q P N C FA c ⇒ = τ ∫ ( ) P Q d D + γ 1 2 C 0 c

Performance Comparison Chaotic vs. Binary PN (Sync) P FA =0.01 and N =1000 0 10 Engergy Coherent-binary Noncoherent-binary Coherent-chaotic Noncoherent-chaotic detection -1 10 -2 10 -30 -25 -20 -15 -10 -5 SNR(chip)

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 ( | ) p r H 1 Λ = ( ( )) I r t Coherent detection ( | ) p H r 0 I ∏ ⎛ ∑ ⎞ N L ⎜ a ( | ( )) ⎟ p r a j , ⎝ I i i ⎠ = = 1 1 i j Λ = ( ( )) r t ∏ N N ( ) L p r , a I n = 1 n ( , | ) p r r H 1 I Q Λ = ( ( )) r t Noncoherent ( , | ) p r r H 0 I Q detection ∏ ∑ ∑ ⎛ ⎞ N L L φ ⎜ a p ( , | ( ), ( )) ⎟ p r r a j p , , ⎝ I i Q i i i ⎠ = = = 1 1 1 i j p Λ = ( ( )) r t ∏ N N N ( , ) L L p r r , , a p I n Q n = 1 n Notice: probability density functions p( • ) is used to select the particles a i ( j ) and φ i ( p ) which are mostly close to the actual amplitude and phase.

ω + φ cos( 0 ) t Antenna Particle-Filtering Based Detector Low Noise Amplifier INITIALIZATION: Particles a j , j = 0, P 0 = 1, P 1 = 1 Synchronous coherent receivers with P FA = 0 . 01 , L a = 50, and N = 1000 + ( 1 ) ∫ j T C jT C 0 10 CALCULATE: Binary seq.: Binary Detection p( r j | a j , H 1 ), p( r j |H 0 ) Logistic seq.: Binary Detection [6] Triangular seq.: Binary Detection [6] Logistic seq.: Particle Filter Triangular seq.: Particle Filter P 1 = P 1 p( r j |H 1 ), Detection probability P 0 = P 0 p( r j |H 0 ) -1 RESAMPLING ( a j + 1 ) 10 j = j +1 YES j < N ? NO -2 10 -30 -25 -20 -15 -10 -5 DECISION P 1 /P 0 SNR γ c (chip)

Asynchronous Detection: Multiple sampling � Obtain multiple observations by multiple sampling at τ n (combat delay uncertainty) ⎛ ⎞ ⎛ ⎞ 1 1 1 1 1 1 L L r r r r r r ⎜ ⎟ ⎜ ⎟ , 1 , 2 , , 1 , 2 , I I I N Q Q Q N ⎜ ⎟ = ⎜ ⎟ = M M O M M M O M R R I Q ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ N N N N N N L L r r r r r r d d d d d d ⎝ ⎠ ⎝ ⎠ , 1 , 1 , , 1 , 1 , I N I I Q Q Q N ⎛ ⎞ 1 1 1 ⎛ ⎞ 1 1 1 L n n n L ⎜ ⎟ n n n , 1 , 2 , ⎜ ⎟ Q Q Q N , 1 , 2 , I I I N ⎜ ⎟ = ⎜ ⎟ M M O M = N M M O M N Q ⎜ ⎟ I ⎜ ⎟ ⎜ ⎟ N N N ⎜ ⎟ L n d n d n d N N N ⎝ ⎠ L n n n , 1 , 1 , d d d Q Q Q N ⎝ ⎠ , 1 , 1 , I N I I

Parallel Detection Algorithm � LRT function ⎧ ⎫ ⎪ n ⎪ ( | ) p r H Λ = 1 = ( ( )) max I , 1 , 2 , , ⎨ ⎬ K r t n N d ⎪ ⎪ n ( | ) ⎩ p H ⎭ n r 0 I ⎧ ⎫ n n ( , | ) ⎪ ⎪ p r r H 1 I Q Λ = = ( ( )) max , 1 , 2 , , ⎨ ⎬ K r t n N d n n ⎪ ( , | ) ⎪ p r r H n ⎩ ⎭ 0 I Q Notice: The row having the minimum delay is automatically selected by probability density functions p( • ) to detect the presence of radio signals.

Numerical Results Asynchronous detectors with various N d , P FA = 0 . 01 , N = 1000 , and parallel detection algorithm. 0 10 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 Detection probability ASYN-NC: Chao. Seq - Bin. Detection -1 10 -2 10 -30 -25 -20 -15 -10 -5 SNR γ c (chip)

Dual-Antenna Approach: Synchronous coherent case Incident Wave ω + φ cos( ) t 0 ∆ = θ θ Antenna cos / d c r 1 ( t ) ψ = π θ λ 2 cos / T d ∫ 0 0 Low Noise Amplifier Decision d ω + φ + ψ cos( 0 ) t r 2 ( t ) T ∫ 0 Signal Model Detection Probability ⎛ ⎞ ⎧ − 1 = ω + φ + − γ ⎪ ( ) 2 ( ) cos( ) ( ) ( ) 2 r t P a t t n t ⎜ Q P N ⎟ 1 0 1 = FA c ⎨ P Q ⎜ ⎟ D ⎪ = ω + φ + ψ + + γ ( ) 2 ( ) cos( ) ( ) 1 4 ⎩ r t P a t t n t ⎝ ⎠ 2 0 2 c

Dual-Antenna Approach: Synchronous noncoherent case Incident Wave θ Antenna r 1 ( t ) T ∫ 0 Low Noise T ∫ Amplifier 0 ω cos( ) 0 t Decision d ω sin( ) 0 t r 2 ( t ) T ∫ 0 T ∫ 0 ⎛ ⎞ − 1 − Ω θ γ ( ) 2 ( ) ⎜ Q P N ⎟ = FA c P Q θ | ⎜ ⎟ D + γ 1 2 ⎝ ⎠ c

Dual-Antenna Approach: Asynchronous case Incident Wave Antenna θ r 1 ( t ) Low Noise Amplifier Decision T d ∫ 0 r 2 ( t ) ⎛ ⎞ − 1 − Ω θ γ ( ) ( ) Q P N ⎜ ⎟ = FA c P Q θ | ⎜ ⎟ D + γ 1 2 ⎝ ⎠ c Ω θ = π θ λ ( ) cos( 2 cos / ) d 0