Chaos- -Based Generation of Optimal Spreading Based Generation of - PowerPoint PPT Presentation

Chaos- -Based Generation of Optimal Spreading Based Generation of Optimal Spreading Chaos Sequences for DS- -UWB Sensor Networks UWB Sensor Networks Sequences for DS Gianluca Setti 12 1 Dep. of Engineering (ENDIF) – University of Ferrara 2 Advanced Research Center on Electronic Systems for Information Engineering and Telecommunications (ARCES) – University of Bologna gianluca.setti@unife.it International Workshop on Complex Systems and Networks (IWCSN2007) Guilin Bravo Hotel, Guilin July 20 th , 2007 University of Bologna University of Ferrara

Acknowledment Collaborators, Students • Michele Balestra , University of Ferrara, Post-Doc • Sergio Callegari , University of Bologna, Assistant Professor • Luca de Michele , University of Bologna, Ph.D. Student • Gianluca Mazzini , University of Ferrara, Associate Professor • Riccardo Rovatti , University of Bologna, Associate Professor • Stefano Santi , University of Bologna, Post-Doc (now with Datalogic s.r.l.) • Stefano Vitali , University of Bologna, Ph.D. Student 2

Outline • Methodology (which is the role of complexity here?) • DS-CDMA System model • Performance in presence of co-channel interference only – Merit figure – Optimal auto-correlation profile • Chaos-Based spreading sequences • Analytical and Experimental Results • Introduction on Wireless Sensor Networks – Impulse Radio communication – System model • Some results for classical UWB pulses • Conclusion 3

Performance improvements by design of stochastic processes performance signals with index in terms tunable Performance of statistical statistical Optimization features of signals features 4

A first example: DS-CDMA performance optimization = ∈ = ( ) 1 y Q x C x M ( x ) M 0.8 + k k k 1 k ( ) Q x Specialization of statistical 0.6 0.4 k 0.2 N ∈ k dynamics theory 0 .2 0 .4 0 .6 .8 0 1 x k ∆ performance signals with Performance Index index in terms Statistical features tunable Performance Chaos-Based in terms of statistical of statistical statistical of chaotic signals DS-CDMA Optimization features features of signals features Ω 1 + Ψ 1 + Ξ 1 ( + 1 ) s T ∫ s s s 1 S s sT 1 1 ) y * ( y k k 2 S ∆ 2 t s 2 Reformulation of classical y k communication theory U S s ∆ U t U y k 5

DS-CDMA: working principle -1 -1 +1 +1 -1 -1 T +1 +1 T/N k s k y k S S y y s S s channel T 1) ∑ sT + s ( de-spreading spreading 6

Asynchronous DS-CDMA Scenario Asynchronous • DS-CDMA: all users transmit on the same Interference band and are distinguished by a code signature (spreading sequence). Useful User Asynchro • Asynchronous model (environement): nous particularly suitable to describe the uplink from Interferen mobile transmitter to a fixed base-station Asynchronous ce • Asynchronous ⇒ channel delays and carrier Interference phases are random 7

DS-CDMA: system model Ω + 1 Ψ 1 + Ξ 1 + T ( 1 ) s s s s ∫ 1 S s sT 1 1 * y ( ) y k k 2 Input of decision S 2 t s block is random 2 y k Multipath MAI U S U t s 1 st receiver channel U y k u ( ) S t • bipolar PAM information signal with period T ; ∞ ∑ = − u u ( ) ( / ) y t y g t sT N • PAM spreading signal with period T/N and symbols / s T N { } =−∞ s ∈ π > u exp( 2 / ) , 2 y x i L complex if L s 8

Asynchronous co-channel interference and performance -I Synchronized useful user (carrier phase offset, timing, time shift compensation) + Ω + Ψ ( 1 ) 1 1 s T ∫ 1 s s S 1 t s sT y t e θ 1 1 ( ) i 1 1 − ω − θ 1 − 1 * ( ) i t ( ) y t t e 0 2 S 2 t Input of decision s block is random 2 t e θ 2 ( ) i y MAI • System performance depends on spreading sequences statistics U S (auto- and cross-correlation) s U t 1 st receiver channel t e θ U U i ( ) y 1 1 1 1 = � BER erfc erfc Standard Gaussian σ − 2 2 2 2 ( 1 ) R U ψ Assumption (Pursley 1982) Expected degradation in Expected Interferce-to-signal ratio Performance by adding a new user per interfering user 9

Classical and maximum performance ⎡ ⎤ ⎛ ⎞ − − ⎡ ⎤ 1 N k 2 4 N = = 2 u ∑ E ( ) 1 A y ⎣ ⎦ = + − 2 2 + ⎢ ( ) ( ) ⎥ ⎜ ⎟ 0 0 R N k A A A − 1 k k k 3 ⎝ ⎠ 3 3 2 ⎣ ⎦ N N = ⎡ ⎤ 1 = k u u E A y y ⎣ ⎦ 0 k k Second-order stationary Sequences auto-correlation sequences Maximal-length, Gold, Kasami, etc., tend to approximate = δ ( ) A k a “white” behavior (Purely Random Sequences) k p p p − 0 m 1 1 2 = R T T T N N N 3 N T T T T k N N N N ′ p ′ p ′ p − 1 0 1 m = ? R Maximum length (m-)sequences min and Gold sequences 10

Interference minimization − − − k N N k large N N a a ≅ = − − a = − k k ( 1 ) ( ) 2 3 A a − k − − N N N k a a − − 2 2 N N large N 3 3 2 a a ≅ = < Minimum R − min + − 2 2 N N 3 2 3 3 N a a N N interference 0.01 At any fixed BER (e.g. 10 -3 ) 0.001 >15% MORE USERS (e.g. 3) 2 = R 0.0001 3 CAN BE ALLOCATED N BER 0.00001 FOR FREE 1. × 10 − 6 3 = R 1. × 10 − 7 3 How can we generate sequences N 1. × 10 − 8 with the optimal autocorrelation profile? 10 12 14 16 18 20 22 24 U What has chaos/complexity to do with it? = σ 2 = 30 0.01 N n 11

Chaos-based generation of spreading sequences Sequences Generation Conceptual Scheme (Off-line) { } 1 = x + = M ( ) x x x M x 1. Chaotic time series generated by + 1 0. 8 k k 1 k k 0. 6 [ ] [ ] → 0. 4 : 0,1 0,1 M where 0. 2 0 .2 0. 4 0 .6 0. 8 1 k ∈ N x k ∆ 2. Repeat a sequence of lenght N extracted from the chaotic time serie, i.e. N -stage shift register = = ± ± … 1, 2, Formally: x x l − …. k lN k / / / T N T N T N + = = … − ( ) 0,1, , 2 x M x k N 1 k k [ ] … Chaos-based 1 , X X 0,1 3. Mapping: L intervals covering ( ) Q x L Spreading ⇒ π ∈ = = 2 ( / ) i j L k ( ) x X y Q x e If k j k k Sequence CB-SS Statistical Properties???? 12

An alternative approach for studying chaos k ∈ N x x = M ( ) x M x + • Chaotic map 1 k k + 1 k k [ ] [ ] [ ] 0 ∈ = � 0 , 1 with and : 0 , 1 0 , 1 x M X ∆ • Evolution of densities • Evolution of points → R + - following single trajectories ρ 0 :[0,1 ] = = � , ( ), ( ), x x M x x M x is difficult 0 1 0 2 1 { } − ≤ ≤ + = ρ Pr / 2 / 2 ( ) x dx x x dx x dx - sensitive dependence on initial conditions 0 0 − = π = π + ' ' ' 4 / 10 , / 10 10 x x 0 0 ρ dx 0 1 0.8 0.6 0.4 1 0 x 0.2 10 20 30 40 50 60 If x 0 is drawn according to ρ 0 , which is the density of x 1 = M ( x 0 ), x 2 = M ( x 1 ),... ? What can we predict with this? 13

Densities instead of trajectories -I = = 0 0 k k = 1 k = 1 k = = − ( ) 4 ( 1 ) x M x x x + k 1 k k k = 2 1 k = 2 k = 3 k = 0 3 k 0 1 1 ρ = ( ) x π − ( 1 ) x x = 4 k = 20 k • The final density does not depend on the initial one 14

Densities instead of trajectories -II = = 0 0 k k = 1 = k 1 k = = − − ( ) 1 2 1 / 2 x M x x + = 1 k k k 2 k 1 = 2 k = 3 k = 3 k 0 0 1 ρ = ( ) 1 x = 4 k = 20 k • Different maps have different densities 15

Densities instead of trajectories -III = 1 k = 1 k = = 0 0 k k = = ( ) 1000 k (mod 1) x M x x + k 1 k ` 1 = 2 k � � � � � � = 2 k 1000 0 0 1 ρ = ( ) 1 x = 20 k ρ • Speed of convergence to depends on the map 16

“Heuristic” remarks • The final density does not depend on the initial one • Different maps have different densities ρ • Speed of convergence to depends Tool for predicting the - on the map evolution of densities - on the initial density is needed... 17

Evolution of densities: Perron-Frobenius operator y = ( x ) M Probability conservation constraint: ρ { } + k 1 dy dy − ≤ ≤ + = Pr y y y y 2 2 { } { } dx dx dx dx − ≤ ≤ + + − ≤ ≤ + Pr Pr x 1 x x 1 x 2 x x 2 2 2 2 2 1 1 1 2 2 2 ρ = ρ + ρ ( ) ( ) ( ) y dy x dx x dx x x x + 1 1 1 2 2 k k k 1 2 ρ ρ ( ) ( ) x x dx dx ρ = + ⇒ ⇒ k 1 k 2 ρ = ρ + ρ ( ) ρ y 1 2 ( ) ( ) ( ) y x x + 1 + k k 1 1 2 k k k ' ( ) ' ( ) M x M x dy dy 1 2 Perron-Frobenius operator P of M ρ = ∑ ( ) x ρ = ρ k ( ) ( ) y P y for maps with “several branches” + 1 k k '( ) M x = ( ) M x y Properties L R ∀ α α ∈ ∀ ∈ , and , f f • P is a (infinite dimensional) linear operator i.e. 1 2 1 2 1 α + α = α + α ( ) P f f P f P f 1 1 2 2 1 1 2 2 = P � f P P f • P has composition properties similar to M 18 M M M M 1 2 1 2