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

SLIDE 1

University of Bologna University of Ferrara

Chaos Chaos-

Based Generation of Optimal Spreading

Based Generation of Optimal Spreading Sequences for DS Sequences for DS-

UWB Sensor Networks

UWB Sensor Networks

Gianluca Setti12

1Dep. of Engineering (ENDIF) – University of Ferrara

2Advanced 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 20th, 2007

SLIDE 2

2

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

SLIDE 3

3

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

SLIDE 4

4

Performance improvements by design of stochastic processes

performance index in terms

f statistical

features of signals

Performance Optimization

signals with tunable statistical features

SLIDE 5

5

A first example: DS-CDMA performance optimization

performance index in terms

f statistical

features of signals

Performance Optimization

signals with tunable statistical features

∫

+ T s sT ) 1 ( 1 s

S

2 s

S

U s

S

2

t ∆

U

t ∆

1 k

y

2 k

y

U k

y

* 1 )

(

k

y

1 1 1 s s s

Ξ + Ψ + Ω

Reformulation of classical communication theory

Performance Index in terms of statistical features

Statistical features

f chaotic signals

Specialization of statistical dynamics theory

) (

1 k k

x M x =

+

.2 .4 .6 .8 1 0.2 0.4 0.6 0.8 1

∆

k

x M

N

∈ k C x Q y

k k

∈ = ) (

( )

k

Q x

Chaos-Based DS-CDMA

SLIDE 6

6

DS-CDMA: working principle

+1

1

+1

1

s

S

k

y +1

1

+1

1

s

S

s k

S y

channel

( 1) s T sT +

∑

de-spreading

T

k

y

spreading

T/N

SLIDE 7

7

Asynchronous DS-CDMA Scenario

Asynchronous Interference Useful User

DS-CDMA: all users transmit on the same

band and are distinguished by a code signature (spreading sequence).

Asynchronous model (environement):

particularly suitable to describe the uplink from mobile transmitter to a fixed base-station

Asynchronous ⇒ channel delays and carrier

phases are random

Asynchro nous Interferen ce Asynchronous Interference

SLIDE 8

8

DS-CDMA: system model channel

∫

+ T s sT ) 1 ( 1 s

S

2 s

S

U s

S

2

t

U

t

1 *

( )

k

y

1st receiver

1 1 1 s s s

Ψ Ω + + Ξ

1 k

y

2 k

y

U k

y

Input of decision block is random

MAI

Multipath

bipolar PAM information signal with period T;
PAM spreading signal with period T/N and symbols

( )

u

S t

/

( ) ( / )

u u s T N s

y t y g t sT N

∞ =−∞

= −

∑

{ }

exp( 2 / ) , 2

u s

y x i L L π ∈ >

complex if

SLIDE 9

9

Asynchronous co-channel interference and performance -I channel 1st receiver

∫

+ T s sT ) 1 ( 1 s

S

2 s

S

U s

S

2

t

U

t

1

1( ) i

y t e θ

Input of decision block is random

1 1 s s

Ω + Ψ

2

2( ) i

y t e θ ( )

U

U i

y t e θ

1

t

1 1

( ) 1 1 *

( )

i t

y t t e

ω θ − −

−

MAI

Synchronized useful user (carrier phase offset, timing, time shift compensation)

System performance depends on

spreading sequences statistics (auto- and cross-correlation) 2

1 1 1 1 BER erfc erfc 2 2 2 1 ( ) U R

ψ

σ = −

Expected Interferce-to-signal ratio

per interfering user Expected degradation in Performance by adding a new user

Standard Gaussian Assumption (Pursley 1982)

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10

Classical and maximum performance

Sequences auto-correlation

E

k k u u

y y A ⎡ ⎤ = ⎣ ⎦

2

E ( ) 1

u

A y ⎡ ⎤ = = ⎣ ⎦

Second-order stationary sequences

( )

k

A k δ =

2 3 R N =

Maximal-length, Gold, Kasami, etc., tend to approximate a “white” behavior (Purely Random Sequences)

p

1

p

1 − m

p

N T N T N T

p′

1

p′

1 −

′

m

p

N T N T N T N T k

min

? R =

Maximum length (m-)sequences and Gold sequences

1 2 1 1 2 3

4 ( ) ( ) 2 3 2 3

N k k k k

R A A A k N k N N N

− = −

⎡ − ⎤ ⎛ ⎞ = + − + ⎢ ⎥ ⎜ ⎟ ⎝ ⎠ ⎣ ⎦

∑

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11

Interference minimization

2 2 2 2 min

3 3 2 2 3 3 3

N N N N N

a a N a N N R a

− −

− = < + −

≅

large

Minimum interference

U

10 12 14 16 18 20 22 24 1.×10−8 1.×10−7 1.×10−6 0.00001 0.0001 0.001 0.01

BER

2

30 0.01

n

N σ = =

2 3 R N = 3 3 R N =

At any fixed BER (e.g. 10-3) >15% MORE USERS (e.g. 3) CAN BE ALLOCATED FOR FREE

How can we generate sequences with the optimal autocorrelation profile? What has chaos/complexity to do with it?

2 3 a = − 1 ( ) ( )

k N N k k k k N N N

A a N a N a k a a

− − −

− = − − − −

≅

large

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12

Chaos-based generation of spreading sequences

Sequences Generation Conceptual Scheme (Off-line)

.2 0. 4 .6 0. 8 1 0. 2 0. 4 0. 6 0. 8 1

1 + k

x

∆

k

x M k ∈N

1. Chaotic time series

generated by

{ }

k

x x =

1

( )

k k

x M x

+ =

where

[ ] [ ]

: 0,1 0,1 M →

/ T N / T N / T N

….

N-stage shift register

2. Repeat a sequence of lenght N extracted

from the chaotic time serie, i.e.

1

( ) 0,1, , 2

k k

x M x k N

+ =

= − … 1, 2,

k lN k

x x l

−

= = ± ± …

Formally:

3. Mapping: L intervals

covering

1, L

X X …

[ ]

0,1

If

2 ( / )

( )

i j L k j k k

x X y Q x e

π

∈ ⇒ = =

Chaos-based Spreading Sequence

CB-SS Statistical Properties????

( )

k

Q x

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13

An alternative approach for studying chaos

Chaotic map

with and

k∈N ) (

1 k k

x M x =

+

[ ] [ ]

1 , 1 , :

=

X M

[ ]

1 ,

0 ∈

x

∆

k

x

1 + k

x M

Evolution of points
following single trajectories

is difficult

sensitive dependence on initial conditions
),

( ), ( ,

1 2 1

x M x x M x x = =

4 ' ' '

10 10 / , 10 /

−

+ = = π π x x

10 20 30 40 50 60 0.2 0.4 0.6 0.8 1

Evolution of densities

0:[0,1

] ρ

+

→R

{ } dx x dx x x dx x ) ( 2 / 2 / Pr ρ = + ≤ ≤ −

x

dx

1

ρ

If x0 is drawn according to ρ0, which is the density of x1 =M(x0), x2 =M(x1),...?

What can we predict with this?

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14

Densities instead of trajectories -I

) 1 ( 4 ) (

1 k k k k

x x x M x − = =

+

) 1 ( 1 ) ( x x x − = π ρ 1 = k 2 = k 3 = k 20 = k 1 = k 2 = k 3 = k 4 = k = k = k 1 1

The final density does not depend on the initial one

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15

Densities instead of trajectories -II

Different maps have different densities

2 = k 2 / 1 2 1 ) (

1

− − = =

+ k k k

x x M x 1 ) ( = x ρ 1 = k 3 = k 20 = k 1 = k 3 = k 4 = k = k = k 1 1 2 = k

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16

Densities instead of trajectories -III

Speed of convergence to depends on the map

2 = k 1) (mod k

k k

x x M x

` 1

1000 ) ( = =

+

ρ 1 = k 1 = k = k = k 1 1 2 = k

1000

1 ) ( = x ρ 20 = k

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17

“Heuristic” remarks

The final density does not depend on the initial one
Different maps have different densities
Speed of convergence to depends
on the map
on the initial density

ρ

Tool for predicting the evolution of densities is needed...

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18

{ }

Pr 2 2 dy dy y y y − ≤ ≤ + =

Evolution of densities: Perron-Frobenius operator

) (x M y= x

k

ρ

1 + k

ρ

2

x

1

x y

Probability conservation constraint:

{ } { }

1 1 2 2 1 1 1 2 2 2

Pr Pr 2 2 2 2 dx dx dx dx x x x x x x − ≤ ≤ + + − ≤ ≤ +

1 1 1 2 2

( ) ( ) ( )

k k k

y dy x dx x dx ρ ρ ρ

+

= + dy dx x dy dx x y

k k k 2 2 1 1 1

) ( ) ( ) ( ρ ρ ρ + =

+

) ( ' ) ( ) ( ' ) ( ) (

2 2 1 1 1

x M x x M x y

k k k

ρ ρ ρ + =

+

⇒ ⇒

1 ( )

( ) ( ) ( ) '( )

k k k M x y

x y y M x ρ ρ ρ

+ =

= = ∑ P

Perron-Frobenius operator P of M for maps with “several branches”

1 2 1 2 1

, and , f f α α ∀ ∈ ∀ ∈ R

L

1 1 2 2 1 1 2 2

( ) f f f f α α α α + = + P P P

P is a (infinite dimensional) linear operator i.e.
P has composition properties similar to M

2 1 1 2

M M M M

f f = P P P

Properties

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19

A Global Linearization

∆

k

x

( ) ( ) k-times

k k

x M M M x M x = =

)

(

1 k k

x M x =

+

☺ Finite dimension (one)

Highly Nonlinear
Extremely Complex

∆

k M k

ρ ρ P =

+1 k

ρ

M

P

k

k k M M

ρ ρ ρ = = P P

☺ Linear ☺ “Simple” (Regular) Behavior

Infinite Dimensions

(not always…)

Are we able to give a complete statistical characterization of the signal generated by the map (our way to generate a stochastic process)?

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20

Statistical characterization for an interesting class of maps DEF

M

1

X

2

X

3

X

4

X

1

X

2

X

3

X

4

X

1/4 1/4 1/4 1/4 1/3 1/3 1/3 1/3 1/3 1/3 1/4 1/4 1/4 1/4 ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ =⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ K

2

X

4

X

1

X

2

X

3

X

3

X

Kneading matrix

( ) ( )

1(

)

j k jk j

X M X X µ µ

−

∩ = K

Fraction of Xj which is mappend in Xk

=

Transition matrix

1

X

2

X

3

X

4

X

µ-Probability that x moves

from Xj to Xk given that it is in Xj

=

) (

k j

X M X ⊆

r

( ) ,

j k

X M X j k ∩ =∅ ∀

Piece-Wise Affine Markov (PAWM) maps

Affine Affine Affine Affine

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21

Some interesting properties

K is the restriction of P to the space of simple functions

i

X

χ

1

1 e e e = = K

( )

1

( ) ( )

( )

n i i i i

x e X X x ρ µ

χ

=

=∑

1. If M is ergodic the invariant density can be computed as
2. The spectrum of K coincides with the discrete part
f the one of P

1 1 ,2, ,

i

i l γ < =

eigenvalues of K

max

i

r γ =

define

max{ , } r r r = =

mix ess

rate of decay of correlation

3. If K is primitive then M is mixing (exact)

⇔ an index l exists such that

l

K

has no null entry

1/ ess

1 lim sup

k k k x

r D M

→∞

⎡ ⎤ = ⎢ ⎥ ⎣ ⎦

SLIDE 22

22

Characterization of Processes Generated by PWAM

( )

k k

y Q x = ∈ C

k

x

1 k

x +

M=? Q

∆

M

1

X

2

X

3

X

4

X

1

X

2

X

3

X

4

X

PWAM with finite partition

Probability density (i.e. how often a certain value appears in the process history)
Rate of mixing (i.e. how fast the probability describing the distribution of the state converges to the invariant
ne)
Exact finite time crosscorrelation profile (i.e. how each realization of the process is related to other

realizations of the same process)

Exact finite time autocorrelation (i.e. the short-time power spectral density of the process)
Asymptotic trend of autocorrelation (i.e. power spectral density at low frequencies: exponential trend,

polynomial trend, combinations)

Higher order moments/correlation (i.e. how multiple samples from the process relate to each other)

What can we analize/design?

SLIDE 23

23

Going on with the statistical characterization… correlations - I

Computing the statistical features of chaotic sequences is generally a very difficult task
Computing is easy for PWAM maps. Is it possible to easily give a full statistical

characterization, i.e. compute (any-order) correlations of chaotic trajectories?

[ ]

1 m 2 1 2 ix

( ) ( ) sup sup

k k k

C E x x r ϕ ϕ ϕ ϕ = ≤Λ

2 1 2

, ([0,1]) ϕ ϕ ∀ ∈

฀

L

For exact maps M, only bounds can be achieved, i.e.

mix and

r ρ

Yes if we add a is properly chosen quantization function

.2 0. 4 .6 0. 8 1 0. 2 0. 4 0. 6 0. 8 1

1 + k

x

∆

k

x M k ∈N

( ) Q i

k

y

Partition

1

X

2

X

3

X

11

Q

12

Q

13

Q

1

X

2

X

3

X

21

Q

22

Q

23

Q

1

X

2

X

3

X

31

Q

32

Q

33

Q

p2 p1 p3

' '' 1

' '' ( ) 0 [0,1 ] ,

n j j j j

j j X X X µ

=

≠ ⇒ ∩ = =

∪

Quantization at time pi

:[0,1 ] ( )

i j i ij

Q x X Q x Q ∈ ⇒ = , C

Vector of quantized values at time pi

1 2

( , , , )

i i in i

Q Q Q Q =

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24

Going on with the statistical characterization… correlations - II

Factorization in the computation of higher-order correlations

2 1 2 2 3

, , 1 2

( ) ( ) , ,

i m m m

q m i q q q q q q q i

E Q x Q x ρ µ

−

= − −

⎡ ⎤= = ⎢ ⎥ ⎣ ⎦

∏

K

K K Q H Q

Conditioned probability m’ indexes m’’ indexes m’+m’’-1 indexes

=C

A B

1

' '' 1 1 ' ' ' '' 1

, , , ,

m m m m m m

j j j j j j

+ − + −

= C A B

Chain product of tensors
2-order correlation

probability All possible values

( )

2 2 2

1 2 1 2 1 2 1 1

( ) ( ) ,

j q q j n k n q j j k j k

E Q x Q x X Q Q Q Q µ µ ρ ρ

= =

⎡ ⎤ ⎡ ⎤= = ⎣ ⎦ ⎣ ⊗ ⎦

∑∑

K K

Matrix

m-order correlation

2 2

1 2 , , , 2 , 1( )

( ) , ,

m m i

q q q q m m i q i

E Q x Q x Q Q Q

=

⊗ ⎡ ⎤= = ⎢ ⎥ ⊗ ⎦ ⊗ ⎣

∏

H

H Q

Tensor

Generalization of K SDTT

Symbolic Dynamics Tracking Tensor

SLIDE 25

25

(n,t)-tailed shifts: 2nd order correlation

binary quantization

alternating signs

[ ]

( ) ( )

k k

t E Q x Q x n t ⎛ ⎞ = − ⎜ ⎟ − ⎝ ⎠

t n − t

Any (n,t)-tailed shifts a PWAM Map with equal

Xj and ρ =1

1

X

2

X

3

X

4

X

5

X

6

X

7

X

8

X

9

X

10

X

1

X

2

X

3

X

4

X

5

X

6

X

7

X

8

X

9

X

10

X 10 = n 2 = t

1 − 1 +

1/( ) 1/ n t t

⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠

− = K

n t − t t t n −

t r n t =− −

1 + k

x

∆

k

x

M

k ∈N

( ) Q i

1 k

y +

5 10 15 20

0.2

0.2 0.4 0.6 0.8 1

numerical, 100 trials theoretical

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26

Chaos-based spreading is (nearly) optimal

What so far?

– Spreading sequences with minimize R (with N large enough) – Chaos-based SS generated by (n,t)-tailed shifts have so that

( 1 )k

k k

A a = − , w ith ( )

k k

A r r t n t =− − =

For any n, set t as the integer closest to
Set to get optimal profile

| | 0.2679 r a = ≈ ( 10 2, 0.25) (1 ) na n t r a = ⇒ = =− + 70, 10, 2 N U L = = =

r

r

Very good match between analytical and

numerical results

(10,2)-tailed shift with

10, 2, 2 128 U L N = = ≤ ≤

Chaos-based Spreading is nearly optimal 15% expected improvement in U with the same BER

SLIDE 27

27

What so far?

Application of nonlinear dynamics to DS-CDMA Systems Optimization

– Co-channel interference: chaos-based spreading is (nearly) optimal: 15% average zero-cost improvement (with peak of 60%) with respect to purely random sequences – This results can be obtained thanks to the possibility of compute in closed form the statistical features of signals generated by chaotic systems

Can we apply the same path also to:

Multipath environement?
Spreading sequences synchronization?
Use of different receivers (Rake)?
….

SLIDE 28

28

Multipath channel model

Random secondary rays Deterministic main ray

0.2 0.4 0.6 0.8 1

l

β

∑

∞ = −

− + =

1

) ( ) ( ) (

l i l l

N T l t e a t t h

lδ

β δ β

φ

al Rayleigh φl uniform distributed

Exponential rays power decay

SLIDE 29

29

phase,

DS-CDMA: co-channel interference and multipath

1

( ) ( ) ( )

u k

i u u k k k

T h t t a e t k N

φ

β δ β δ

∞ − =

= + −

∑

channel

∫

+ T s sT ) 1 ( 1 s

S

2 s

S

U s

S

2

t

U

t

1 *

( )

k

Q

1st receiver

1 1 1 s s s

Ψ Ω + + Ξ

1 k

Q

2 k

Q

U k

Q

Input of decision block is random

Asynchronous Interference Selective Fading

Assume receiver synchronization
Channel impulse response
random power attenuation

2 2

( )

u k k

a β

u k

φ / kT N delay of the k-th ray

Rayleigh distributed real RV

u k

a

u k

φ [0,2 ] π

uniform RV in

Define Rice factor

;

Power of first ray Power of secondary rays

K =

Exponential vanishing rays

Bl l

Ae β

−

=

1 1

= useful component

s s

S Ω ∝

1 , 1

( , ) (other = cross-in users primary and secondar terfere y r nce ays)

u N s

F x x

τ

⎡ ⎤ = ⎦ Ψ Γ ⎣

Integrate-and-dump output

1 , 1 1

( , ) (useful user sec = self-interferenc

ndary rays of previous symbols)

e

s N

G x x

τ

⎡ ⎤ = Γ ⎣ ⎦ Ξ

partial cross- and autocorrelation

SLIDE 30

30

Self-interference

sum of independent RV

Cross-interference

sum of independent RV

1 s

Ψ

2

Gaussian RV

estim ateσ Ψ

⇒ ⇒

Multipath propagation: system modeling

2 2

1 BER erfc 2 8( 1)( ) K K σ σ Ψ

Ξ

= + +

1 s

Ξ

2

Gaussian RV

estim ateσ Ξ

⇒ ⇒

For sequences generated by (n,t)-tailed shifts analytical computation of high-order correlation terms

2 1 2 2

( , , ) ( , , , , )

f analytical expressio

r

a d n n

f N r Q f N r Q B K σ σ Ξ

Ψ =

=

3rd, 4th order correlations

( ) ( )

2 2 2 2 , 1 2 , 2 2

, , 8 (1 )

N B N N N BN

A E v v E v v e N e

τ τ τ τ

σΞ

− − − =

⎧ ⎫ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ = Γ + Γ ⎨ ⎬ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ − ⎩ ⎭

∑

{

2 * ,0 ,0 ,1 3 1 2 * , 2 , , 1 1

1 ( , ) Re ( , ) ( , ) 12 2 ( , ) Re ( , ) ( , )

u v u v N N N N u v u v N N N

U E u v E u v u v N E u v E u v u v

τ τ τ τ

σ

≠ ≠ − ≠ = Ψ ≠ +

− ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ = Γ + Γ Γ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎫ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ + Γ + Γ Γ ⎬ ⎣ ⎦ ⎣ ⎦ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦⎭

∑

SLIDE 31

31

Multipath propagation: system modeling - II

Ex: for binary sequences

{ }

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 2 2 1

( ) ( ) 8 (1 ) ( ) ( ) ( )

BN N B N B BN N B N N B N N B

A N e NF e F r e N e F r e r F r e r F r e σ

− − − − − − − − Ξ −

= + + − − − + −

}

2 2 2 2 3 2 2 2 2 1 2 2 1 2 2

1 2 ( 1) ( 1) (1 ) 12 (1 ) 2 ( 2) 1 ( )(2 )

N N N N N

U r N N N rN N r N r N r N r r r F r r r σ

− − − Ψ

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

where

1 1 1 1

( ) , ( ) ( )

M M M q l M q l l l

F x l x F x M l x

− − = =

= = −

∑ ∑

SLIDE 32

32

Model verification and performance optimization

70, 10, 2 N U L = = =

70,

10, 2, 1 ( )

sim ilar varying

N U L K K = = = =

2

0.2679 1. minimum for

2. Chaos - based better than

a nd Gold r m

σ Ψ

− −

2

| | | | 1. if maps have large means low power in secondary rays but firsts inability of maps with large

f breaking correlation
2. Chaos - based worse than

and Go ld r B r m

σ Ξ ↑

↑ ⇒ ⇒ −

2 1 2 2 2 2 2

pt

0.2679 and ( , )

( , , ) ( , , , , )

Due to BER non minimum for Analytical expression of find which minimizes sequences generated by corresponding

TS

r r r n t

f N r Q f N r Q B K σ σ σ σ σ

Ψ Ψ Ξ Ξ Ξ

− =

= = ⇒ + ⇒

5% to 15% zero cost improvement

SLIDE 33

33

Prototype System

useful TX RX channel interfering TXs interfering TXs useful TX RX Interfering TXs

6 Transmitters (useful + 5 interferers)

SLIDE 34

34

Chaos- based Spreading reduces BER by a factor 4

Spreading factor N=15 or N=20, M=256
Relative delays change
Mean over 10 trials of 106 bits
Binary sequences (optimal profile)

1

i

u =

m-sequences 2.66 10-4 16.77 10-4 Purely-random 2.66 10-4 3.81 10-4 Chaos-based 0.98 10-4 1.46 10-4

N=20

Theory Measure Gold sequenc. 13.5 10-4 21.8 10-4 Purely-random 13.5 10-4 16.2 10-4 Chaos-based 6.3 10-4 6.5 10-4

N=15

Theory Measure

Measurement results

SLIDE 35

35

Other results on chaos-based DS-CDMA optimization-I

1. 10% reduction in terms of average lost bit at link start up, [20] 2. Optimal perfomance can be achieved with chaos-based SS with autocorrelation decay [32]-[35] 2nd, 3rd, 4th

rder moments
f SS

Synch mechanism

Synch. of the

useful link 1. Average zero zero-

cost

cost improvement between 5% and 15% depending on the channel condition, [17],[22],[23] 2nd, 3rd, 4th

rder moments
f SS

Multipath Propagation 1. Nearly optimal average zero zero-

cost

cost 15% (60% peak ) improvement over i.i.d. 2. Experimental confirmation [2],[17],[18],[19],[21] 3. Optimal sequences are non-binary ( [36],[37]) 2nd order moments of SS Co-channel Interference Main Results Necessary tool

1 s

S

1 k

y

1(

)

C

R ω j

∫

+ T s sT ) 1 ( 1 k

y

1

t ∆

1(

) C ω j

2 3 − +

SLIDE 36

36

Other results on chaos-based DS-CDMA optimization-II

1. Semianalytical optimization of pulses shape (non rectangular) and SS. Improvement over raised cosine and i.i.d sequences [31] 2nd, 3rd, 4th

rder moments
f SS

Joint

ptimization

sequences and pulse shape 1. Chaos-based spreading is asymptotically optimal (N→¶) in terms

f Shannon Capacity [29],[30]

2ndorder moments of SS Model of the system Shannon Capacity 1. Joint optimization of rake receiver taps and spreading sequences, given the

channel. Improves Perr by a factor 4,

[24] 2nd, 3rd, 4th

rder moments
f SS

Rake receiver Main Results Necessary tool

2 2

1 log det C I HH σ

+

⎛ ⎞ = + ⎜ ⎟ ⎝ ⎠

q Hb ν = +

T N T N T N

∫

+ T s sT ) 1 (

Recent Directions: CDMA Multicarrier (PAPR), Multicode, UWB for WSNs

SLIDE 37

37

Wireless Sensors Networks

Enviromental events WSN Gateway Data collector Sensor nodes must provide processing, storage and communication capabilities. No global structure or authority is assumed. ALARM ! node

SLIDE 38

38

Impulse Radio & Regulation

Federal Communications Commission (FCC) definition of

UWB signals:

total bandwidth constraint fractional bandwidth constraint

1 1.5 2 3 4 5 6 7 8 9 10

80
75
70
65
60
55
50
45
40

Frequency (GHz) EIRP Spectral Density (dBm / MHz)

Frequency MHz Frequency MHz EIRP dBm / MHz EIRP dBm / MHz 960-1610 1610-1990 1990-3100 3100-10600 Above 10600 960-1610 1610-1990 1990-3100 3100-10600 Above 10600

75.3
53.3
51.3
41.3
51.3
75.3
53.3
51.3
41.3
51.3

Part 15 Limit

Total Average Power Max =

41.3 + 10 Log (10.6-3.1) + 30 =
2.5 dBm
FCC assigned 3.1 to 10.6 GHz spectrum for unlicensed use.
Wideband, very low power spectral density

⇒To achieve this short (<1 ns) pulses are used. Shape is the n-th derivative of a Gaussian

Modulation can be PPM or PAM (as in our case); Multiple access capability is needed

Advantages: low probability of intercept, low transmitted power (long battey life)

⇒ IEEE802.11 TG4 has choosen UWB as a condidate for WSNs

SLIDE 39

39

Transmitter structure

Nodes employ Direct-Sequence to share communication channel.
Gaussian pulse and its derivatives are used.

Pulse generator Node Signature

LNA

Typically the spreading factor ranges from 32 up to 256. Attenuation and derivative models the effect of the antenna

SLIDE 40

40

Receiver structure

Simplest possible decoder (due to energy constraints): correlate-and-dump
The receiver is fed with all the incoming signals plus a noise term.

Template pulse generator Node Signature

LNA

Attenuations due to antennas and channel Hp: received pulse is perfectly reconstructed at the receiver in terms

f timing. Note that since antennas act

as differentiators the received pulse is the 2nd derivative of a Gaussian pulse

SLIDE 41

41

Other nonidealities: MAI and ICI

Multiple Access Interference (MAI)

Node ( ) Node ( ) Node ( ) Receiver

Inter Chip Interference (ICI)

Increase means decrease the chip time . Features: Nodes are fixed ⇒ synchronous DS-CDMA system No global structure or central authority ⇒ codes cannot be orthogonal! ⇒ PN sequences are used Code are internally generated in each node. Do PN sequences provide the best performance, or can we do better here as well ?

Desired communication Interfering users

ICI INCREASES

SLIDE 42

42

System performance -I

Assumptions:
all the nodes have identical antenna and air interface
all the transmitter are located at the same distance

from the receiver (not necessary)

all the transmitters deliver the same power

(equal to the maximum one – normalized to 1)

node position doesn’t change in time

Bit-synchronous environment

LNA

U transmitters and 0 is the useful one: if the

bit is transmitted then MAI, Interchip interference, thermal noise are Gaussian R.V. with

2 2 2 2 2 2

, ,

C S n C S n

σ σ σ σ σ σ +

SLIDE 43

43

System performance -II

Performance figure: Signal to Interference Ratio (SIR)
SIR is a function of the number of active nodes (U), the pulse shape

(g(t)) and spreading sequence statistic.

TARGET

Assigned N,U,g(t), and ρmin find the sequence auto-correlation which

maximizes ρ or the BR at which each node may transmit information.

Unfortunately, we cannot solve the problem in an analytic way, since the

dependence of ρ from r dependence can’t be exploited in a simple way.

We must rely on numerical evidence

SLIDE 44

44

Node signature generation

PWAM map (can be studied through Markov Chains).

“y=+1” “y=-1” For the map generates low pass correlated symbols. For the map generates high pass correlated symbols. For the map generates i.i.d. symbols.

SLIDE 45

45

Results -I

Numerical simulation with Gaussian pulse
The Bit-Rate is maximized with positive correlated sequences.
Only for small Bit-Rates i.i.d. sequences provide better performance.
Performance is stable with respect to the number of active users.

P A R A M E T E R S

SLIDE 46

46

Results -II

Numerical simulation with 2-nd derivative of the Gaussian pulse
The Bit-Rate is maximized with positive correlated sequences.
Only for small Bit-Rates i.i.d. sequences provide better performance.
Performance is stable with respect to the number of active users.

P A R A M E T E R S

SLIDE 47

47

System Performance -III

Gaussian Approximation for interference terms has been verified numerically. Examples refer to Gaussian pulse with and . Numerical simulation have been performed with other pulse shape and sequences.

SLIDE 48

48

Pulse Shaper Hardware Implementation - I

Modulator: generates a sequence of 1-st derivative Gaussian-shaped

pulses, controlled by both the spreading sequences and the bits to be transmitted – The simple Pulse-shaper is based on the classical scheme used to obtain a glitch from a NAND gate. τ1 depends both

n τ and Ctune

– A Delay-selector comes before the pulse-shaper: it allows to obtain the proper shape for the final pulse, and at the same time to get a sequence composed of equidistant pulses: if Sel = 0, then signal In is taken to the output with no delay; otherwise with a delay equal to T (given by I1 and I2)

SLIDE 49

49

Pulse Shaper Hardware Implementation - II

Whole modulator, two branches:

– pulse generator – delay selector – some additional logic – simulated circuit also includes equivalent models for antennas.

Rising edge of input clock Clk:

– P1, P2, respond with a glitch – the TX antenna performs the difference, P1-P2 – depending on the value of Sel, the signal which gets delayed is either P1 or P2

Two antipodal shapes are

possible for the output pulse, both 1-st derivative Gaussian shaped

SLIDE 50

50

Pulse at the input of the TX

antenna

pulse transmitted (comparison

with theoretical)

Circuit simulation, CMOS UMC130 1.2V: Results

Comparison of spectra:

– FCC mask – Theoretical Gaussian 2-nd derivative – Simulated pulse

SLIDE 51

51

Results: reference examples

N = 255, SIRmin = 9.55 (Perr = 10-3);

Z(U) = 24

– Comparison between the SIR achieved using i.i.d. and chaos- based spreading, for different values

f BR

– Maximum achievable BR is evalueted intersecting with SIR=SIRmin

Result: always BRoptimum >= BRi.i.d.
Speed Up = 100 (BRo/BRi-1) =

33%

Trend of SIR when Z (U) sweeps from

8 to 33:

– SIR from chaos-based optimum sequences always outperform SIR of i.i.d. sequences, independently of the number of nodes

SLIDE 52

52

Conclusion

Systematic methodology for characterizing one-dimensional chaotic

maps from a statistical point of view

Performance optimization for DS-CDMA system in several

asynchronous environments (MAI only, multipath, rake receiver…) for negatively correlated spreading sequence

Performance improvement of DS-UWB (synchronous) environment by

means of positively correlated sequences.

SLIDE 53

53

Sources of further information

“Chaotic Electronics in Telecommunications” M.P. Kennedy, R. Rovatti, G. Setti, eds. CRC Press, Boca Raton, Florida, 2000

“Application of Nonlinear Dynamics to Electronic and Information Engineering” M.Hasler, G. Mazzini, M. Ogorzalek, R. Rovatti, G. Setti, eds. Special Issue Proceedings of the IEEE, May 2002

G. Mazzini, R. Rovatti, G. Setti

Tutorial ISCAS 2001 (http://ieeexplore.ieee.org) Tutorial ISCAS 2003 Tutorial Globecomm 2003

SLIDE 54

54 Statistical Approach to Chaos

1.

A. Lasota and M. C. Mackey, “Chaos, Fractals, and Noise” New York: Springer-Verlag, 1994.

2.

G. Setti, G. Mazzini, R. Rovatti, S. Callegari, “Statistical Modeling and Design of Discrete Time Chaotic Processes: Basic Finite-

Dimensional Tools and Applications,” Proceedings of the IEEE, vol. 90, pp. 662-690, May 2002. 3.

G. Keller, “On the rate of convergence to equilibrium in one-dimensional systems,” Commun. Math. Phys., vol. 96, pp. 181–193,

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Poincaré—Physique Théorique, vol. 62, pp. 251–265, 1995. 8.

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9.

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Proceedings, Newton Institute, Cambridge, 1998, pages 283-324, Birkhauser, 2000 11. M.l Dellnitz, G. Froyland, S. Sertl, “On the isolated spectrum of the Perron-Frobenius operator,” Nonlinearity, 13(4):1171-1188, 2000 12. M Goetz, W. Schwarz, “Statistical analysis of chaotic Markov systems with quantised output” Proceedings. ISCAS 2000 Geneva, 2000, pp. 229 -232 vol.5 13. T.Kohda, A.Tsuneda, “Statistics of Chaotic Binary Sequences”, IEEE Trans. Inf. Theory, vol. 43, pp. 104-112, 1997 14. T.Kohda, A.Tsuneda, “Explicit Evaluations of Correlation Functions of Chebyshev Binary and Bit Sequences Based on Perron- Frobenius Operator”, IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, vol. E77-A,

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T. Kohda, H. Fujisaki, “Variances of multiple access interference code average against data average” Electronics Letters, vol 36

Issue: 20 , 28 Sept. 2000 16.

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Fundamentals, vol. 81-A, n. 9, pp. 1769-1776, September 1998.

Some bibliography - I

SLIDE 55

55 Chaos-Based DS-CDMA

17.

M. P. Kennedy, R. Rovatti, G. Setti (Eds.), “Chaotic Electronics in Telecommnunications,” CRC Press, Boca Raton, June 2000

18.

G. Mazzini, G. Setti, R. Rovatti, “Chaotic Complex Spreading Sequences for Asynchronous DS-CDMA – Part I: System Modeling

and Results,” IEEE Transactions on Circuits and Systems- Part I, vol. 44, n.10, pp.937-947, October 1997. 19.

R. Rovatti, G. Setti, G. Mazzini “Chaotic Complex Spreading Sequences for Asynchronous DS-CDMA – Part II: Some Theoretical

Performance Bounds” IEEE Transactions on Circuits and Systems- Part I, vol. 45, n. 4, pp. 496-506, April 1998. 20.

G. Setti, R. Rovatti, G. Mazzini, “Synchronization Mechanism and Optimization of Spreading Sequences in Chaos-Based DS-CDMA

Systems,” IEICE Transactions on Fundamentals, vol. 82, n. 9, September 1999. 21.

G. Mazzini, R.Rovatti, G. Setti, “Interference Minimization by Auto-correlation Shaping in Asynchronous DS-CDMA Systems: Chaos-

based Spreading is Nearly Optimal,” IEE Electronics Letters, vol. 35, n. 13, Jun. 24 1999, pp. 1054-1055. 22.

R. Rovatti, G. Mazzini, G. Setti, “A Tensor Approach to higher-order Expectation of Quantized Chaotic Trajectories - Part I: General

Theory and Specialization to Piecewise Affine Markov Systems,” IEEE Transactions on Circuits and Systems- Part I, vol. 48, November 2000. 23.

G. Mazzini, R. Rovatti, G. Setti, “A Tensor Approach to higher-order Expectation of Quantized Chaotic Trajectories - Part II:

Application to Chaos-based DS-CDMA in Multipath Environments,” IEEE Transactions on Circuits and Systems- Part I, vol.48, November 2000. 24.

R. Rovatti, G. Mazzini, G. Setti, “Enhanced Rake Receivers for Chaos-Based DS-CDMA,” IEEE Transactions on Circuits and

Systems- Part I, vol. 48, pp. 818-829, 2001. 25.

G. Setti, R. Rovatti, G. Mazzini, “Tensor-Based Theory for Quantized Piecewise-Affine Markov Systems: Analysis of Some Map

Families,” IEICE Transactions on Fundamentals, vol. 84, n. 9, pp. 2090-2100, September 2001. 26.

G. Mazzini, R. Rovatti, G. Setti, “Chaos-Based Asynchronous DS-CDMA Systems and Enhanced Rake Receivers: Measuring the

Improvements,” IEEE Transactions on Circuits and Systems- Part I, vol. 48, n. 12, pp. 1445-1453, 2001. 27.

R. Rovatti, G. Mazzini, G. Setti, “Shannon Capacities of Chaos-Based and Conventional Asynchronous DS-CDMA Systems over

AWGN Channels,” IEE Electronics Letters, vol. 38, n. 10, pp. 478-480, 9 May 2002. 28.

G. Mazzini, R. Rovatti, G. Setti, “Capacity of Chaos-based Asynchronous DS-CDMA Systems with Exponentially Vanishing

Autocorrelations,” IEE Electronics Letters, vol. 38, n. 25, pp. 1717-1718, December 2002. 29.

R. Rovatti, G. Mazzini, G. Setti, “On the Ultimate Limits of Chaos-Based Asynchronous DS-CDMA - Part I: Basic Definitions and

Results,” IEEE Transactions on Circuits and Systems- Part I, vol. 52, n. 7, pp. 1336 – 1347, July 2004

Some bibliography - II

SLIDE 56

56

30.

R. Rovatti, G. Mazzini, G. Setti, “On the Ultimate Limits of Chaos-Based Asynchronous DS-CDMA - Part II: Analytical Results and

Asymptotics,” IEEE Transactions on Circuits and Systems- Part I, vol. 52, n. 7 pp. 1348 – 1364, July 2004 31.

G. Setti, R. Rovatti, G. Mazzini, “Performance of Chaos-Based Asynchronous DS-CDMA with Different Pulse Shapes,” IEEE

Communications Letters, vol. 8, n. 7, pp. 416-418, July 2004 32.

Y. Jitsumatsu, N. Eshima, T. Kohda, “Superiority of Markovian Spreading Sequences in Code Acquisition Performance,” Proc.

NOLTA2004, pp. 694-698, 2004 33.

T. A. Khan, N. Eshima, Y. Jitsumatsu, T. Kohda, “Code Acquisition in Asynchronous DS/CDMA Systems, Markov and i.i.d. Codes in

Multiuser Scenario,” Proc. NOLTA2004, pp. 689-693 34. Eshima, N.; Jitsumatsu, Y.; Kohda, T.; ”Markovian SS codes imply inversion-free code acquisition in asynchronous DS/CDMA systems”, Proc. ISCAS04 pp. IV-617 - IV-620 35. Khan, T.A.; Eshima, N.; Jitsumatsu, Y.; Kohda, T.; “A Novel Code Acquisition Algorithm and its Application to Markov Spreading Codes,”, Proc. ISCAS05, pp. 2056 - 2059 36.

H. Fujisaki, “On correlation values of M-phase spreading sequences of Markov chains,” IEEE Transactions on Circuits and Systems-

Part I, vol. 49, n. 12, Dec. 2002 pp. 1745 – 1750 37.

H. Fujisaki, “On optimum M-phase spreading sequences of Markov chains,” Spread Spectrum Techniques and Applications, 2004

IEEE Eighth International Symposium on, 2004 pp. 699 - 703

EMI Reduction

38.

G. Setti, M. Balestra, R. Rovatti, “Experimental verification of enhanced electromagnetic compatibility in chaotic FM clock signals,”
Proceedings. ISCAS 2000 Geneva, 2000, pp. 229 – 232, vol.3

39.

M. Balestra, A. Bellini, S. Callegari, R. Rovatti, G. Setti, “Chaos-Based Generation of PWM-Like Signal for Low-EMI Induction Motor

Drives: Analysis and Experimental Results,” IEICE Transactions on Electronics, pp. 66-75, vol. 87-C, n. 1, January 2004 40.

S. Santi, R. Rovatti, G. Setti, “Zero Crossing Statistics of Chaos-based FM Clock Signals,” IEICE Transactions on Fundamentals, pp.

2229-2240, vol. 88-A, n. 9, September 2003 41.

S. Callegari, R. Rovatti, G. Setti, “Chaos-Based FM Signals: Application and Implementation Issues,” IEEE Transactions on Circuits

and Systems- Part I, vol. 50, pp. 1141-1147, August 2003. 42.

S. Callegari, R. Rovatti, G. Setti, “Generation of Constant-Envelope Spread-Spectrum Signals via Chaos-Based FM: Theory and

Simulation Results,” IEEE Transactions on Circuits and Systems- Part I, vol. 50, pp. 3-15, 2003 (Recipient of the IEEE CAS Society Darlington Award)

Some bibliography - III

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57

43.

R. Rovatti, G. Setti, S. Callegari, “Limit Properties of Folded Sums of Chaotic Trajectories,” IEEE Transactions on Circuits and

Systems- Part I, vol. 49, pp. 1736-1744, December 2002. 44.

S. Callegari, R. Rovatti, G. Setti, “Chaotic Modulations Can Outperform Random Ones in EMI Reduction Tasks,” IEE Electronics

Letters, vol. 38, n. 12, pp. 543-544, June 2002. 45.

F. Pareschi, L. De Michele, R. Rovatti, G. Setti, “A chaos-driven PLL based spread spectrum clock generator”, EMC Zurich 2005,

(Recipient of the Best Student Paper Award)

Random Numbers Generation

46.

S. Callegari, R. Rovatti, G. Setti, “ADC-based Design of Chaotic Truly Random Sources'', Proceedings of IEEE NDES200, pp. 5-9--

5-12, Izmir, Turkey, June 2002 47.

S. Callegari, R. Rovatti, G. Setti, “Efficient Chaos-based Secret Key Generation Method for Secure Communications,” Proceedings
f NOLTA2002, X'ian, Cina, October 2002

48.

S. Poli, S. Callegari, R. Rovatti, G. Setti, “Post-Processing of Data Generated by a Chaotic Pipelined ADC for the Robust Generation
f Perfectly Random Bitstreams,'' Proceedins of ISCAS2004, pp. IV-585--IV-588, Vancouver, May 2004

49.

S. Callegari, R. Rovatti, G. Setti, “First Direct Implementation of a True Random Source on Programmable Hardware,” International

Journal of Circuit Theory and Applications, vol. 29, 2005, to appear 50.

S. Callegari, R. Rovatti, G. Setti, “Embeddable ADC-Based True Random Number Generator Exploiting Chaotic Dynamics,” IEEE

Transactions on Signal Processing, Feb 2005

Some bibliography - IV

SLIDE 58

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