Ternopil’ State Technical University named after Ivan Ternopil’ State Technical University named after Ivan Pul’uiPul’ui

U K R A I N EU K R A I N E

International Conference on Inductive Modelling 2008, Kyiv

National University “Lvivs’ka Politechnica”National University “Lvivs’ka Politechnica”

Reconstruction of Algorithms for Spread Reconstruction of Algorithms for Spread Spectrum Signals Detection into a Frame of Spectrum Signals Detection into a Frame of Inductive Modeling MethodsInductive Modeling Methods

Bohdan YaBohdan Yavorskyyvorskyy, Yaroslav Dragan, Lubomyr , Yaroslav Dragan, Lubomyr

SicoraSicora

Kaf_BT@tu.edu.te.uaKaf_BT@tu.edu.te.ua

Can weCan we to to explainexplainby an Inductive Modeling by an Inductive Modeling

MethodMethoda succesful a succesful detectiondetection

of a Spread Spectrum Signalof a Spread Spectrum Signalwith unknown spectrum with unknown spectrum

spreadsspreads

??

2L)x,x()t(x (t)s(t))t(x ,

Spread Spectrum Signal after wide-band ADCSpread Spectrum Signal after wide-band ADC

/s

Signal to Noise Ratio (SNR)

Introduction backgroundsIntroduction backgrounds

Optimum detectors has been expressed in a coordinate free way in terms Optimum detectors has been expressed in a coordinate free way in terms of RKHS inner productsof RKHS inner products [Kailath T, Poor H.V. Detection of Stochastic Processes// [Kailath T, Poor H.V. Detection of Stochastic Processes// IEEE, Trans. Information Theory, vol. IT-44, pp. 2230-2299, 1998].IEEE, Trans. Information Theory, vol. IT-44, pp. 2230-2299, 1998].

Orthonormal expansions for second-order stochastic processes, a general Orthonormal expansions for second-order stochastic processes, a general expression for the reproducing kernel inner product in terms of the expression for the reproducing kernel inner product in terms of the eigenvalues and eigenfunctions of a certain operator has been analyzed ineigenvalues and eigenfunctions of a certain operator has been analyzed in [Parzen E. Extraction and Detection Problems and Reproducing Kernel Hilbert [Parzen E. Extraction and Detection Problems and Reproducing Kernel Hilbert Spaces// J. SIAM Control, vol. 1, pp. 35-62, 1962].Spaces// J. SIAM Control, vol. 1, pp. 35-62, 1962].

A some problems in signal detection applications were designedA some problems in signal detection applications were designed [Oya A., [Oya A., Ruiz-Molina J.C., Navarro-Moreno J. An approach to RKHS inner products evaluations. Ruiz-Molina J.C., Navarro-Moreno J. An approach to RKHS inner products evaluations. Application to signal detection problem// ISIT-2002, Lausanne, Switzerland, June 30-Application to signal detection problem// ISIT-2002, Lausanne, Switzerland, June 30-July 5, p. 214, 2002]July 5, p. 214, 2002]

Detection methods for either stationary Gaussian noise of known Detection methods for either stationary Gaussian noise of known autocorrelation or of noise plus a FHS of known hop epoch, unknown autocorrelation or of noise plus a FHS of known hop epoch, unknown phase or energy above a minimum levels are based on [1-3] had been phase or energy above a minimum levels are based on [1-3] had been developeddeveloped  [Taboada F., Lima A., Gau J., Jarpa P., Pace P.C. Intercept receiver signal [Taboada F., Lima A., Gau J., Jarpa P., Pace P.C. Intercept receiver signal processing techniques to detect low probability of intercept radar signals, ICASSP.-processing techniques to detect low probability of intercept radar signals, ICASSP.-2002]2002]

A factor of fatal increasing of a complexity and decreasing of a quality of A factor of fatal increasing of a complexity and decreasing of a quality of detection of completely unknown FHS in the ADC of radioradiation by the detection of completely unknown FHS in the ADC of radioradiation by the RKHS method was declined in RHS in a Hilbert space over Hilbert space RKHS method was declined in RHS in a Hilbert space over Hilbert space (HSoHS) (HSoHS) [Yavorskyy B. Vyyavlennya skladnyh syhnaliv z nevidomymy [Yavorskyy B. Vyyavlennya skladnyh syhnaliv z nevidomymy parametramy v radiovyprominyuvannyah// Radioelektronica ta telecomunicatsii.- parametramy v radiovyprominyuvannyah// Radioelektronica ta telecomunicatsii.- № 508, 2004.-с. 58-64]№ 508, 2004.-с. 58-64]

Signal Detection  Signal Detection   [Котельніков, Cameron, Martin, Middelton, Peterson, Siegert, Jacobs, Wald, Woodward,

Wozencraft]

Threshold for detection at a given - fault probability

Ф (·) - standard function, , - dispersion and expectation for signal 

Probability of detection

:QDL 2 ),t(sD )(tQ

2

2200

2

21

2 2

0 2L

L

)s,s(

),s()x(

L

L es

),s()Hx(l

01

0 1 M)P(Vv f

'' V)Mv(Pd 1

(1)

(2)

(3) 2

1L

)x,x(T

,

(4)

fP

0V 0M

Signal Signal RepresentationRepresentation in in [J.Fourier-Н.А. Колмогоров-N.Wiener-Karhunen-Loév-E.Parzen]

)t;()t(x kkk

2L)t(x)t(xminarg

2),(

Llk

lk

lkCdttt

T

lk ,0

,)()(

; : e

2L*kk

2ts

2ts ),x(,LRKHSxR ,LxU:

(5)

(6)

(7)

(8)

2L

2Lts )st(x),t(x)}t(x{R:A

)st(x}t(x{U:A ts

SHIFT operator

CORRELATION operator

22

Ltj )e,x(

The Narrow Band SignalThe Narrow Band SignalRepresentation (1.5)Representation (1.5)

t

Hz,

t

0 0.1 0.2 0.3 0.4 0.5 0.60

2000

4000

6000

8000

10000

12000

t

ss

The Signal with Known Spectrum The Signal with Known Spectrum Spreads, Representation (1.5)Spreads, Representation (1.5)

Schema of detection

Characteristics of Detection (1.4 ) Characteristics of Detection (1.4 ) of the Known Spread Spectrum Signalof the Known Spread Spectrum Signal

dP

0010.Pf

010.Pf

10.Pf

Representation (1.5) of the SignalRepresentation (1.5) of the Signalwith Unknown Spread Spectrumwith Unknown Spread Spectrum

a wide-band ADC of SSSa wide-band ADC of SSS

Characteristics (1.4) Characteristics (1.4) of Detection of the Spread Spectrum of Detection of the Spread Spectrum

Signal Signal

(a wide-band ADC of SSS)(a wide-band ADC of SSS)

dP

0010.Pf

010.Pf 10.Pf

SHIFT operator

CORRELATION operator

? 2L) ,x(

?

?

The Function Representation in the The Function Representation in the HSoHSHSoHS

— stochastic measure

— spectral measure

— probability measure

(D-ergodisity)

(K-isomorphism)

)d(Z)tjexp()t(x x )d(Z x

),()(exp))()((M),( ddFtsjststR

)(M)()( ttt

))d()d)d,d(F (M(

M )F,(),(

2L )P,(),( 2

0L

)d(P E )d(P

1)(t))(t(M(t)),(tPr tF,(L2

0

ξτξξτξ)

)F,( 2L KJ )P,( 20L

(14)

(9)

(10)

(11)

)),(),((E)(exp)),(),((E 2121 ddtsjst

(12)

(13)

Rigged Hilbert Space Rigged Hilbert Space with Reproduced Correlation Kernelwith Reproduced Correlation Kernel

Ordering of representations: S > O – [S.Vatanabe], S > C – [Ya.Dragan]

R

Conditions of ExistenceConditions of Existence [Vitali]:

[Розанов]:

[Драґан]:

2R

xxV )d,d(FFvar

xV},{

Fvarmaxarg

2R)(,)(

xR )d,d()()(Fvar sup11

xR)()()}(),({},{

Fvarmaxarg

Dvarmaxarg,

Dvar FvarFvar FV HDC HRHL CC

(15)dt)t(x L

lim)R(DvarK

L

-LL

2

21

,

The Likelihood Ratio and Detection The Likelihood Ratio and Detection Test Test StatisticStatistic

— RHS with RKHS as an one of rigging spaces is over Hilbert space

K - frequencies components

)d,d(2

)q,d()x(2

1

0

200

ed2

)q,d()Hx(l

(16)

(17) d)(SK k

K

k

1

1

11M

1k

2

0

k2 )()M

k1(M

k

M

k M

kV

1

1

)1(2

     - spectral density; ,   - number of spectral

components

Energy is concentrate on , - spectral band of SSS

k M,1k M

21 Mk /12

Methods & EquationsMethods & Equations

- an optimal estimation of spectra , ; by method with parameter

Zk

tT

jk

kx euBu,tb2

dueuBSR

jukk

21

)Svar( maxarg ),n,i(x

k,j

),n,i(xS I,i 1

N,1n i

(18)

(19)

GenerationGeneration of the Indexesof the Indexes

R  — cycle shift register of indexing (m-sequence), М – period of correlation (SSS epoch), N – quantity of correlation components (is determined by relation between periods of spectra harmonics and hops)

)(t

MNT

1 0 0 0 0 0

1 2 3 M

0 1 0 0 0 0

……….

0 0 0 0 0 1

АЦП

1R

2R

NR NMT

MT2

MT1

n

NMdT

ADC)t(x

Computation Computation of of The ExpectationThe Expectation

(а) — component’s (b) — process

Algorithm of Befitting DetectionAlgorithm of Befitting Detection

6. SA

of stationary components

9. Statistics

of detection

7. VAR

of stationary components

8. Adaptation?

2. ADC

5

1 Receiving

& settings

3 . Determine of parameters

ADC & SA

10. Detection

4

No Yes

Results of Befitting ComputationResults of Befitting Computationof spectral componentsof spectral components

0102030405060708090

100

1 2 3 4 5

)}k;(S{var xD 21 43 5

Caracteristics (1.4) of Befitting Caracteristics (1.4) of Befitting

DetectionDetection

0010.Pf 010.Pf

10.Pf dP

ConclusionConclusionDetectionof s(t) in ADC of x(t)

Spectra of x(t)

Basesfunction

Eigen functionof common Shift operator

Eigen function of operator for spectra Spreading

Conditionsof existence

22

Ltj )e,x(

),( 2 tjex

2)()(minarg,

Ltxtx

)ˆ(var maxarg ),,(

,,

ni

xDni

S

?

?