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U K R A I N E. Ternopil’ State Technical University named after Ivan Pul’ui. International Conference on Inductive Modelling 2008, Kyiv. National University “Lvivs’ka Politechnica”. Reconstruction of Algorithms for Spread Spectrum Signals Detection into a Frame of Inductive Modeling Methods.

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U k r a i n e

U K R A I N E

Ternopil’ State Technical University named after Ivan Pul’ui

International Conference onInductive Modelling 2008, Kyiv

National University “Lvivs’ka Politechnica”


Ternopil state technical university named after ivan pul ui

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

Bohdan Yavorskyy, Yaroslav Dragan, Lubomyr Sicora

[email protected]


Ternopil state technical university named after ivan pul ui

Can we to explainby an Inductive Modeling Methoda succesful detectionof a Spread Spectrum Signalwith unknown spectrum spreads?


Ternopil state technical university named after ivan pul ui

Spread Spectrum Signal after wide-band ADC

Signal to Noise Ratio (SNR)


Introduction backgrounds

Introduction backgrounds

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

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

  • A some problems in signal detection applications were designed [Oya A., 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-July 5, p. 214, 2002]

  • Detection methods for either stationary Gaussian noise of known 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 developed[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.-2002]

  • 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 RKHS method was declined in RHS in a Hilbert space over Hilbert space (HSoHS) [Yavorskyy B. Vyyavlennya skladnyh syhnaliv z nevidomymy parametramy v radiovyprominyuvannyah// Radioelektronica ta telecomunicatsii.- № 508, 2004.-с. 58-64]


Signal detection 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

Probabilityof detection

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

,

(1)

(2)

(3)

(4)


Signal representation in j fourier n wiener karhunen lo v e parzen

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

(5)

(6)

(7)

(8)


Ternopil state technical university named after ivan pul ui

SHIFT operator

CORRELATION operator


The narrow band signal representation 1 5

The Narrow Band SignalRepresentation (1.5)

s

s


The signal with known spectrum spreads representation 1 5

The Signal with Known Spectrum Spreads, Representation (1.5)

Schema of detection


Characteristics of detection 1 4 of the known spread spectrum signal

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


Representation 1 5 of the signal with unknown spread spectrum

Representation (1.5) of the Signalwith Unknown Spread Spectrum

a wide-band ADC of SSS


Characteristics 1 4 of detection of the spread spectrum signal

Characteristics (1.4) of Detection of the Spread Spectrum Signal

(a wide-band ADC of SSS)


Ternopil state technical university named after ivan pul ui

SHIFT operator

?

CORRELATION operator

?

?


The function representation in the hsohs

The Function Representation in the HSoHS

— stochastic measure

— spectral measure

— probability measure

(D-ergodisity)

(K-isomorphism)

(9)

(10)

(11)

(12)

(13)

(14)


Rigged hilbert space with reproduced correlation kernel

Rigged Hilbert Space with Reproduced Correlation Kernel

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


Conditions of existence

Conditions of Existence

[Vitali]:

[Розанов]:

[Драґан]:

,

(15)


The likelihood ratio and detection test statistic

The Likelihood Ratio and Detection Test Statistic

(16)

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

K-frequencies components

(17)


Spectral density number of spectral components energy is concentrate on spectr al band of sss

- spectral density; , - numberof spectral componentsEnergyis concentrate on ,-spectral band of SSS


Methods equations

Methods & Equations

(18)

- an optimal estimation of spectra , ;

by method with parameter

(19)


Generation of the indexes

Generationof the Indexes

ADC

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)


Computation of the expectation

Computation of The Expectation

(а) — component’s (b) — process


Algorithm of befitting detection

Algorithm of Befitting Detection


Ternopil state technical university named after ivan pul ui

Results of Befitting Computation

of spectral components


Caracteristics 1 4 of befitting detection

Caracteristics (1.4) of Befitting Detection


Ternopil state technical university named after ivan pul ui

Conclusion

Detection

of s(t) in

ADC of x(t)

Eigen function of operator

for spectra Spreading

Spectra

of x(t)

Bases

function

Eigen function

of common

Shift operator

Conditions

of existence

Inner

product

functional

?

?


Ternopil state technical university named after ivan pul ui

Thank You


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