Ternopil State Technical University named after Ivan Pului

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Ternopil State Technical University named after
Ivan Pului
International Conference on Inductive Modelling
2008, Kyiv
National University Lvivska Politechnica

• U K R A I N E

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

• Bohdan Yavorskyy, Yaroslav Dragan, Lubomyr
  Sicora
• 
  
• Kaf_BT_at_tu.edu.te.ua

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Can we to explainby an Inductive Modeling
Methoda succesful detection of a Spread
Spectrum Signalwith unknown spectrum spreads
?
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Spread Spectrum Signal after wide-band ADC
Signal to Noise Ratio (SNR)
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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.-?. 58-64

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Signal Detection   ???????????,
Cameron, Martin, Middelton, Peterson, Siegert,
Jacobs, Wald, Woodward, Wozencraft
,
(1)

• Threshold for detection at a given -
  fault probability
• ? () - standard function, , -
  dispersion and expectation for signal
•  
• Probability of detection

(2)
(3)
(4)
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Signal Representation in J.Fourier-?.?. ??????
????-N.Wiener-Karhunen-Loév-E.Parzen
(5)
(6)
(7)
(8)
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SHIFT operator
CORRELATION operator
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The Narrow Band SignalRepresentation (1.5)
s
s
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The Signal with Known Spectrum Spreads,
Representation (1.5)
Schema of detection
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Characteristics of Detection (1.4 ) of the Known
Spread Spectrum Signal
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Representation (1.5) of the Signalwith Unknown
Spread Spectrum

• a wide-band ADC of SSS

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Characteristics (1.4) of Detection of the
Spread Spectrum Signal

• (a wide-band ADC of SSS)

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SHIFT operator
?
CORRELATION operator
?
?
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The Function Representation in the HSoHS

• stochastic measure
• spectral measure
• probability measure
• (D-ergodisity)
• (K-isomorphism)

(9)
(10)
(11)
(12)
(13)
(14)
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Rigged Hilbert Space with Reproduced Correlation
Kernel

• Ordering of representations S gt O 
  S.Vatanabe, S gt C  Ya.Dragan

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Conditions of Existence

• Vitali
• ???????
• ??????

,
(15)
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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)
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   - spectral density ,
  - number of spectral components
Energy is concentrate on ,
- spectral band of SSS
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Methods Equations
(18)
(19)

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

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Generation of the Indexes
ADC

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

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Computation of The Expectation

• (?)  components (b)  process

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Algorithm of Befitting Detection
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Results of Befitting Computation of spectral
components
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Caracteristics (1.4) of Befitting Detection
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• 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
?
?
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Thank You