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Advanced signal processing Dr. Mohamad KAHLIL Islamic University of Lebanon

Advanced signal processing Dr. Mohamad KAHLIL Islamic University of Lebanon. Outline. Random variables Histogram, Mean, Variances, Moments, Correlation, types, multiple random variables Random functions Correlation, stationarity, spectral density estimation methods

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Advanced signal processing Dr. Mohamad KAHLIL Islamic University of Lebanon

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  1. Advanced signal processing Dr. Mohamad KAHLIL Islamic University of Lebanon

  2. Outline • Random variables • Histogram, Mean, Variances, Moments, Correlation, types, multiple random variables • Random functions • Correlation, stationarity, spectral density estimation methods • Signal modeling: AR, MA, ARMA, • Detection and classification in signals • Advanced applications on signal processing: • Time frequency and wavelet

  3. Chapter 4: detection and classification in random signals • Detection • Definition • Statistical tests for detection • Likelihood ratio • Example of detection when change in mean • Example of detection when change in variances • Multidimensional detection

  4. Detection: definition Hypotheses : estimated Known or unknown

  5. Gaussian distributions Normal distributions

  6. Chi2 distributions Loi du Chi 2 (Khi-two of Pearson) 10 dof 15 dof  chi2 with k degree of freedom E[chi2]=k Variance of Chi2=2k

  7. Fisher Test Student distribution F(6,7) F(6,10) Student with k degree of freedom Fisher-Snédécor Distribution Fisher with k and l degree of freedom Example: Detection in signals

  8. Detection: definition Hypotheses : estimated Known or unknown

  9. Parameters definition • False alarm • Detect H1, H0 is correct • Detection • Detect H1, H1 is correct • Miss detection • Detect H0, H1 is correct

  10. Likelihood ratio • Detection in signals

  11. Variation in mean • Detection in mean H0: z(t) = 0 + b(t) = b(t) H1: z(t) = m + b(t)

  12. Detection in variance • Detection in variance

  13. Parameters • False Alarm probability • Detection probability

  14. Parameters

  15. Neymen pearson method • Fix the probability of false alarm • Estimate the threshold

  16. Detection: multidimensional case • Multidimensional case

  17. Distribution de Fisher-Snedecora = 0,05

  18. DISTRIBUTION DU KHI-DEUX

  19. DISTRIBUTION DU KHI-DEUX (suite)

  20. LOI NORMALE CENTRÉE RÉDUITE

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