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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, • Advanced applications on signal processing: • Time frequency and wavelet • Detection and classification in signals

  3. Chapter 2: Random functions • Definitions • Probability density functions • Cumulative density functions • Moments of a random functions • Covariance • Stationary process • Statistical auto and inter-correlation • Spectral density estimation • From autocorrelation • Bank filters method • Periodogram • White noise analysis

  4. Random functions • Let us consider the random process : measure the temperature in a room • Many measurements can be taken simultaneously using different sensors (same sensors, same environments…) and give different signals z1 t t1 t2 z2 Signals obtained when measuring temperature using many sensors z3

  5. PDF and CDF of Random Process • Probability density function is fX(x;t) or f(x1,x2,x3,…xn;t1,t2,…tn) • Cumulative density function • We can write

  6. Mean and correlations

  7. Correlation coefficient

  8. Stationarity

  9. Stationarity

  10. ergodic

  11. Ergodic

  12. Ergodic

  13. Ergodic

  14. White noise

  15. White noise effect

  16. Noise

  17. Autocorrelation properties

  18. Cross correlation properties

  19. Jointly Stationary Properties • Properties • Uncorrelated: • Orthogonal: • Independent: if x(t1) and y(t2) are independent (joint distribution is product of individual distributions)

  20. Cross correlation

  21. Spectral density

  22. Spectral density

  23. Spectral density

  24. Cross spectral density

  25. Simulation

  26. Simulation

  27. Simulation

  28. Simulation

  29. Filtering of random signals

  30. Filtering of random signals

  31. Filtering of random signals

  32. Filtering of random signals

  33. Filtering of random signals

  34. Filtering of random signals

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