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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 3: Signal modelling • Definition • AR modeling • Expression • Spectral density estimations • Coefficients calculation • MA modeling • Expression and spectral density estimation • ARMA modeling • Expression and spectral density estimation

  4. Numerical filtering • FIR Filter • IIR Filter

  5. Filter realization • Non recursive, using delay elements, multiplication, addition x(n) Z-1 Z-1 Z-1 Z-1 b1 b0 b2 b3 bM y(n) +

  6. Filter realization y(n) w(n) x(n) • Recursive realization + - Z-1 Z-1 Z-1 Z-1

  7. Example: Equivalent numerical RC filter R Analog Differential Equation x(t) y(t) C Numerical approximation Equation of differences N =1, M= 0, ao=RC+1, a1= -RC, bo=1 Filter realization Recursive equation Computer algorithm Numerical filter y(k) * x(k) x(k-1) x(k) G(z)=z-1  + bo /ao * z-1 -a1/ao  y(k-1)

  8. Modeling?

  9. AR Modeling • The aim is to represent a stochastic signal using a parametric model. An autoregressive signal of order p is written as indicated in the following equation: • A sample at instant n can be estimated from its p previous parameters. The difference between the estimated value and the original value is a white noise v(n). V(z)

  10. AR modeling • AR realization: generation of signal from white noise • Problem: Determine -Order -coefficients x(n) Z-1 Z-1 Z-1 Z-1 a1 a0=1 a2 a3 ap v(n) +

  11. AR model : Coefficients

  12. AR modeling

  13. Moving average (MA) model • A signal is MA modeled of order q when the signal can be written as: V(n) is a white noise Problem: Determine - structure of the filter -Order -coefficients

  14. MA model: realization v(n) Z-1 Z-1 Z-1 Z-1 a1 a0=1 a2 a3 ap x(n) +

  15. ARMA Model A signal is ARMA modeled (AutoRegressive-Moving Average), order p and q, if the signal can be written as: V(n) is a white noise Problem: Determine - structure of the filter -Order -coefficients

  16. ARMA realization • Model v(n) x(n) + - Z-1 Z-1 Z-1 Z-1

  17. Spectral Density Spectral density Properties : By inverse Fourier transform:   Energy Frequency distribution of the signal is independent of the phase of the signal (Arg[X(f)])  Not sensible to the delay of the signal

  18. Spectral density: Periodogram • A periodogram is a method used to compute the spectral density, using parts from the original signal x(t) t T       = Random Limitations: • Width and window used, • Duration of measures, number ofX(f,T)

  19. spectral density: periodogram

  20. G(f) fn Spectral density: Banc of filters Principle f1, B1 Multi Channel Display x(t) f2, B2 fn, Bn selectif filters Bn

  21. Spectral density: from autocorrelation Wiener-Khinchine theorem: For stationary signals : For ergodic signals: • Unique definition of the spectral density, if it is random or deterministic. Experiment: T finite x(t) FFT A/D  retard

  22. Spectral density: Spectrum analyzer Principle: xm(t) x(t) Selective filter B, fo Power measurements Display  Scanning Commanded Oscillator B |Xm(f)| > < -fo fo f

  23. Spectral density: After modelling

  24. Spectral density: After modulation

  25. Estimation of parameters of signals, Statistical parameters

  26. Estimation of parameters from spectral density Spectral moment formula 1- Power of the signal : M0 2- Mean frequency: MPF=M1/M0 3- Dissymmetry coefficient: CD

  27. Estimation of parameters from spectral density 4- Kurtosis (pate coefficients) 5- Median frequency Fmed: compose the surface under S(f) into 2 equals area 6- Peak of frequency 7-relative energy by frequency band

  28. Estimation of parameters from spectral density 8- Ratio H/L (High/Low): 9- Percentiles or fractiles fk: 10- Spectral Entropy H

  29. Estimation of parameters from spectral density

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