Advanced signal processing Dr. Mohamad KAHLIL Islamic University of Lebanon

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. Marks • Partial : 20/100 • About April 29, 2008 • Lab advanced: 30/100 • Final exam: 50/100 • About June 25,2008

4. Chapter 1: Random variables • Random variables • Probability • Histogram or probability density function • Cumulative function • Mean • Variance • Moments • Some representations of random variables • Bi-dimensional random variables • Marginal distributions • Independence • Correlations • Gaussian expression of multiple random variables • Changing random variables

5. Introduction signal = every entity which contains some physical information Examples: Acoustic waves Music, speech, ... Electric current given by a microphone Light source (star, …) ... Light waves Current given by a spectrometer Number series Physical measurements ... Photography

7. Signal processing = procedure used to:  extract the information (filtering, detection, estimation, spectral analysis...)  Adapt the signal (modulation, sampling….) (to transmit it or save it)  pattern recognition In physics: TS Physical system signal Analysis Transmission Detection  interpretation Noise source

8. Exemples: Astronomy: Electromagnetic waves  information concerning stars • Sig. Process.: •  sampling • filtering •  spectrale analysis • ... signal V(t) Atmosphere  noise Transmitted light Signal processing:  Spectral analysis  Synchronous detection ... I(t) Light rays incident detector Sample test Periodic opening

9. Classification of signals : Number of free variables. Dimensional classification : Examples : Electrical potential V(t) = Unidimensional signal Statistic image black and white  brightness B(x,y) = bi-dimensional signal Black and white film  B(x,y,t) = tri-dimensional signal ...  The signal theory is independent on the physic phenomenon and the types of variables. Phenomenological Classification Random or deterministic evolution Deterministic signal : temporal evolution can be predicted or modeled by an appropriate mathematical mode Random signal : the signal cannot be predicted statistical description  Every signal has a random component (external perturbation, …)

10. Morphological classification: [Fig.2.10,(I)]

13. Probability

14. Probability If two events A and B occurs, P(B/A) is the conditional probability If A and B are independent, P(A,B)=P(A).P(B)

15. Random variable and random process • 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

16. Random variable and random process • The random process is represented as a function • Each signal x(t), for each sensor, is a random signal. • At an instant t, all values at this time define a random variable z1 t t1 t2 z2 Signals obtained when measuring temperature using many sensors z3

17. Probability density function (PDF) • The characteristics of a random process or a random variable can be interpreted from the histogram N(m, ti) = number of events: "xi = x +Dx" Precision of measurment N(m) x Nmes= total number of measurments (m+1)Dx m Dx

18. PDF properties • Id Δx=dx (trop petit) so, the histogram becomes continuous. In this case we can write:

19. Histogram or PDF • Sine wave : Random signal f(x) x -1 1 Uniform PDF

20. Cumulative density function

21. examples

22. Expectation, variance Every function of a random variable is a random variable. If we know the probability distribution of a RV, we can deduce the expectation value of the function of a random variable: Statistical parameters : Average value : Mean quadratic value: Variance : Standard deviation :

23. Moments of higher order • The definition of the moment of order r is: • The definition of the characteristic function is: We can demonstrate:

24. Exponential random variable

25. Uniform random variable f(x) c x a b

29. Gaussian random variable

35. f(x) a c b F(x) a c b x Triangular random variable

36. f(x) a c b F(x) a c b x Triangular random variable

37. Bi-dimensional random variable • Two random variables X and Y have a common probability density functions as : • (X,Y) fXY(x,y) is the probability density function of the couple (X,Y) • Example:

39. Bi-dimensional Random variables • Cumulative functions: • Marginal cumulative distribution functions • Marginal probability density functions

42. Bi-dimensional Random variables • Moments of a random variable X If X and Y are independents and in this case

46. Covariance

47. Covariance

48. Correlation coefficient

49. Correlation coefficient

50. Correlation coefficient