1. 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

2. 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

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

4. 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: Physical system signal TransmissionDetection Analysis Noise source TS interpretation 

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

6. Classification of signals : Dimensional classification : Number of free variables. : 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... 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  The signal theory is independent on the physic phenomenon and the types of variables.  Every signal has a random component (external perturbation, …)

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

10. Probability

11. 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)

12. 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 t 11 22 33 t1t1 t2t2 Signals obtained when measuring temperature using many sensors

13. 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 t 11 22 33 t1t1 t2t2 Signals obtained when measuring temperature using many sensors

14. N(m, t i ) = number of events: "x i = x +  x" x m  x (m+1)  x N(m) Precision of measurment N mes = total number of measurments Probability density function (PDF) The characteristics of a random process or a random variable can be interpreted from the histogram

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

16. Histogram or PDF Random signal 1 f(x) x Sine wave : Uniform PDF

17. Cumulative density function

18. examples

19. Statistical parameters : Average value : Mean quadratic value: Variance : Standard deviation : 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:

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

21. Exponential random variable

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

26. Gaussian random variable

32. Triangular random variable ab f(x) c ab F(x) c x

33. Triangular random variable ab f(x) c ab F(x) c x

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

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

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

43. Covariance

45. Correlation coefficient

55. PDF of a transformed RV Suppose X is a continuous RV with known PDF Y=h(X) a function of the RV X What is the PDF of Y?

58. PDF of a transformed RV: exercises X is a uniform random variable between -2 and 2. –Write the expression on pdf of X –Find the PDF of Y=5X+9

59. Exercise Let us consider the bidimensional RV: 1.Find c 2.Compute the CDF of f(x,y) 3.Compute G X (x) and G Y (y) 4.Compute the moments of order 1 and 2

60. Sum of 2 RVs