1. Stochastic Signal Processing PRESENTED BY ILA SHARMA

2. OUTLINE 4/22/2011 12:01:10 AM 2 Introduction to probability Random variables Moments of random variables Stochastic or Random processes Basic types of Stochastic Processes

3. PROBABILITY THEORY 4/22/2011 12:01:10 AM 3 Probability theory begins with the concept of a probability space, which is a collection of three items(Ω,F, P); Ω = Sample space F = Event space or field F, P = Probability measure. This (Ω,F, P) is collectively called a probability space or an experiment.

4. AXIOMATIC DEFINATION OF PROBABILITY 4/22/2011 12:01:10 AM 4 Given a sample space Ω, and a field F of events defined on Ω, we define probability Pr[.] as a measure on each event E belongs to F, such that: Pr[E]>= 0, Pr[Ω] = 1, Pr[E U F] = Pr[E] + Pr[F], if EF = Ø.

5. RANDOM VARIABLE 4/22/2011 12:01:10 AM 5 A Real Random Variable X(.) is a mapping from sample space(Ω) to the real line, which assigns a number X(ç) to every outcome ç belongs to sample space(Ω).

6. MEAN AND VARIANCE 4/22/2011 12:01:10 AM 6 The expected value (or mean) of an RV is defined as: The variance of an RV X is defined as:

7. VARIANCE AND CORRELATION 4/22/2011 12:01:11 AM 7 The variance of an RV X is defined as: We can define the covariance between two random variables as:

8. CONTINUED………… 4/22/2011 12:01:11 AM 8 For a discrete random variable representing the samples of a time series, we can estimate this directly from the signal as: Two random variables are said to be uncorrelated if

9. RANDOM PROCESS 4/22/2011 12:01:12 AM 9

10. AUTO CORRELATION FUNCTION 4/22/2011 12:01:12 AM 10

11. BASIC TYPES OF RANDOM PROCESS 4/22/2011 12:01:12 AM 11 GAUSSIAN PROCESS MARKOV PROCESS STATIONARY PROCESS WHITE PROCESS

12. GAUSSIAN PROCESS 4/22/2011 12:01:12 AM 12 A random process X(t) is a Gaussian process if for all n and for all , the random variables has a jointly Gaussian density function, which may expressed as Where -> : n random variables : mean value vector : nxn covariance matrix

13. MARKOV PROCESS 4/22/2011 12:01:12 AM 13 Markov process X(t) is a random process whose past has no influence on the future if its present is specified. If , then Or if

14. STATIONARY PROCESS 4/22/2011 12:01:12 AM 14 Definition of Autocorrelation Where X(t1),X(t2) are random variables obtained at t1,t2 Definition of stationary A random process is said to stationary, if its mean(m) and covariance(C) do not vary with a shift in the time origin A process is stationary if

15. WHITE PROCESS 4/22/2011 12:01:12 AM 15 A random process X(t) is called a white process if it has a flat power spectrum. If Sx(f) is constant for all f It closely represent thermal noise Sx(f) f The area is infinite (Infinite power !)

16. REFERENCES 4/22/2011 12:01:12 AM 16 Stark & Woods : Probability and Random Processes with Applications to Signal Processing, Chapters 1-3 &7. Edward R. Dougherty : Random process for image and signal processing, Chapters 1-2. T. Chonavel : Stochastic signal processing. Robert M. Gray & Lee D. Davisson: An Introduction to Statistical Signal Processing.

17. THANK YOU 4/22/2011 12:01:12 AM 17