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### Dsp presentation

1. 1. Stochastic Signal Processing<br /> PRESENTED BY ILA SHARMA<br />
2. 2. OUTLINE<br />4/22/2011 12:01:10 AM<br />2<br />Introduction to probability<br />Random variables<br />Moments of random variables<br /> Stochastic or Random processes<br />Basic types of Stochastic Processes<br />
3. 3. PROBABILITY THEORY<br />4/22/2011 12:01:10 AM<br />3<br />Probability theory begins with the concept of a probability space, which is a collection of three items(Ω,F, P);<br />Ω = Sample space F = Event space or field F,<br /> P = Probability measure.<br /> This (Ω,F, P) is collectively called a probability space or an experiment.<br />
4. 4. AXIOMATIC DEFINATION OF PROBABILITY<br />4/22/2011 12:01:10 AM<br />4<br />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:<br />Pr[E]>= 0,<br /> Pr[Ω] = 1,<br />Pr[E U F] = Pr[E] + Pr[F], if EF = Ø.<br />
5. 5. RANDOM VARIABLE<br />4/22/2011 12:01:10 AM<br />5<br />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).<br />
6. 6. MEAN AND VARIANCE<br />4/22/2011 12:01:10 AM<br />6<br />The expected value (or mean) of an RV is defined as:<br />The variance of an RV X is defined as:<br />
7. 7. VARIANCE AND CORRELATION <br />4/22/2011 12:01:11 AM<br />7<br />The variance of an RV X is defined as:<br />We can define the covariance between two random variables as:<br />
8. 8. CONTINUED…………<br />4/22/2011 12:01:11 AM<br />8<br />For a discrete random variable representing the samples of a time series, we can estimate this directly from the signal as:<br />Two random variables are said to be uncorrelated if <br />
9. 9. RANDOM PROCESS<br />4/22/2011 12:01:12 AM<br />9<br />
10. 10. AUTO CORRELATION FUNCTION<br />4/22/2011 12:01:12 AM<br />10<br />
11. 11. BASIC TYPES OF RANDOM PROCESS<br />4/22/2011 12:01:12 AM<br />11<br />GAUSSIAN PROCESS<br />MARKOV PROCESS<br />STATIONARY PROCESS<br />WHITE PROCESS<br />
12. 12. GAUSSIAN PROCESS<br />4/22/2011 12:01:12 AM<br />12<br />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<br />Where -> <br />: n random variables<br />: mean value vector<br />: nxn covariance matrix<br />
13. 13. MARKOV PROCESS<br />4/22/2011 12:01:12 AM<br />13<br />Markov process X(t) is a random process whose past has no influence on the future if its present is specified.<br />If , then<br />Or if <br />
14. 14. STATIONARY PROCESS<br />4/22/2011 12:01:12 AM<br />14<br />Definition of Autocorrelation<br />Where X(t1),X(t2) are random variables obtained at t1,t2<br />Definition of stationary<br />A random process is said to stationary, if its mean(m) and covariance(C) do not vary with a shift in the time origin<br />A process is stationary if<br />
15. 15. WHITE PROCESS<br />4/22/2011 12:01:12 AM<br />15<br />A random process X(t) is called a white process if it has a flat power spectrum.<br />If Sx(f) is constant for all f<br />It closely represent thermal noise<br />Sx(f)<br />f<br />The area is infinite<br />(Infinite power !)<br />
16. 16. REFERENCES<br />4/22/2011 12:01:12 AM<br />16<br />Stark & Woods : Probability and Random Processes with Applications to Signal Processing, Chapters 1-3 &7.<br />Edward R. Dougherty : Random process for image and signal processing, Chapters 1-2.<br />T. Chonavel : Stochastic signal processing.<br />Robert M. Gray & Lee D. Davisson: An Introduction to Statistical Signal Processing.<br />
17. 17. THANK YOU<br />4/22/2011 12:01:12 AM<br />17<br />