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EcE 5013 Digital Signal Processing

EcE 5013 Digital Signal Processing. Random Signals. Consider signal as deterministic signal In pratical, many process as Random Signals Develop the analysis tools for random signals

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EcE 5013 Digital Signal Processing

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  1. EcE 5013 Digital Signal Processing

  2. Random Signals Consider signal as deterministic signal In pratical, many process as Random Signals Develop the analysis tools for random signals Having a good probabilities model, can solve various useful estimation problems. Eg. Remove noise and enhance the image quality

  3. Events and Probability • An outcome of the experiment • All possible outcomes of the experiment is called sample space • Set of sample space is called event • A probability measure is a function which assigns a probability

  4. Properties of probability • It is non-negative: • The probability of whole sample is one • P(Ω )=1 • It is countably additive

  5. Conditional Probability • Observe outcome of one event A is influenced by that of another event B • A may occur whenever B does • A never occur whenever B does

  6. Properties of conditional probability • 0 P(A\B) 1 • If ,then P(A/B)=0 • If , P( A / B) =1 • If A1, A2, …….. are mutually exclusive • Total probability theorem • If Ai are mutually exclusive and

  7. Statistical Independence • If , then the events A and B are said to be statistically independent • P( A\B) = P(A) if P(B) ≠ 0 • P(B\A) = P(B) if P(A) ≠ 0 • A1, A2,----------,An are statistically independent if

  8. Random Variables • A random variable is an assignment if a value to every possible outcome • A random variable is discrete if its range is finite . • Probability mass function pmf of a discrete random variable X is Px(X) = P (X=x) X is continuous random variable where fx(x) is probability density function of X (pdf of X)

  9. Continued • For discrete random variable • The cumulative distribution function cdf of X is

  10. Some properties of cumulative distribution function • FX(x) is non decreasing • FX(x) approach to 0 as x approach to • FX(x) approach to 1 as x approach to • If X is continuous random variable,

  11. Continued • When g(X) is continuous • When g(X) is discrete • Expected value of X

  12. continued • Second moment of x • Variance of X

  13. Two Random Variables • The joint cumulative distribution function for two random variables X and Y defined as • The joint probabilities density function

  14. Continued • Marginal pdf

  15. End of Lesson

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