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10 Intro. to Random Processes

10 Intro. to Random Processes. A random process is a family of random variables – usually an infinite family; e.g., { X n , n=1,2,3,... }, { X n , n=0,1,2,... }, { X n , n=...,-3,-2,-1,0,1,2,3,... } or { X t , t ≥ 0 }, { X t , 0 ≤ t ≤ T }, { X t , -∞ < t < ∞ }.

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10 Intro. to Random Processes

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  1. 10 Intro. to Random Processes • A random process is a family of random variables – usually an infinite family; e.g., • { Xn, n=1,2,3,... }, { Xn, n=0,1,2,... }, • { Xn, n=...,-3,-2,-1,0,1,2,3,... } • or • { Xt, t ≥ 0 }, { Xt, 0 ≤ t ≤ T }, { Xt, -∞ < t < ∞ }.

  2. Recalling that a random variable is a function of the sample space Ω, note that Xn is really Xn(ω) and Xt is really Xt(ω). So, each time we change ω, the sequence of numbers Xn(ω) or the wave- form Xt(ω) changes... A particular sequence or waveform is called a realization, sample path, or sample function.

  3. Xn(ω) for different ω

  4. Zn(ω) for different ω

  5. 5sin(2πfn) + Zn(ω)for different ω

  6. Yn(ω) for different ω

  7. Xt(ω) = cos(2πft+Θ(ω)) for different ω

  8. Nt(ω) for different ω

  9. Brownian Motion = Wiener Process

  10. 10.2 Characterization of Random Process • Mean function • Correlation function

  11. Properties of Correlation Fcns • symmetry: RX(t1,t2)=RX(t1,t2)

  12. Properties of Correlation Fcns • symmetry: RX(t1,t2)=RX(t1,t2) since

  13. Properties of Correlation Fcns • symmetry: RX(t1,t2)=RX(t1,t2) since • RX(t,t) ≥ 0

  14. Properties of Correlation Fcns • symmetry: RX(t1,t2)=RX(t1,t2) since • RX(t,t) ≥ 0 since

  15. Properties of Correlation Fcns • symmetry: RX(t1,t2)=RX(t1,t2) since • RX(t,t) ≥ 0 since • Bound:

  16. Properties of Correlation Fcns • symmetry: RX(t1,t2)=RX(t1,t2) since • RX(t,t) ≥ 0 since • Bound: follows by Cauchy-Schwarz inequality:

  17. Second-Order Process • A process is second order if

  18. Second-Order Process • A process is second order if • Such a process has finite mean by the Cauchy-Schwarz inequality:

  19. How It Works • You can interchange expectation and integration. If • then

  20. How It Works • You can interchange expectation and integration. If • then

  21. Example 10.12 If then

  22. Similarly, and then

  23. SX(f) must be real and even:

  24. SX(f) must be real and even: integral of odd function between symmetic limits is zero.

  25. SX(f) must be real and even:

  26. SX(f) must be real and even: This is an even function of f.

  27. 10.4 WSS Processes through LTI Systems

  28. 10.4 WSS Processes through LTI Systems

  29. 10.4 WSS Processes through LTI Systems

  30. Recall What if Xt is WSS?

  31. Recall What if Xt is WSS? Then which depends only on the time difference!

  32. Since

  33. 10.5 Power Spectral Densities for WSS Processes

  34. 10.5 Power Spectral Densities for WSS Processes

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