10 Intro to Random Processes A random process

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 < ∞ }.

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 waveform Xt(ω) changes. . . A particular sequence or waveform is called a realization, sample path, or sample function.

Xn(ω) for different ω

Zn(ω) for different ω

5 sin(2πfn) + Zn(ω) for different ω

Yn(ω) for different ω

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

Nt(ω) for different ω

Brownian Motion = Wiener Process

10. 2 Characterization of Random Process Mean function Correlation function

Properties of Correlation Fcns symmetry: RX(t 1, t 2)=RX(t 1, t 2)

Properties of Correlation Fcns symmetry: RX(t 1, t 2)=RX(t 1, t 2) since

Properties of Correlation Fcns symmetry: RX(t 1, t 2)=RX(t 1, t 2) since RX(t, t) ≥ 0

Properties of Correlation Fcns symmetry: RX(t 1, t 2)=RX(t 1, t 2) since RX(t, t) ≥ 0 since

Properties of Correlation Fcns symmetry: RX(t 1, t 2)=RX(t 1, t 2) since RX(t, t) ≥ 0 since Bound:

Properties of Correlation Fcns symmetry: RX(t 1, t 2)=RX(t 1, t 2) since RX(t, t) ≥ 0 since Bound: follows by Cauchy-Schwarz inequality:

Second-Order Process A process is second order if

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

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

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

Example 10. 12 If then

Similarly, and then

SX(f) must be real and even:

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

SX(f) must be real and even:

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

10. 4 WSS Processes through LTI Systems

10. 4 WSS Processes through LTI Systems

10. 4 WSS Processes through LTI Systems

Recall What if Xt is WSS?

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

Since

10. 5 Power Spectral Densities for WSS Processes

10. 5 Power Spectral Densities for WSS Processes