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EEE461Pres8

# EEE461Pres8 - Lecture Notes 8 Random Processes EE278 Prof B...

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Unformatted text preview: Lecture Notes 8 Random Processes EE278 Prof. B. Prabhakar Statistical Signal Processing Autumn 02-03 • Deﬁnition and simple examples • Discrete Time Random Processes – IID – Random Walk Process • Mean and Autocorrelation Function • Gaussian Random Processes Copyright c 2000–2002 Abbas El Gamal 8-1 EE 278: Random Processes Random Process • A random prccess (also called stochastic process) is an inﬁnite collection of random variables {X (t) : t ∈ T }, deﬁned over a common probability space • The parameter t is typically time, but can also be a spatial dimension, etc • A random process can be viewed as a function X (t, ω ) of two variables, time t ∈ T and the outcome of the underlying random experiment ω ∈ Ω • Thus – For a ﬁxed t: X (t, ω ) is a r.v. – For a ﬁxed ω : X (t, ω ) is a deterministic function of t, denoted by a sample function EE 278: Random Processes 8-2 • Sample functions: X (t, w1) t X (t, w2) t X (t, w3) t1 t2 X (t1, w) X (t2, w) t 8-3 EE 278: Random Processes Discrete Time Random Process • A random process is said to be discrete time if T is a countably inﬁnite set, e.g., {1, 2, . . .}, {. . . , −2, −1, 0, 1, 2, . . .} • In this case the process is denoted by Xn, for n ∈ N , a countably inﬁnite set, and is simply an inﬁnite sequence of random variables • A sample function for a discrete time process is denoted by sample sequence or sample path EE 278: Random Processes 8-4 Example • Let Z ∼ U[0, 1], and deﬁne the discrete time process Xn = Z n, for n ≥ 1 • Sample paths: 1 2 xn Z= 1 2 xn Z= 1 4 1 4 1 4 1 16 1 8 1 16 n 1 64 n xn Z=0 0 1 0 2 0 ... 345 6 7 ... n 8-5 EE 278: Random Processes • First order pdf of the process: For any ﬁxed n, Xn = Z n is a r.v., its pdf is called the ﬁrst order pdf of the process xn 1 z 0 1 Since Xn is a diﬀerentiable function of the continuous r.v. Z , we can ﬁnd its pdf as 11 fXn (x) = x n −1, for 0 ≤ x ≤ 1 n EE 278: Random Processes 8-6 Continuous Time Random Process • A random process is continuous time if T is uncountably inﬁnite • Example: Consider the sinusoidal signal with random phase X (t) = α cos(ωt + Θ), for t ≥ 0, where α and ω are constant and Θ ∼ U[0, 2π ] • Sample functions: x(t) α Θ=0 π 2ω π ω 3π 2ω 2π ω t x(t) Θ= π 4 t x(t) Θ= π 2 t 8-7 EE 278: Random Processes • The ﬁrst order pdf of the process is the pdf of X (t) = α cos(ωt + Θ), for a constant t, and is given by fX (t)(x) = 1 , for x ∈ [−α, α] x 1 − ( α )2 απ fX (t)(x) 1 απ −α EE 278: Random Processes 0 α x 8-8 ...
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