Unformatted text preview: Lecture Notes 8
Random Processes
EE278
Prof. B. Prabhakar Statistical Signal Processing
Autumn 0203 • 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 81 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 82 • Sample functions:
X (t, w1) t
X (t, w2) t
X (t, w3) t1
t2
X (t1, w) X (t2, w) t 83 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 84 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 85 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 86 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 87 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 88 ...
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 Winter '09
 Eggers
 Correlation, Signal Processing, Probability theory, Stochastic process, Random Processes, Abbas El Gamal

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