Lecture Notes 6 Random Processes • Definition and Simple Examples • Important Classes of Random Processes ◦ IID ◦ Random Walk Process ◦ Markov Processes ◦ Independent Increment Processes ◦ Counting processes and Poisson Process • Mean and Autocorrelation Function • Gaussian Random Processes ◦ Gauss–Markov Process EE 278: Random Processes Page 6–1 Random Process • A random process (RP) (or stochastic process) is an infinite indexed collection of random variables {X(t): t ∈ T }, defined over a common probability space • The index parameter t is typically time, but can also be a spatial dimension • Random processes are used to model random experiments that evolve in time: ◦ Received sequence/waveform at the output of a communication channel ◦ Packet arrival times at a node in a communication network ◦ Thermal noise in a resistor ◦ Scores of an NBA team in consecutive games ◦ Daily price of a stock ◦ Winnings or losses of a gambler EE 278: Random Processes Page 6–2 Questions Involving Random Processes • Dependencies of the random variables of the process ◦ How do future received values depend on past received values? ◦ How do future prices of a stock depend on its past values? • Long term averages ◦ What is the proportion of time a queue is empty? ◦ What is the average noise power at the output of a circuit? • Extreme or boundary events ◦ What is the probability that a link in a communication network is congested? ◦ What is the probability that the maximum power in a power distribution line is exceeded? ◦ What is the probability that a gambler will lose all his captial? • Estimation/detection of a signal from a noisy waveform EE 278: Random Processes Page 6–3 Two Ways to View a Random Process • 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 ◦ For fixed t, X(t, ω) is a random variable over Ω ◦ For fixed ω, X(t, ω) is a deterministic function of t, called a sample function X(t, w1) t X(t, w2) t X(t, w3) t t1 t2 X(t1, w) X(t2, w) EE 278: Random Processes Page 6–4 Discrete Time Random Process • A random process is said to be discrete time if T is a countably infinite set, e.g., ◦ N = {0, 1, 2,...} ◦ Z = {..., −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 called a sample sequence or sample path • A discrete-time process can comprise discrete, continuous, or mixed r.v.s EE 278: Random Processes Page 6–5 Example n • Let Z ∼ U[0, 1], and define the discrete time process Xn = Z for n ≥ 1 • Sample paths: x 1 n 2 1 1 Z = 2 4 1 8 1 16 n xn 1 1 Z = 4 4 1 16 1 64 n xn Z = 0 00 0 ... n 1234567 ... EE 278: Random Processes Page 6–6 n • First-order pdf of the process: For each n, Xn = Z is a r.v.; the sequence of pdfs of Xn 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 1 1 1 f (x)= = xn−1 , 0 ≤ x ≤ 1 Xn nx(n−1)/n n EE 278: Random Processes Page 6–7 Continuous Time Random Process • A random process is continuous time if T is a continuous set • Example: Sinusoidal Signal with Random Phase X(t)= α cos(ωt + Θ) , t ≥ 0 where Θ ∼ U[0, 2π] and α and ω are constants • Sample functions: x(t) α Θ = 0 π π 3π 2π t 2ω ω 2ω ω x(t) Θ = π 4 t x(t) Θ = π 2 t EE 278: Random Processes Page 6–8 • The first-order pdf of the process is the pdf of X(t)= α cos(ωt + Θ). In an earlier homework exercise, we found it to be 1 fX(t)(x)= , −α<x< +α απ 1 − (x/α)2 The graph of the pdf is shownp below fX(t)(x) 1 απ x −α 0 +α Note that the pdf is independent of t. (The process is stationary) EE 278: Random Processes Page 6–9 Specifying a Random Process • In the above examples we specified the random process by describing the set of sample functions (sequences, paths) and explicitly providing a probability measure over the set of events (subsets of sample functions) • This way of specifying a random process has very limited applicability, and is suited only for very simple processes • A random process is typically specified (directly or indirectly) by specifying all its n-th order cdfs (pdfs, pmfs), i.e., the joint cdf (pdf, pmf) of the samples X(t1),X(t2),...,X(tn) for every order n and for every set of n points t1, t2,...,tn ∈T • The following examples of important random processes will be specified (directly or indirectly) in this manner EE 278: Random Processes Page 6–10 Important Classes of Random Processes • IID process: {Xn : n ∈ N } is an IID process if the r.v.s Xn are i.i.d. Examples: ◦ Bernoulli process: X1,X2,...,Xn,... i.i.d. ∼ Bern(p) ◦ Discrete-time white Gaussian noise (WGN): X1,...,Xn,... i.i.d. ∼ N (0, N) • Here we specified the n-th order pmfs (pdfs) of the processes by specifying the first-order pmf (pdf) and stating that the r.v.s are independent • It would be quite difficult to provide the specifications for an IID process by specifying the probability measure over the subsets of the sample space EE 278: Random Processes Page 6–11 The Random Walk Process • Let Z1,Z2,...,Zn,... be i.i.d., where 1 +1 with probability 2 Zn = 1 (−1 with probability 2 • The random walk process is defined by X0 = 0 n Xn = Zi , n ≥ 1 i=1 X • Again this process is specified by (indirectly) specifying all n-th order pmfs • Sample path: The sample path for a random walk is a sequence of integers, e.g., 0, +1, 0, −1, −2, −3, −4,... or 0, +1, +2, +3, +4, +3, +4, +3, +4,... EE 278: Random Processes Page 6–12 Example: xn 3 2 1 n 1 2 3 4 5 6 7 8 9 10 −1 −2 −3 zn : 1 1 1 −1 1 −1 −1 −1 −1 −1 • First-order pmf: The first-order pmf is P{Xn = k} as a function of n. Note that k ∈ {−n, −(n − 2),..., −2, 0, +2,..., +(n − 2), +n} for n even k ∈ {−n, −(n − 2),..., −1, +1, +3,..., +(n − 2), +n} for n odd Hence, P{Xn = k} = 0 if n + k is odd, or if k < −n or k > n EE 278: Random Processes Page 6–13 Now for n + k even, let a be the number of +1’s in n steps, then the number of −1’s is n − a, and we find that n + k k = a − (n − a) = 2a − n ⇒ a = 2 Thus 1 P{Xn = k} = P{2(n + k) heads in n independent coin tosses} n −n = n+k · 2 for n + k even and −n ≤ k ≤ n 2 P{Xn = k}, n even kk ... −6 −4 −2 0 +2 +4 +6 ... EE 278: Random Processes Page 6–14 Markov Processes • A discrete-time random process Xn is said to be a Markov process if the process future and past are conditionally independent given its present value • Mathematically this can be rephrased in several ways. For example, if the r.v.s {Xn : n ≥ 1} are discrete, then the process is Markov iff n−1 n x pXn+1|X (xn+1|xn, )= pXn+1|Xn(xn+1|xn) for every n • IID processes are Markov • The random walk process is Markov. To see this consider n n n n P{Xn+1 = xn+1 | X = x } = P{Xn + Zn+1 = xn+1 | X = x } = P{Xn + Zn+1 = xn+1 | Xn = xn} = P{Xn+1 = xn+1 | Xn = xn} EE 278: Random Processes Page 6–15 Independent Increment Processes • A discrete-time random process {Xn : n ≥ 0} is said to be independent increment if the increment random variables Xn1, Xn2 − Xn1, ..., Xnk − Xnk−1 are independent for all sequences of indices such that n1 < n2 < ··· < nk • Example: Random walk is an independent increment process because n1 n2 nk Xn1 = Zi , Xn2−Xn1 = Zi , ..., Xnk−Xnk−1 = Zi i=1 i=n +1 i=n +1 X X1 Xk−1 are independent because they are functions of independent random vectors • The independent increment property makes it easy to find the n-th order pmfs of a random walk process from knowledge only of the first-order pmf EE 278: Random Processes Page 6–16 • Example: Find P{X5 = 3, X10 = 6, X20 = 10} for random walk process {Xn} Solution: We use the independent increment property as follows P{X5 = 3, X10 = 6, X20 = 10} = P{X5 = 3, X10 − X5 = 3, X20 − X10 = 4} = P{X5 = 3}P{X5 = 3}P{X10 = 4} 5 5 10 = 2−5 2−5 2−10 = 3000 · 2−20 4 4 7 • In general if a process is independent increment, then it is also Markov. To see this let Xn be an independent increment process and define n T ∆X =[X1, X2 − X1,...,Xn − Xn−1] Then n n n n x X x pXn+1|X (xn+1 | ) = P{Xn+1 = xn+1 | = } n n = P{Xn+1 − Xn + Xn = xn+1 | ∆X = ∆x , Xn = xn} = P{Xn+1 = xn+1 | Xn = xn} • The converse is not necessarily true, e.g., IID processes are Markov but not independent increment EE 278: Random Processes Page 6–17 • The independent increment property can be extended to continuous-time processes: A process X(t), t ≥ 0, is said to be independent increment if X(t1), X(t2) − X(t1),..., X(tk) − X(tk−1) are independent for every 0 ≤ t1 < t2 <...<tk and every k ≥ 2 • Markovity can also be extended to continuous-time processes: A process X(t) is said to be Markov if X(tk+1) and (X(t1),...,X(tk−1)) are conditionally independent given X(tk) for every 0 ≤ t1 < t2 <...<tk < tk+1 and every k ≥ 2 EE 278: Random Processes Page 6–18 Counting Processes and Poisson Process • A continuous-time random process N(t), t ≥ 0, is said to be a counting process if N(0) = 0 and N(t)= n, n ∈ {0, 1, 2,...}, is the number of events from 0 to t (hence N(t2) ≥ N(t1) for every t2 > t1 ≥ 0) Sample path of a counting process: n(t) 5 4 3 2 1 0 t 0 t1 t2 t3 t4 t5 t1, t2,..