tutorial 4

ELEG3410 Random Process & DSP Tutorial # 4 Random Variable: Definition Probability Density Function and Probability Distribution Function Moment Transformation of Variables Central Limit Theorem Two Dimensional Distribution Stochastic Process / Random Process The outcome of a trial (throwing a dice) has only one value (1, 2, …, 6). In a random process, the outcome is a function of time t. For example, if the random process is defined as the noise voltages of telephone lines in EWB, then for each telephone line, there will be a noise voltage output which depends on time t. 1 For the case above, “statistically determined” means if we are given any time t and noise voltage x, we know the probability of the noise of all telephone lines which is lower than x. st1 order p.d.f. nd2 order joint p.d.f. thn order joint p.d.f.: st1 Moment: ∞µ()t = E[]x()t = xp()x,t dx ∫ − ∞Autocorrelation: ∞ ∞R()t ,t = E[]x(t)x(t)= x x p()x , x ;t ,t dx dx1 2 1 2 1 2 1 2 1 2 1 2∫∫ − ∞ − ∞Autocovariance: 2C()t ,t = E[]()x(t)− µ (t ) (x (t ) − µ (t ) )1 2 1 1 2 2 = Rt ,t − µ(t)µ(t)1 2 1 2 Variance: ∞22σ ()t =()xt − µt p(x,t)dx = C(t,t)∫ − ∞ X (t ) = A cos (2 πt )E.g.: Let , where A is some random variable. Find the mean, autocorrelation, and autocovariance of X(t). Mean: µ()t = E[]Acos(2 πt) = E [A ]cos (2 πt ) Autocorrelation: ()()()R t ,t = E X t X t1 2 1 2= E[]Acos()2 πt Acos(2 πt) 1 22= EA cos2 ...