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Random Signals Pedro M. Q. Aguiar January 2012 Parts of this document are based on powerpoint slides by Jorge Salvador Marques. Motivation Many signals processed by computers can be considered as random. Examples: speech, audio, video, digital communication, medical, biological, and economic signals. speech ECG Is this a random signal? 0 x(n) = A cos (ωn+φ) Yes, for example if A, ω, or φ are random variables! finite length – random vectors random signals infinite length – random processes (stochastic processes) A finite length signal x1, x2 ,..., xn can be considered as n-dimensional vector x = [ x1, x2 ,..., xn ]T several realizations of x Full description x is characterized by the joint probability density function (pdf) p(x)=p(x1, x2, …, xn) p ( x) ≥ 0 , ∀x ∫ p( x )dx = 1 Rn Pr{x ∈ C} = ∫ p ( y )dy C If x is discrete, it is characterized by a joint probability function P(x)=P(x1, x2, …, xn) P( x) ≥ 0 , ∀x ∑ P ( x) = 1 x Pr{x ∈ C} = ∑ P( y ) y∈C Pr{x1 = i1 ,..., xn = in } = P(i1 ,..., in ) The pdfs of variables xi do not contain information about their mutual relationships (dependence). Example: x=[x1 x2] p(x1) p(x2) x1 x2 x2 x2 x1 x1 (both joint pdfs p(x1, x2) have the same marginal pdfs p(x1), p(x2)) Independent random variables N random variables x1, …, xn are independent iif p ( x1,..., xn ) = p( x1 ) p ( x2 )... p( xn ) (in this case, the joint pdf can be obtained from the marginal pdfs) A sequence of independent random variables is called white noise 2nd order description A random vector x can be (partially) characterized by E{x1} mean vector µ = E{x} = M E{x } n covariance matrix R11 R12 L R1n R R L R 2n 21 22 T R = E{( x − µ )( x − µ ) } = M M Rij = E{( x − µi )( x − µ j )} Rn1 Rn 2 M Rnn covariance matrix R is: • symmetric • semipositive definite n • can be expressed as R = ∑ λi vi viT i =1 λi – eigen value vi – (normalized) eigen vector Covariance of independent variables The covariance matrix of a set of n independent random variables is a diagonal matrix Rij = E{( xi − µi )( x j − µ j )} = E{xi − µi } E{x j − µ j } = 0 Independence (i ≠ j) σ12 0 L 0 2 0 σ2 L 0 R= M M 2 0 0 M σn Rij = σ i2 δ (i − j ) Normal distribution Completely defined by the 1st and 2nd order statistics µ and R Normal distribution x~N(µ,R) p ( x) = √λ1v1 1 ( 2π )n / 2 |R|1 / 2 e − 1 ( x − µ )T R −1 ( x − µ ) 2 quadratic form constant √λ2v2 the level surfaces are elipsoids centered at µ with axis determined by the eigenvalues and eigenvectors of R µ Nice property: a linear combination of normal variables is normal (if x is a normal variable, y=Ax is a normal variable) Infinite length signals A random signal with infinite length { xn , n ∈ Z } is called a random, or stochastic, process Naturally, its complete characterization is more complex than the one of a vector In general, a random process is characterized through the pdfs of all finite subsets of samples xk1 , xk2 ,..., xkn 2nd order description Partial description of a random process, based on the 1st and 2nd order statistics mean µi = E{xi } covariance function c(i, j ) = E{( xi − µi )( x j − µ j )} Gaussian processes A random process is Gaussian iif any subset of samples follows a Normal distribution It is completely characterized by the 2nd order description White noise A sequence of independent random variables is called white noise The covariance function of a white noise is c(i,j)=σi2 δ(i,j) If, in addition, each variable follows a normal distribution, the process is called a white Gaussian noise It is very simple to generate a realization of a Gaussian white noise in the computer Other processes We can synthesize processes with non-independent samples (colored noise) by filtering white noise: xi ~ N (0,1) (white noise) autoregressive (AR) model yi = a1 yi −1 + ... + a p yi − p + xi moving average (MA) model yi = b0 xi + ... + bq xi −q autoregressive moving average (ARMA) model yi = a1 yi −1 + ... + a p yi − p + b0 xi + ... + bq xi − q yi = .98 yi −1 + xi