Random Procss Part A random procss t (, ζ is a signal or wavform in tim. t : tim ζ : outcom in th sampl spac Each tim w rapat th xprimnt, a nw wavform is gnratd. ( W will adopt t for short. Tim sampls of a random procss ( t constitut a squnc of random variabls:,,, or ( t, ( t, ( t, t t t3 3 Statistical Rlationship among Random Sampls In gnral, a random procss is compltly charactrizd by th joint cdf F,,, ( x, x,, xk Pt x, t x,, t x k t t tk k for any choic of t < t < < t k and for any k Howvr w may nd a simplr modl that is clos to th ral world tractabl to analysis. Frquntly usd assumptions ar: Stationary and wid-sns stationary random procsss for analyzing signals Markov procsss for analyzing quus
Classs of Random Procss Stationary Random Procss: tim-shift-invariant ( t is a stationary random procss if F ( x, x,, x F ( x, x,, x,,, k,,, k t t tk t+ t+ tk+ for any, for any choic of t < t < < t, and for any k Wid-Sns Stationary Random Procss ( t is a wid-sns stationary (WSS random procss if k Man: ( t m for all t Autocorrlation: R ( t, t R (, t t for any t and t Markov procss Mmorylss: t t for any F ( x x x,, x F ( x x,,, k+ k, k k+ k tk+ tk tk t tk+ tk k and for any choic of t < t < < t < t k k+
3 Momnts of a Random Procss Man m ( t ( t Varianc { } σ ( t ( t m ( t Auto-corrlation t R ( t, t t t Auto-covarianc ( t t { }{ } C ( t, t m ( t m ( t t t R ( t, t m ( t m ( t cov, is an altrnativ notation.
4 Exampls of Random Procsss Random Phas Signal t ( cos( ωt+θ, whr Θ is a random variabl, uniform in th intrval (0, π. Th man of th random procss is m ( t ( t cos( ωt+θ cos( ωt+ θ f ( θ dθ Θ π cos( ωt+ θ dθ 0 π 0 for all t. Th auto-corrlation is R ( t, t ( t ( t cos( ωt +Θ cos( ωt +Θ ωt + θ ωt + θ dθ π π cos( 0 cos( π cos( ωt + ωt + θ + cos( ωt ωt π 0 0 + cos ( { ω t t } dθ cos ω R ( t, t is a function of t t. Th random phas signal is a wid-sns stationary random procss Rf. cos( A+ B + cos( A B cos Acos B, cos( A+ B cos Acos B sin Asin B
5 Random Tlgraph Signal ( t taks on on of th two valus for any t. For simplicity, assum th signal lvl is ithr + or at any tim. Th tim btwn succssiv transitions is xponntially distributd with man. α Th procss bgins at tim t 0. (0 ± with qual probabilitis 0.5 W claim [ ] [ ] P ( t P ( t 0.5 for any t ( ( ( ( ( ( ( P t P 0 P t 0 P 0 P t 0 + P[ an odd numbr of transitions] + P[ an vn numbr of transitions] P[ any numbr of transitions] Howvr w not [ ] k k 0 k 0 P an vn numbr of transitions duringintrval t P ( t ( αt αt k ( αt k 0 αt αt + αt αt P[ an odd numbr of transitions during intrval t] αt αt αt ( k! k ( k! αt + αt
6 Naturally m ( t 0 and σ ( t for allt. W claim α R ( t, t whr t t. R ( t, t ( t ( t ( P[ ( t ( t ] ( P[ ( t ( t ] + + [ an vn numbr of transitions during ] [ an odd numbr of ] P t t P α t t α t t + α t t α whr t t Th autocorrlation dcays with. Th random tlgraph signal is a wid-sns stationary random procss.
7 Winr Procss and Brownian Motion Initially (0 0. Th procss maks a transition vry Δ in tim, up or down by h with qual probabilitis. W claim that m ( t 0 for all t, and σ ( t αt as Δ 0. Modl th succssiv transitions as iid random variabls D, D, D3,. Thn j σd j D 0 and h αδ.. How many transitions occur during (0, t? Ans. t Δ Thrfor and ( t D + D + + D m ( t 0, t σ ( t αδ αt Δ t Δ According to th cntral limit thorm, as Δ 0, ( t bcoms a gaussian random variabl N(0, αt.
8 W claim that R ( t, t α min( t, t Suppos t< t. Thn R ( t, t ( t ( t D+ D + + D t D + D + + D t Δ Δ noting DD D D 0 for i j i j i j t Δ D + D + + D t α Δ Δ αt α min( tt, Th Winr procss is not a wid-sns stationary random procss.
9 Exampls of Stationary Random Procsss To prov ( t is a stationary random procss, w must show th tim-shift-invarianc: F ( x, x,, x F ( x, x,, x,,, k,,, k t t tk t+ t+ tk + for any, any choic of t < t < < t, and for any k A random tlgraph signal is stationary k W hav shown [ ] [ ] P ( t P ( t 0.5 for any t + p ( θ P[ an vn numbr of transitions during a tim intrval θ] po ( θ P[ an odd numbr of transitions during a tim intrval θ] Lt k b th numbr of sampls. For k, f ( x ± with prob 0.5, for any t. t That is quivalnt to say f ( x f ( x for any t+ t αθ αθ For k, f ( x, x (, with prob 0.5 p ( t t for any., t+ t+ (, 0.5 p ( t t (, 0.5 p ( t t (, 0.5 p ( t t o o
0 f, t+ t+ ( x, x dos not dpnd on. f ( x, x f ( x, x for any.,, t+ t+ t t For k 3, f ( x, x, x (,, with prob 0.5 p ( t t p ( t t,, 3 3 t+ t+ t3+ (,, 0.5 p ( t t p ( t t o 3 (,, 0.5 p ( t t p ( t t o 3 (,, 0.5 p ( t t p ( t t 3 (,, 0.5 p ( t t p ( t t 3 (,, 0.5 po( t t po( t3 t (,, 0.5 p ( t t p ( t t o o o o 3 (,, 0.5 p ( t t p ( t t 3 for any. f ( x, x, x dos not dpnd on.,, 3 t+ t+ t3+ f ( x, x, x f ( x, x, x for any.,, 3,, 3 t+ t+ t3+ t t t3 In gnral, w can s that f ( x, x,, x dos not dpnd on for any k,,, k t+ t+ t k + and for any sampling instants t,, t. k
A random phas signal is stationary t ( cos( ωt+θ, whr Θ is uniform ovr (0, π is stationary. In fact, on can show any priodic signal with th random phas uniformly distributd ovr th priod is stationary [rf. Wozncraft and Jacobs, Principls of Communications Enginring, pp37]
Wid-sns Stationary Random Procss Thorm. A stationary random procss is a wid-sns stationary random procss. wid-sns stationary stationary ( Suppos t Howvr is stationary. Thn ( ( f x, x f x, x for any t, t and s.,, t t t+ s t+ s R ( t, t t t applying tim shift of s t R t t 0 t t (, xx f x x dxdx, (, xx f x x dxdx, ( whr t t. Som proprtis of wid-sns stationary random procsss:. Avrag powr dos not vary with th tim t t t+ 0 R (0, which do not vary with t.. Autocorrlation is an vn function R ( R ( R ( t t R t ( t intrchanging th ordr
3 3. Max Valu R (0 R ( for any From th Cauchy-Schwartz inquality, for any rvs and Y, Y Y., Thrfor Howvr That says t t+ t t+ (0 t t+ R R ( t t+ for any. R (0 R ( for any Sinc R (0 0, R (0 R (. Also R ( R ( 4. Bound on th rat of chang { R R } (0 ( P t+ t > ε. ε Markov Inquality: For any non-ngativ random variabl, P[ > ε] u( ε / ε ε ( P P t+ t > ε t+ t > ε ( t+ ε { R R } (0 ( t ε from th Markov inquality
4 5. If R (0 R ( d, thn ( Using th Cauchy-Schwartz inquality, ( ( R is priodic with priod d, and ( 0 t+ + d t+ t t+ + d t+ t for any t+ d t t t+ + d t t+ t+ + d + t+ t+ t+ + d t for any { + } { } R ( d R ( R (0 R ( d R (0 { } If R (0 R ( d, R ( + d R ( 0. Thrfor R ( + d R ( for any. ( Also + R (0 R ( d 0 t+ d t t+ d t t t+ d 6. If ( t m+ N( t and Nt ( 0 and lim R N ( 0, thn lim R ( m. R ( t t+ ( m N ( m N + + ( ( Thrfor lim R ( m t N t+ m + m N + N + N N m + R t t+ t t+ S LG (3 rd Edition pp.54 figur9.3 for som plots of autocorrlation.