Given a r.v Y with characteristic function ϕ(ω)=E($e^{jωY})$ and a random process

defined by X(t) = cos(λt+Y), show that X(t) is stationary is wide sense if ϕ(1)=ϕ(2)=0

Mumbai University > Electronics and Telecommunication > Sem5 > Random Signal Analysis

Marks: 10M

Year: May 2015

A random process {X(t)} with finite first and second order moments is called as weakly stationary process or wide sense stationary (WSS) if its mean is constant and the autocorrelation depends only on the difference

i.e. if $E(X(t)=μ)$ and $E(X(t)×X(t-τ))=R(τ)$

Given: X(t) = cos(λt+Y)

$$∴E(X(t))=E(cos⁡(λt)+Y)$$

$$ =cos⁡(λt) cos⁡(Y)-sin⁡(λt) sin⁡(Y)$$

$$=cos⁡(λt)×E(cosY)-sin⁡(λt)E(sinY)----- (1)$$

$$ϕ(ω)=E(e^{jωY} )$$

$$=E(cos⁡(ωY)+jsin(ωY))$$

Given ϕ(1)=0

$$E(cosY)=E(sinY)=0-----(2)$$

Using eq(2) in eq(1) we get

E(X(t))=0

$R(t_1,t_2 )=E(X(t_1 )×X(t_2 ))=E(cos⁡(λt_1+Y)×cos⁡(λt_2+Y)))$

$=E((cos⁡(λt_1 )cosY-sin⁡(λt_1 )sinY))×(cos⁡(λt_2 )cosYsin⁡(λt_2)sinY))$

$=cos⁡(λt_1 ) cos⁡(λt_2 )E(cos^2⁡Y )+sin⁡(λt_1 ) sin⁡(λt_2 )E(sin^2⁡Y )-sin⁡(λ(t_1+t_2))E(cosYsinY)$

$=cos⁡(λt_1 ) cos⁡(λt_2 )E(1/2+cos2Y/2)+sin⁡(λt_1 ) sin⁡(λt_2 )E(1/2-cos2Y/2)-1/2 sin⁡(λ(t_1+t_2 ))E(sin2Y)-------(4)$

Given ϕ(2)=0

E(cos2Y)=E(sin2Y)=0 -----(5)

Using eq(5) in eq(4)

$R(t_1,t_2 )=E(X(t_1 )×X(t_2 ))=1/2[cos⁡(λt_1 ) cos⁡(λt_2 )+sin⁡(λt_1 ) sin⁡(λt_2 )]$

$∴R(t_1,t_2 )=\frac{1}2 cosλ(t_1-t_2)------(6)$

From (3) and (6) we can say that X(t) is a WSS process