Exam zone
E x a mc py lc l o s t a t i o n a Consider
X ( t ) that X ( t, ) = A ( ω) cos(ωo t ) + B ( ω) sin ωo t ) , t 2 R , ω 2 R ,
where A _ N ( µ1 , at to 2 R w
2 1)
2 2)
and B _ N ( µ2 ,
and
If
X ( t o , ) = X ( ω) = A ( ω) cos(ωo t o ) + B ( ω) sin(ωo t o ) . Since a
X ( ω) ha
A and B are µx ( t ) = µ1 cos ωo t ) + µ2 sin(ωo t ) .
The
X ( t ) is σx2 ( t ) = E _( A cos ωo t ) + B sin(ωo t )) 2
Using
( µ1 cos(ωo t ) + µ 2 sin(ωo t )) 2 .
A and B are σx2 ( t ) = σ12 cos2 ( ωo t ) + σ22 sin2 ( ωo t )
Since this w
To = 2ωπo t1, 2 2 R
t v If X 1 and X 2 defin _ X1 _ _ cos ωo t 1 ) sin(ωo t 1 ) _ _ A _ = . X2 cos ωo t 2 ) sin(ωo t 2 ) B | {z } M
The µx = M The auto-co co
fu C xx :
ass
Cxx ( t 1 , 2 ) = Substitu
t 1 = t and t 2 = t
with this p σ12 + σ22 cos ωc ( t 2 2
is p
i obtai
t 1 )) +
σ12
σ22 2
2 2)
M T. as th
o_-diagonal elemen of the
cos ωc ( t 1 + t 2 )) .
τ w
Cxx ( t, ) = This expre other p pro The is
_ µ1 _ , C xx = M diag σ12 , µ2
σ12 + σ22 σ2 σ22 cos ωc τ ) + 1 cos ωc t 2 2
in the t v X ( t ) is
ωc τ ) .
with a fu p of To R xx ( t, ) and ρxx ( t, ) are The
=
2π 2 ωo
. Cons
all
tv Z C˜ xx ( τ ) =
To
/2
Cxx ( t, ) dt = To
/2
σ12 + σ22 cos(ωc τ ) . 2
W will re thi sp form again when w lo at the Rice repr for b pro It su_ces to s that in general the pro X ( t ) is not s but cyclostation F the sp cas where µ1 = µ2 = 0 and σ22 = σ22 = σ2 w ob the sp cas where µx ( t ) = 0, σx2 ( t ) = σ2 and 2 R xx ( τ ) = σ cos ωo τ ). In X (t ) b is 1