Wiley signals and systems e book TLFe BO 429

17. 13escribing Randoiii Signals

414

Example 17.6
The weak stationay random signal x ( t ) from Example 1'7.5 i s modulated by a
deterministic signal r n ( f ) = sin ~ v tso
, a new randnrii signal ;y(t)is created:

y(tj = m(tj.r(t)-- sin(wt)-c(t).

= ?72(tl)m(t2)piI ( t l , ti').

It,, t z ) only deperids 011 thr differeiicc between the observativri times z = t~-tz
(see F:xaryIiple 17.5), but, this is riot true for

p,

nz(tl)in(t2)= sirid1 sinwtz.
Therefore y ( t ) is neither st&ioriaiy nor weak stationary. For cxarnple, the mean
square at Liine t = & is

but at lime C = 0, it is

WC will now return t o the qurstion of the conditions under which wr ci>n esyrcss
the rnsenible averagcs by the time average. W e h s t define t,liP first-order time
(17.18)
cud the second-order time average
(17.1'3)