sm8_14

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Problem 8.14 A discrete-time random process {Yn: n = …,-1,0,1,2, …} is defined by 
 
110 −+= nnn ZZY αα 
 
where {Zn} is a random process with autocorrelation function . What is the 
autocorrelation function 
)()( 2 nnRZ δσ=[ ]mnY YYmnR E=),( ? Is the process {Yn} wide-sense stationary? 
 
Solution 
 
We implicitly assume that Zn is stationary and has a constant mean µZ. Then the mean of 
Yn is given by 
 [ ] [ ]
( ) Z
nnn ZZY
µαα
αα
10
11 ][
+=
+= −0 EEE 
The autocorrelation of Y is given by 
 [ ] ( )( )[ ]
[ ] [ ] [ ] [ ]
( ) ( ) ( ) (( )
( ) ( ) ( ) ( )[ ]11
1111
2
10
22
1
2
0
2
1
2
10
2
01
22
0
11
2
1110101
2
0
110100
−−+−−+−+=
−−−+−−+−−+−=
+++=
)
++=
−−−−
−−
nmmnmn
nmmnnmmn
ZZZZZZZZ
ZZZZYY
nmmnmnmn
mmnnmn
δδσααδσαα
δαδσααδσααδσα
αααααα
αααα
EEEE
EE
 
 
Since the autocorrelation only depends on the time difference n-m, the process is wide-
sense stationary with 
 ( ) ( )( ))1(1)()( 21022120 ++−++= nnnnRY δδσααδσαα 
 
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