sm8_31

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Problem 8.31 A discrete-time random process is defined by 
 
 nnn WYY += −1α n = …, -1, 0, +1, … 
 
where the zero-mean random process Wn is stationary with autocorrelation function RW(k) = σ2δ(k). What is the 
autocorrelation function Ry(k) of Yn? Is Yn a wide-sense stationary process? Justify your answer. 
 
Solution 
We partially address the question of whether Yn is wide-sense stationary (WSS) first by 
noting that 
 
 
[ ] [ ]
[ ] [
[ ]1
1
1
−
−
−
=
+=
+=
n
nn
nnn
Y
WY
WYY
E
EE
EE
α
α ]
α
 
 
since E[Wn] = 0. To be WSS, the mean of the process must be constant and consequently, 
we must have that E[Yn] = 0 for all n, to satisfy the above relationship. 
 
We consider the autocorrelation of Yn in steps. First note that RY(0) is given by 
[ ] [ ]2)0( nnnY YYYR EE == 
 
and that RY(1) is 
 [ ]
[ ][ ] [ ]nnn nnn
nnY
WYY
WYY
YYR
EE
E
E
+=
+=
= +
2
1
)(
)1(
α
α 
 
Although not explicitly stated in the problem, we assume that Wn is independent of Yn, 
thus E[YnWn] = E[Yn]E[Wn] = 0, and so 
 
)0()1( YY RR α= 
 
We prove the result for general positive k by assuming RY(k) = αkRY(0) and then noting 
that 
 [ ]
[ ]
[ ] [ knnknn
knknn
knnY
WYYY
WYY
YYkR
++
++
++
+=
+=
]
=+
EE
E
E
α
α )(
)1( 1
 
 
Continued on next slide 
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to students enrolled in courses for which the textbook has been adopted. Any other reproduction or translation of this work beyond that 
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page…8-37 
Problem 8.31 continued 
 
To evaluate this last expression, we note that, since 
 
L=
+++=
++=
+=
−−−
−−
−
nnnn
nnn
nnn
WWWY
WWY
WYY
12
2
3
3
12
2
1
ααα
αα
α
 
 
we see that Yn only depends on Wk for k ≤ n. Thus E[YnWn+k] = 0. Thus, for positive k, we 
have 
 
)0(
)()1(
1
Y
k
YY
R
kRkR
+=
=+
α
α
 
 
Using a similar argument, a corresponding result can be shown for negative k. Combining 
the results, we have 
 
)0()( Y
k
Y RkR α= 
 
Since the autocorrelation only depends on the time difference k, and the process is wide-
sense stationary. 
 
 
 
 
Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or instructional purposes only 
to students enrolled in courses for which the textbook has been adopted. Any other reproduction or translation of this work beyond that 
permitted by Sections 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. 
page…8-38