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Massachusetts Institute of Technology
Department of Mechanical Engineering
2.160 Identification, Estimation, and Learning
Spring 2006
Problem Set No. 2
Out: February 22, 2006
Due: March 1, 2006
Problem 1
A stationary random process
X(t)
has a mean value of
m
and an autocorrelation
function of the form:
R
X
(
τ
)
=
σ
2
e
−
τβ
Another random process
Y(t)
is related to
X(t)
by the deterministic equation:
t
Y
)
=
t
X
a
)
+
b
(
(
where
a
and
b
are known constants.
(a). What is the autocorrelation function for
Y(t)
?
(b). What is the crosscorrelation function
R
XY
(
)?
Problem 2
Two random processes are defined by
t
X
)
=
A
sin(
ω
t
+
θ
)
(
t
Y
)
=
B
sin(
t
+
)
(
where
is a random variable with uniform distribution between 0 and 2
π
, and
is a
2
),
0(
N
and are correlated to each other with a correlation coefficient
ρ
. Show that the cross
known constant. The
A
and
B
coefficients are both normal random variables
,
)(
R
XY
(
)
=
1
ρσ
2
cos(
t
) .
2
Assume
A
and
B
are independent of
.
Problem 3
Wearable medical sensors monitor a patient’s health conditions anytime,
anywhere, and continuously. These sensors are expected to revolutionize health care
services, including diagnosis and treatment in the home for diabetes, hypertension, and
pulmonary disorders. However, the wearable sensors must be robust against disturbances,
in particular, the motion of the wearer, since the patients move around for daily activities
correlation function
R
is given by:
XY
1
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View Full Documentrather than staying in a hospital. Wearable sensors often cause false signals due to motion
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 Spring '06
 HarryAsada
 Mechanical Engineering

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