Massachusetts Institute of Technology
Department
of
Electrical
Engineering
and
Computer
Science
6.011:
Introduction
to
Communication,
Control
and
Signal
Processing
QUIZ
1,
March
15,
2005
Answer
Booklet
Your Full Name:
Recitation Instructor & Time :
at
o’clock
•
This
quiz
is
closed book
,
but
two
sheets
of
notes
are
allowed.
Calculators
will
not
be
necessary
and
are
not
allowed.
•
Put
your
name
in
the
space
indicated
above,
and
your
recitation
time
next
to
the
name
of
your
instructor.
•
Check
that
this
answer
booklet
has
pages
numbered
up
to
16.
The
booklet
contains
spaces
for
all
relevant
reasoning
and
answers.
•
Neat work and clear explanations count; show all relevant work and reasoning!
You
may
want
to
Frst
work
things
through
on
scratch
paper
and
then
neatly
transfer
to
this
booklet
the
work
you
would
like
us
to
look
at.
Let
us
know
if
you
need
additional
scratch
paper.
Only
this
booklet
will
be
considered
in
the
grading;
no additional an
swer or solution written elsewhere will be considered.
Absolutely
no
exceptions!
•
There
are
two problems
,
weighted as indicated on the question booklet
.
•
DO NOT DISCUSS THIS QUIZ WITH 6.011 STUDENTS WHO HAVE NOT
YET TAKEN IT TODAY!
Problem
Your Score
1 (20 points)
2 (30 points)
Total (50 points)
1
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Quiz 1, March 15, 2005
Problem 1 (20 points)
Suppose
x
(
t
)=
y
(
t
)
cos(
ω
o
t
+
Θ),
where:
y
(
t
)
is
a
widesense
stationary
(WSS)
process
with
mean
µ
y
and
autocovariance
function
C
yy
(
τ
);
ω
o
is
a
known
constant;
and
Θ
is
a
random
variable
that
is
independent
of
y
(
·
)
and
is
uniformly
distributed
in
the
interval
[0
,
2
π
].
Do
part
(a)
below
especially
carefully,
because
(b)
and
(c)
depend
on
it
to
some
extent!
You
might
Fnd
it
helpful
in
one
or
more
parts
of
the
problem
to
recall
that
1
cos(
A
)
cos(
B
)=
2
[cos(
A
+
B
)+cos(
A
−
B
)]
.
(a)
(8
points)
±ind
the
mean
µ
x
(
t
)
and
autocorrelation
function
E
[
x
(
t
+
τ
)
x
(
t
)]
of
the
process
x
(
·
).
Also
Fnd
the
crosscorrelation
function
E
[
y
(
t
+
τ
)
x
(
t
)].
Explain
precisely
what
features
of
your
answers
tell
you
that:
(i)
x
(
·
)
is
a
WSS
process;
and
(ii)
x
(
·
)and
y
(
·
)
are
jointly
WSS.
Begin
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 Spring '10
 AmrNady
 Cos, Autocorrelation, Syy

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