2.171
Analysis
and
Design
of
Control
Systems
Prof.
David
Trumper
Final
Exam
Problem
1:
This
problem
considers
a
system
for
which
the
impulse
response
h
(
k
)
is
equal
to
1
for
N
= 0
,
1
, ..., N
−
1,
and
equal
to
zero
elsewhere.
That
is,
N
−
1
h
(
k
) =
δ
(
k
−
i
)
(1)
i
=0
The
system
has
an
input
u
and
an
output
y
.
a)
For
this
system,
solve
for
the
transfer
function
H
(
z
).
Clearly
show
your
reasoning.
b)
Find
a
diﬀerence
equation
in
terms
of
u
and
y
which
has
this
impulse
response
h
(
k
).
Explain
why
this
type
of
system
is
often
referred
to
as
an
Npoint
averager.
(It
may
also
be
called
a
boxcar
±lter,
because
of
the
shape
of
the
impulse
reponse.)
c)
Show
that
the
frequency
response
of
this
system
is
given
by
H
(
e
j
Ω
)
=
sin(
N
Ω
/
2)
e
−
j
Ω(
N
−
1)
/
2
(2)
sin(Ω
/
2)
Sketch
a
Bode
plot
for
this
system
for
the
value
N
=
4.
Use
a
linear
frequency
and
magnitude
axis,
and
sketch
from
Ω
=
0
to
Ω
=
π
.
Be
sure
to
include
both
magnitude
and
phase,
and
to
label
the
axes
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 Fall '06
 DavidTrumper
 Unit Circle, Root Locus, Prof. David Trumper, Heq

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