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ECEN 487
QUIZ 2
NAME:_________KEY_____________________
1.
a)
Define a sequence
ˆ
x
[
n
]
with ztransform
ˆ
X
(
z
)
=
ln{
X
(
z
)}
, with an
X
(
z
)
that has
poles at
z
=
1.5 and
z
=
3
j
, and a single zero at
z
=
1/ 4
.
Is there an R.O.C. choice
for which
ˆ
x
[
n
]
is two sided and for which the corresponding Fourier transform,
)
(
ˆ
ω
j
e
X
, exists (i.e.
)
(
ˆ
j
e
X
is bounded for all
!
)?
If so, specify the R.O.C. or
explain why not.
(Hint: Recall that for any complex
a
,
}
arg{
}
ln{
}
ln{
a
j
a
a
+
=
).
ˆ
X
(
z
)
z
=
1/4
=
ln
X
(
z
)
{ }
z
=
1/4
=
ln{0
}
=
"#
.
So the zero at
z
= 1/4 becomes a pole.
Thus a two sided sequence with a Fourier
transform requires the R.O.C. to include the unit circle, so
1/ 4
!
z
!
3/ 2
.
Therefore
ˆ
x
[
n
]
can be both right sided and have a Fourier transform,
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This note was uploaded on 03/16/2012 for the course ECEN 487 taught by Professor Dr.brianjeffs during the Winter '12 term at BYU.
 Winter '12
 Dr.BrianJeffs

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