ECE 108 Hw#10 — Solution
10.7
Find the poles and zeros of the transfer function.
In general to determine the
possible Regions of Convergence and signals it is not necessary to consider the zeros.
However, there is always the possibility that one of the numerator factors (zero) will
cancel one of the denominator factors (poles). Thus it is necessary to factor both the
numerator and denominator.
( )
2
2
1
2
1
1
4
1
5
3
1
1
4
4
8
z
X z
z
z
z
−
−
−
−
−
=
+
+
+
To find the roots we must write this in positive powers of
z.
This can be done by
multiplying the numerator and denominator by
, i.e.,
4
z
( )
2
2
2
2
1
4
1
5
3
4
4
8
z
z
X z
z
z
z
−
=
+
+
+
The roots of the numerator are
.
These are the locations of the zeros.
The roots of
1
0,
2
±
the first term in the denominator
are
, and the roots of the second term in
2
1
4
z
+
1
2
j
±
the denominator,
are
.
Thus the apparent poles are at
2
5
1
4
8
z
z
+
+
1
3
and
2
4
−
−
1
1
1
3
,
,
, and
2
2
2
4
j
j
−
−
−
However, the pole at ½ is canceled by the zero at ½.
This leaves only three poles
located in the zplane as shown below
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2
Now we see there are three different signals that could have the same transform
1
2
3
1
2
4
3
4
Signal #1.
Its ROC is
.
This is a leftsided signal
Signal #2.
Its ROC is
.
This is a twosided signal
Signal #3.
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 Spring '08
 Li
 JayZ, Mathematics in medieval Islam, Polish American, Beanie Sigel

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