Solution to ECE Test #12 S07 #1
()
Refer to the system diagram below with transfer function H z =
(a)
( ).
X(z)
Y z
Write out the transfer function in terms of the a’s and b’s. (If you don’t remember,
Y z
Y z Y z
Y z
it may be helpful to think of it as H z = 1
and find 1
and
X z Y1 z
X z
Y1 z
separately.)
() ()
() ()
()
(b)
()
()
()
()
()
Let a1 = 1.5 , a2 = 0.8 , b0 = 0 , b1 = 0 , b2 = 2 . What is the numerical value of
H 1 ?
()
H z =
b0 z 2 + b1 z + b2
z + a1 z + a2
2
=
()
2
2
H 1 =
= 0.6061
1 + 1.5 + 0.8
z + 1.5z + 0.8
2
(c)
Let a1 = 1.5 , a2 = 0.8 , b0 = 3 , b1 = 0 . What numerical value of b2 would
make the transfer function have zeros on the unit circle?
()
H z =
b0 z 2 + b1 z + b2
z 2 + a1 z + a2
=
3z 2 + b2
z 2 + 1.5z + 0.8
For zeros on the unit circle
3z 2 + b2 = 0 z = ± j b2 / 3 and z = 1
Therefore
± b2 / 3 = 1 b2 = 3.
b0
b1
b2
X(z)
z2Y(z)
1
1
z
zY(z)
1
1
z
a1
a2
Y(z)
1
Y(z)
Solution to ECE Test #12 S07 #2
()
Refer to the system diagram below with transfer function H z =
(a)
( ).
X(z)
Y z
Write out the transfer function in terms of the a’s and b’s. (If you don’t remember,
Y z
Y z Y z
Y z
it may be helpful to think of it as H z = 1
and find 1
and
X z Y1 z
X z
Y1 z
() ()
() ()
()
()
()
()
()
separately.)
(b)
Let a1 = 1.5 , a2 = 0.8 , b0 = 0 , b1 = 0 , b2 = 3. What is the numerical value
()
of H 1 ?
()
H z =
b0 z 2 + b1 z + b2
z + a1 z + a2
2
=
3
3
H 1 =
= 0.9091
1 + 1.5 + 0.8
z + 1.5z + 0.8
()
2
(c)
Let a1 = 1.5 , a2 = 0.8 , b0 = 0.5 , b1 = 0 . What numerical value of b2 would
make the transfer function have zeros on the unit circle?
()
H z =
b0 z 2 + b1 z + b2
z 2 + a1 z + a2
=
0.5z 2 + b2
z 2 + 1.5z + 0.8
For zeros on the unit circle
0.5z 2 + b2 = 0 z = ± j 2b2 and z = 1
Therefore
± 2b2 = 1 b2 = 0.5 .
b0
b1
b2
X(z)
z2Y(z)
1
1
z
zY(z)
1
1
z
a1
a2
Y(z)
1
Y(z)
Solution to ECE Test #12 S07 #3
()
Refer to the system diagram below with transfer function H z =
(a)
( ).
X(z)
Y z
Write out the transfer function in terms of the a’s and b’s. (If you don’t remember,
Y z
Y z Y z
Y z
it may be helpful to think of it as H z = 1
and find 1
and
X z Y1 z
X z
Y1 z
() ()
() ()
()
()
()
()
()
separately.)
(b)
()
Let a1 = 1.5 , a2 = 0.9 , b0 = 0 , b1 = 0 , b2 = 5 . What is the numerical value of
H 1 ?
()
H z =
b0 z 2 + b1 z + b2
z + a1 z + a2
2
=
()
5
5
H 1 =
= 1.471
1 + 1.5 + 0.9
z + 1.5z + 0.9
2
(c)
Let a1 = 1.5 , a2 = 0.8 , b0 = 7 , b1 = 0 . What numerical value of b2 would
make the transfer function have zeros on the unit circle?
()
H z =
b0 z 2 + b1 z + b2
z 2 + a1 z + a2
=
7z 2 + b2
z 2 + 1.5z + 0.9
For zeros on the unit circle
7z 2 + b2 = 0 z = ± j b2 / 3 and z = 1
Therefore
± b2 / 3 = 1 b2 = 7 .
b0
b1
b2
X(z)
z2Y(z)
1
1
z
zY(z)
1
1
z
a1
a2
Y(z)
1
Y(z)