Solution to ECE Test #4 Su06
1.
Z
If x [ n ] ←
→ X(z) =
z
fill in the table below with numbers.
z + 0.81
2
Z
α n sin ( Ω0 n ) u [ n ] ←
→
α z sin ( Ω0 )
z − 2α z cos ( Ω0 ) + α 2
2
α = 0.81 = 0.9 , 2α cos ( Ω0 ) = 0 ⇒ Ω0 = π / 2
X(z) =
zα sin ( Ω0 )
1
1
α n sin ( Ω0 n ) u [ n ]
⇒ x[n] =
2
α sin ( Ω0 ) z + 0.81
α sin ( Ω0 )
x [ n ] = ( 0.9 )
n −1
sin (π n / 2 ) u [ n ]
n
x[n]
1
1
3
−0.81
11 −0.3487
21 0.1216
2.
On the right-hand graph coordinates, sketch the region in the z plane corresponding
to the given shaded region in the s plane, using the mapping relationship z = esTs with
Ts = 1 .
ω
ω
[s]
0.5
[z]
2
1
-0.5
0.5
σ
-2
-1
1
2
σ
-1
-0.5
-2
The range of magnitudes of z will be e−0.25 = 0.7788 to e+0.25 = 1.284 . The range of angles
will be from -0.4 radians to +0.4 radians (-22.9° to +22.9°).
z
. It is excited by a
z − 0.6
suddenly-applied sinusoid of the form x [ n ] = 4 cos ( 2π n / 16 ) u [ n ] . After a long time
( n → ∞ ) the response of the system approaches a sinusoid of the form
 2π n

y [ n ] = A cos 
+ C .
Using the result derived in the text
 B

3.
A discrete-time system has a transfer function H ( z ) =
y  n  = Z
−1
()
()
 N1 z 
z
 + H p1 cos Ω0 n +
 D z 
( ) (
( ))
jΩ
H p1 u  n  where p1 = e 0 ,
what are the numerical values of A, B and C? A = 1.9946 , B = 16 , C = -0.4757
Ω0 = π / 8 ⇒ p1 = e jπ / 8
e jπ / 8
0.9239 + j 0.3827
=
jπ / 8
e
− 0.6 0.9239 + j 0.3827 − 0.6
0.9239 + j 0.3827
=
= 1.7731 − j 0.9135 = 1.9946 − 0.4757 = 1.9946 e− j 0.4757
0.3239 + j 0.3827
(
)
H ( p1 ) = H e jπ / 8 =

 2π n
− 0.4757  u  n 
y  n  = 1.9946 cos 

 16
4.
A discrete-time bandpass filter has a transfer function H ( z ) =
(a)
( z − 1) ( z + 1)
z − 1.4 z + 0.98
2
.
Letting z = e jΩ where Ω is real radian frequency, what two numerical values
of Ω in the range −π ≤ Ω < π make the magnitude of the transfer function
a maximum?
Ω = −π / 4 or − 0.785
and
π / 4 or 0.785
This transfer function has two zeros at z = ±1 and two poles at
z = 0.9899 e± j 0.7854 . These two poles are very close to the unit circle and the
maximum response magnitude occurs when z is closest to the poles. That
occurs at an angle of ±π / 4 radians and those are the same as the values of
Ω for a maximum magnitude.
(b)
Letting z = e jΩ where Ω is real radian frequency, what two numerical values
of Ω in the range −π ≤ Ω < π make the magnitude of the transfer function
a minimum?
Ω = −π
and
0
The minimum magnitude occurs at the zeros which are at z = ±1.
z = ±1 ⇒ Ω = −π and 0