Solution of ECE 316 Test #9 S04
1.
Find the magnitude of the transfer function of the systems with these pole-zero plots at the
specified frequencies. (In each case assume the transfer function is of the general
form, H(z) = K
(z − z1)(z − z2 )L(z − zN ) ,where the z’s are the zeros and the p’s are the poles, and let K = 1.)
(z − p1)(z − p2 )L(z − pD )
(a)
@ Ω = 0 , H(e jΩ ) = H(1) = 1 ×
1
=5
0.2
(b)
@ Ω = π , H(e jΩ ) = H(−1) = 1 ×
−1
1
=
= 0.4082
(−1 − (0.4 + j 0.7))(−1 − (0.4 − j 0.7)) 1.565 × 1.565
(a)
(b)
1
1
0.7
0.5
Im(z)
0.5
Im(z)
[z]
[z]
0
0
-0.5
-0.5
-1
-1
0
Re(z)
0.8
-0.7
-1
-1
1
0 0.4
Re(z)
1
2.
For each of the systems with these pole-zero plots find the DT radian frequencies, Ωmax and
Ωmin , in the range, −π ≤ Ω ≤ π for which the transfer function magnitude is a maximum and a
minimum. If there is more than one value of Ωmax or Ωmin , find all such values.
(a)
The minimum magnitude obviously occurs at z = 1 or Ω = 0 because the transfer function has a
zero there and nowhere else. The maximum magnitude occurs when z is nearest one of the poles. That
occurs where Ω is the angle formed by a vector from the origin to either one of the poles. Those Ω’s
 ±0.8 
are tan −1 
 = ±1.012 .
 0.5 
π
(b)
The minimum magnitude occurs at the two zeros at Ω = ± . The maximum magnitude occurs
2
at the closest approach to the pole at z = −0.8 which is at Ω = ±π .
Ωmax = ±π
Ωmax = ±1.012
(a)
(b)
π
Ωmin = 0
Ωmin = ±
2
1
[z]
1
0.8
[z]
0.5
Im(z)
Im(z)
0.5
0
-0.5
0
-0.5
-0.8
-1
-1
0
Re(z)
1
0.5
-1
-1
-0.8
0
Re(z)
1