ECE 513 Exam #2
Instructor: Dr. Cranos Williams
4
3.Frequency Sampling Method: (20 Pts)We want to design a filterh(n) whose magnitude response is 0.5 at|ω|= 4π/7, at-tenuates frequencies (zero response) for|ω|<4π/7, and has unity gain at frequencies|ω|>4π/7, where-π≤ω≤π.(a) Define the values of anN= 7 point DFT,H(k), for all values ofk, which matchesthe response described above.(b) Given that we desire alinearphase andcausalfilter, write a formula forH(k),defined for all values ofk= 0, . . . , N-1.(c) Using the definition of the inverse DFT (h(n) =1N∑N-1k=0H(k)ej2πknN) and theH(k) defined in part b.), write a formula forh(n) whereh(n) is composed ofONLY real components. HINT:h(n) will be of the formh(n) =A1cos(ω1(n-n1)) +A2cos(ω2(n-n2))(7)
Solution:
ECE 513 Exam #2
Instructor: Dr. Cranos Williams
5
(c) (EXTRA CREDIT 5 pts)
Now, we can solve for
h
(
n
) using the inverse DFT.
h
(
n
)
=
1
N
N
-
1
X
k
=0
H
(
k
)
e
j
2
πkn
N
=
1
7
6
X
k
=0
|
H
(
k
)
|
e
-
j
2
πk
3
7
e
j
2
πkn
7
=
1
7
6
X
k
=0
|
H
(
k
)
|
e
j
2
πk
(
n
-
3)
7
=
1
7
|
H
(2)
|
e
j
2
π
2(
n
-
3)
7
+
|
H
(3)
|
e
j
2
π
3(
n
-
3)
7
+
|
H
(4)
|
e
j
2
π
4(
n
-
3)
7
+
|
H
(5)
|
e
j
2
π
5(
n
-
3)
7
=
1
7
1
2
e
j
2
π
2(
n
-
3)
7
+
e
j
2
π
3(
n
-
3)
7
+
e
j
2
π
4(
n
-
3)
7
+
1
2
e
j
2
π
5(
n
-
3)
7
(11)
You can multiply any term by
e
-
j
2
π
7(
n
-
3)
7
