1.
Solution of ECE 315 Test 3 F06
Below is graphed exactly one period of a periodic function harmonic function
X k
(with
T F
= T
0
) can be written as x t
( )
. Its
X k = A g
( ) e jck
.
What is the name of the function g
( )
?
g is
2
What are the numerical values of A , b and c ?
A = b = c =
( ) tri
( )
( ) tri
( )
6 tri
( )
20
T
0
( )
FS f
0 sinc 2
0
20
( )
FS
(
1 / 20
) sinc 2
(
6 k / 20
)
( )
FS
( ) sinc 2
(
3 k / 10
)
6 tri
( ( )
/ 6
)
6 tri
( ( )
/ 6
) x
( )
= 6 tri
( t 2 / 6
)
20
20
( )
FS
( ) sinc 2
(
3 k / 10
) e j 2 k 2 / 20
20
( )
FS
( ) sinc 2
(
3 k / 10
) e j k /5
10 x( t )
10 t
-10
-10

2.
Below is graphed exactly one period of a periodic function harmonic function X k (with
N
F
= N
0 x n
. Its
) can be written in the form
X k =
( jbk e jck + De jdk
)
Find the numerical values of the constants.
A = B = b = , c =
D = d = x n = 6
20 n + 8 2
20 n 1 + 8
20 n 9
Using
N
F n FS 1 / N
F
and x n n
0
FS X k e j 2 n
0 k / N
F
X k =
( ) e j 16 k / 20 2 1 / 20
) e j 2 k / 20 +
( ) e j 18 k / 20
X k =
( ) (
3 e j 4 k /5 e j k /10 + 4 e j 9 k /10
)
2
0
-2
-4
-6
-8
-10
-10
10
8
6
4
-8 -6 -4 -2 0 n
2 4 6 8 10

3.
2
1
0
-1
-2
-3
0
3
Match the CTFS magnitude graphs to the time functions. (In all cases
T
F
= 4 .)
J C D I G
3 3 3 3
2 t
0
-1
-2
2
1
4
-3
0 2 t
0
-1
-2
2
1
4
-3
0 2 t
0
-1
-2
2
1
4
-3
0 t
2
0
-1
-2
2
1
4
-3
0 t
2 4
A
1
0.8
0.6
0.4
0.2
0
-10 0 k
10
0.8
1
B
0.6
0.4
0.2
0
-10 0 k
10
C
1
0.8
0.6
0.4
0.2
0
-10 0 k
10
D
1
0.8
0.6
0.4
0.2
0
-10 0 k
10
E
1
0.8
0.6
0.4
0.2
0
-10 0 k
10
1
0.8
0.6
0.4
0.2
1
0.8
0.6
0.4
0.2
1
0.8
0.6
0.4
0.2
1
0.8
0.6
0.4
0.2
1
0.8
0.6
0.4
0.2
1.
2.
3.
4.
5.
0
-10 0 k
F
10
0
-10 0 k
G
10
0
-10
H
0 k
10
0
-10
I
0 k
10
0
-10
J
0 k
Second-harmonic sine function. Non-zero impulses only at k
= ± 2
.
One fundamental period of a square wave with an average value of 1.
10
Harmonic function is a sinc function with a non-zero impulse at the fundamental k
= 1 .
Two fundamental periods of a square wave with an average value of 1.
Harmonic function is a sinc function with a zero impulse at the fundamental k
= 1
and a non-zero impulse at k
= 2.
Two fundamental periods of a triangle wave with an average value of 1.
Harmonic function is a sinc squared function with a zero impulse at the fundamental k
= 1
and a non-zero impulse at k
= 2.
The impulses are smaller than for a square wave because of the faster rate of decay of sinc squared compared with sinc.
Fourth-harmonic sine function. Non-zero impulses only at k
= ± 4 .

4.
-1
-2
-3
0
2
1
0
3
Match the DTFS magnitude graphs to the time functions. (In all cases
N
F
= 16
G
.)
D E
3 3 3 3 n
10
-1
-2
-3
0
2
1
0 n
10
-1
-2
-3
0
2
1
0 n
10
-1
-2
-3
0
2
1
0 n
10
-1
-2
-3
0
2
1
0 n
10
1.5
A
1.5
B C
1.5
D
1.5
E
1.5
1 1 1 1 1
1
0.5
0.5
0
0
1.5
k
10
1
0.5
0.5
0
0
1.5
k
10
1
0.5
0.5
0
0
1.5
k
10
1
0.5
0.5
0
0
1.5
k
10
1
0.5
0.5
0
0
1.5
k
10
1.
2.
3.
4.
5.
0
0 k
10
F
0
0 10 k
G
0
0
H k
10
0
0
I k
10
0
0
J k
10
Second-harmonic sine function. Non-zero impulses only at k
= ± 2 repeated periocidally with period 16.
Pulse train with average value of 1/4. Harmonic function has a Dirichlet shape (periodically-repeated sinc functions).
Square wave with an average value of zero and fundamental period equal to representation time. Zero at periodically with period 16.
k
= 0 and non-zero at k
= ± 1 repeated
Pulse train with average value of 1/2. Harmonic function has a Dirichlet shape (periodically-repeated sinc functions) narrower than #2 because the time-domain pulse is twice as wide.
Fourth-harmonic sine function. Non-zero impulses only at k
= ± 4 repeated periodically with period 16.

1.
Solution of ECE 315 Test 3 F06
Below is graphed exactly one period of a periodic function harmonic function
X k
(with
T
F
= T
0
) can be written as x t
( )
. Its
X k = A g
( ) e jck
.
What is the name of the function g
( )
?
What are the numerical values of A , b and c ?
A = b = c = x
( )
= 8tri
( t 2 / 4
)
( ) tri
( )
( ) tri
( )
T
0
( )
FS f
0 sinc 2
0
20
( )
FS
(
1 / 20
) sinc 2
(
4 k / 20
)
8tri
( )
20
( )
FS
( ) sinc 2
( )
8tri
( ( )
/ 4
)
8tri
( ( )
/ 4
)
20
20
( )
FS
( ) sinc 2
( ) e j 2 k 2 / 20
20
( )
FS
( ) sinc 2
( ) e j k /5
10 x( t )
10 t
-10
-10

2.
Below is graphed exactly one period of a periodic function harmonic function X k (with
N
F
= N
0 x n
. Its
) can be written in the form
X k =
( jbk + Ce jck + De jdk
)
Find the numerical values of the constants.
A = b = , C = -3 , c =
D = d = x n = 2
20 n + 7 6
20 n 1 + 8
20 n 5
Using
N
F n FS 1 / N
F
and x n n
0
FS X k e j 2 n
0 k / N
F
X k =
( ) e j 14 k / 20 6 1 / 20
) e j 2 k / 20 +
( ) e j 10 k / 20
X k =
( ) ( e j 7 k /10 3 e j k /10 + 4 e j k / 2
)
2
0
-2
-4
-6
-8
-10
-10
10
8
6
4
-8 -6 -4 -2 0 n
2 4 6 8 10

3.
2
1
0
-1
-2
-3
0
3
Match the CTFS magnitude graphs to the time functions. (In all cases
T
F
= 4 .)
I
3 3 3 3
2 t
0
-1
-2
2
1
4
-3
0 2 t
0
-1
-2
2
1
4
-3
0 2 t
0
-1
-2
2
1
4
-3
0 t
2
0
-1
-2
2
1
4
-3
0 t
2 4
A
1
0.8
0.6
0.4
0.2
0
-10 0 k
10
0.8
1
B
0.6
0.4
0.2
0
-10 0 k
10
C
1
0.8
0.6
0.4
0.2
0
-10 0 k
10
D
1
0.8
0.6
0.4
0.2
0
-10 0 k
10
E
1
0.8
0.6
0.4
0.2
0
-10 0 k
10
1
0.8
0.6
0.4
0.2
1
0.8
0.6
0.4
0.2
1
0.8
0.6
0.4
0.2
1
0.8
0.6
0.4
0.2
1
0.8
0.6
0.4
0.2
1.
2.
3.
4.
5.
0
-10 0 k
F
10
0
-10 0 k
G
10
0
-10
H
0 k
10
0
-10
I
0 k
10
0
-10
J
0 k
Second-harmonic sine function. Non-zero impulses only at k
= ± 2
.
One fundamental period of a square wave with an average value of 1.
Harmonic function is a sinc function with a non-zero impulse at the fundamental k
= 1 .
Two fundamental periods of a square wave with an average value of 1.
Harmonic function is a sinc function with a zero impulse at the fundamental k
= 1
and a non-zero impulse at k
= 2.
Two fundamental periods of a triangle wave with an average value of 1.
Harmonic function is a sinc squared function with a zero impulse at the fundamental k
= 1
and a non-zero impulse at k
= 2.
The impulses are smaller than for a square wave because of the faster rate of decay of sinc squared compared with sinc.
Fourth-harmonic sine function. Non-zero impulses only at k
= ± 4 .
10

4.
-1
-2
-3
0
2
1
0
3
Match the DTFS magnitude graphs to the time functions. (In all cases
N
F
= 16
E
.)
A
3 3 3 3 n
10
-1
-2
-3
0
2
1
0 n
10
-1
-2
-3
0
2
1
0 n
10
-1
-2
-3
0
2
1
0 n
10
-1
-2
-3
0
2
1
0 n
10
1.5
A
1.5
B C
1.5
D
1.5
E
1.5
1 1 1 1 1
1
0.5
0.5
0
0
1.5
k
10
1
0.5
0.5
0
0
1.5
k
10
1
0.5
0.5
0
0
1.5
k
10
1
0.5
0.5
0
0
1.5
k
10
1
0.5
0.5
0
0
1.5
k
10
1.
2.
3.
4.
5.
0
0 k
10
F
0
0 10 k
G
0
0
H k
10
0
0
I k
10
0
0
J k
10
Second-harmonic sine function. Non-zero impulses only at k
= ± 2 repeated periocidally with period 16.
Pulse train with average value of 1/4. Harmonic function has a Dirichlet shape (periodically-repeated sinc functions).
Square wave with an average value of zero and fundamental period equal to representation time. Zero at periodically with period 16.
k
= 0 and non-zero at k
= ± 1 repeated
Pulse train with average value of 1/2. Harmonic function has a Dirichlet shape (periodically-repeated sinc functions) narrower than #2 because the time-domain pulse is twice as wide.
Fourth-harmonic sine function. Non-zero impulses only at k
= ± 4 repeated periodically with period 16.