Solution of ECE 315 Test 2 F09
1.
Find the numerical values of the constants in these Laplace transform pairs.
A
s−a
What is the ROC? ROC is ___________________________________
10
L
10e2t u ( t ) ←⎯
→
, σ >2
s−2
(a)
L
10e2t u ( t ) ←⎯
→
L
5δ ( 3t − 1) ←⎯
→ Ae− as
(b)
What is the ROC? ROC is ___________________________________
L
5δ ( 3t − 1) = 5δ ( 3( t − 1 / 3)) = ( 5 / 3)δ ( t − 1 / 3) ←⎯
→ ( 5 / 3) e− s / 3 , ROC is all s.
(c)
L
Aeat u ( bt ) + Cect u ( dt ) ←⎯
→
s −1
, σ >0
s ( s + 2)
s −1
−1 / 2 3 / 2
=
+
, σ >0
s ( s + 2)
s
s+2
L
⎡⎣ − (1 / 2 ) + ( 3 / 2 ) e−2t ⎤⎦ u ( t ) ←⎯
→
−1 / 2 3 / 2
+
, σ >0
s
s+2
s −1
, −2<σ <0
s ( s + 2)
s −1
L
→
, −2<σ <0
(1 / 2 ) u ( −t ) + ( 3 / 2 ) e−2t u (t ) ←⎯
s ( s + 2)
(d)
L
Aeat u ( bt ) + Cect u ( dt ) ←⎯
→
2.
Find the numerical values of the constants in these Fourier transform pairs.
(a)
F
−2 tri ( t − 4 ) ←⎯
→ Asinc 2 ( af ) ebf
F
−2 tri ( t − 4 ) ←⎯
→ −2sinc 2 ( f ) e− j 8 π f
(b)
F
Asin ( at ) sinc ( bt ) ←⎯
→ j ⎡⎣δ ( f − 3) − δ ( f + 3) ⎤⎦ ∗ 4 rect ( 2 f )
F
−4sin ( 6π t ) sinc ( t / 2 ) ←⎯
→ j ⎡⎣δ ( f − 3) − δ ( f + 3) ⎤⎦ ∗ 4 rect ( 2 f )
or
F
4sin ( −6π t ) sinc ( t / 2 ) ←⎯
→ j ⎡⎣δ ( f − 3) − δ ( f + 3) ⎤⎦ ∗ 4 rect ( 2 f )
(c)
F
A rect ( at ) ∗ rect ( bt ) ←⎯
→ (14 / 3) sinc ( f / 3) sinc ( 2 f )
F
7 rect ( 3t ) ∗ rect ( t / 2 ) ←⎯
→ (14 / 3) sinc ( f / 3) sinc ( 2 f )
3.
If a CTFT in the f form is 24δ 3 ( f ) and in the ω form it is Aδ b ( cω ) , what are
numerical values of A, b and c?
∞
⎡⎣ 24δ 3 ( f ) ⎤⎦ f →ω /2 π = 24δ 3 (ω / 2π ) = 24 ∑ δ (ω / 2π − 3k ) = 48π
k = −∞
⎡⎣ 24δ 3 ( f ) ⎤⎦ f →ω /2 π = 48πδ 6 π (ω )
∞
∑ δ (ω − 6π k )
k = −∞
Because of the way the problem was worded, I also accepted the solution
24δ 3 (ω / 2π )
even though that was not what I intended.
the
Below are some pole-zero diagrams and some magnitude frequency responses.
Match the frequency responses to the pole-zero diagrams by writing the letter
designation of the frequency response in the blank space provided above the polezero diagram. If there is no match, just write "None".
D
A
C
E
H
B
____________
5
[s]
____________
5
[s]
____________
5
[s]
0
0
0
0
-5
-4 -2 0 2
-5
-4 -2 0 2
-5
-4 -2 0 2
-4 -2 0 2
____________
5
[s]
____________
5
[s]
____________
5
[s]
____________
5
[s]
0
0
0
0
E
20
0
F
I
20
0
-20
0
J
|H|
0
ω
20
0
-20
0
-20
20
0
-20
20
20
0
K
0
20
0
ω
20
H
0
-20
20
L
1
1
0
-20
20
0.2
0.5
0.5
0
ω
0
0.4
1.5
1
0.05
G
0.2
2
0.1
0
|H|
|H|
0
0
-20
20
0.2
0.4
1
0.05
|H|
0.5
2
0.1
0
-20
0
-20
0.4
|H|
|H|
|H|
0
-4 -2 0 2
D
1
0.5
0.1
0
-20
C
1
0.2
0
-20
B
-5
-4 -2 0 2
|H|
A
-5
-4 -2 0 2
|H|
-5
-4 -2 0 2
|H|
-5
|H|
G
F
____________
5
[s]
-5
|H|
4.
0
ω
20
0
-20
Solution of ECE 315 Test 3 F09
1.
Find the numerical values of the constants in these Laplace transform pairs.
A
s−a
What is the ROC? ROC is ___________________________________
7
L
7e3t u ( t ) ←⎯
→
, σ >3
s−3
(a)
L
7e3t u ( t ) ←⎯
→
L
3δ ( 5t − 1) ←⎯
→ Ae− as
(b)
What is the ROC? ROC is ___________________________________
L
3δ ( 5t − 1) = 3δ ( 5 ( t − 1 / 5 )) = ( 3 / 5 )δ ( t − 1 / 5 ) ←⎯
→ ( 3 / 5 ) e− s /5 , ROC is all s.
(c)
L
Aeat u ( bt ) + Cect u ( dt ) ←⎯
→
s−2
, σ >0
s ( s + 1)
s−2
−2
3
=
+
, σ >0
s ( s + 1)
s s +1
2
3
L
⎡⎣ −2 + 3e−t ⎤⎦ u ( t ) ←⎯
→− +
, σ >0
s s +1
(d)
L
Aeat u ( bt ) + Cect u ( dt ) ←⎯
→
L
2 u ( −t ) + 3e−t u ( t ) ←⎯
→
s−2
, −1< σ < 0
s ( s + 1)
s−2
, −1< σ < 0
s ( s + 1)
2.
Find the numerical values of the constants in these Fourier transform pairs.
(a)
F
−4 tri ( t − 2 ) ←⎯
→ Asinc 2 ( af ) ebf
F
−4 tri ( t − 2 ) ←⎯
→ −4sinc 2 ( f ) e− j 4 π f
(b)
F
Asin ( at ) sinc ( bt ) ←⎯
→ j ⎡⎣δ ( f − 5 ) − δ ( f + 5 ) ⎤⎦ ∗ 6 rect ( 3 f )
F
F
−4sin (10π t ) sinc ( t / 3) ←⎯
→ ←⎯
→ j ⎡⎣δ ( f − 5 ) − δ ( f + 5 ) ⎤⎦ ∗ 6 rect ( 3 f )
or
F
F
4sin ( −10π t ) sinc ( t / 3) ←⎯
→ ←⎯
→ j ⎡⎣δ ( f − 5 ) − δ ( f + 5 ) ⎤⎦ ∗ 6 rect ( 3 f )
(c)
F
A rect ( at ) ∗ rect ( bt ) ←⎯
→ ( 24 / 5 ) sinc ( f / 5 ) sinc ( 3 f )
F
8 rect ( 5t ) ∗ rect ( t / 3) ←⎯
→ ( 24 / 5 ) sinc ( f / 5 ) sinc ( 3 f )
3.
If a CTFT in the f form is 18δ 5 ( f ) and in the ω form it is Aδ b ( cω ) , what are the
numerical values of A, b and c?
∞
⎣⎡18δ 5 ( f ) ⎤⎦ f →ω /2 π = 18δ 5 (ω / 2π ) = 18 ∑ δ (ω / 2π − 5k ) = 36π
k = −∞
⎡⎣18δ 5 ( f ) ⎤⎦ f →ω /2 π = 36πδ10 π (ω )
∞
∑ δ (ω − 10π k )
k = −∞
Because of the way the problem was worded, I also accepted the solution
18δ 5 (ω / 2π )
even though that was not what I intended.
Below are some pole-zero diagrams and some magnitude frequency responses.
Match the frequency responses to the pole-zero diagrams by writing the letter
designation of the frequency response in the blank space provided above the polezero diagram. If there is no match, just write "None".
D
F
____________
5
[s]
0
0
-5
-4 -2 0 2
E
A
C
G
____________
5
[s]
____________
5
[s]
0
0
-5
-4 -2 0 2
-5
-4 -2 0 2
-4 -2 0 2
____________
5
[s]
____________
5
[s]
____________
5
[s]
____________
5
[s]
0
0
0
0
E
20
0
F
I
20
0
-20
0
J
|H|
0
ω
20
0
-20
0
-20
20
0
-20
20
20
0
K
0
20
0
ω
20
H
0
-20
20
L
1
1
0
-20
20
0.2
0.5
0.5
0
ω
0
0.4
1.5
1
0.05
G
0.2
2
0.1
0
|H|
|H|
0
0
-20
20
0.2
0.4
1
0.05
|H|
0.5
2
0.1
0
-20
0
-20
0.4
|H|
|H|
|H|
0
-4 -2 0 2
D
1
0.5
0.1
0
-20
C
1
0.2
0
-20
B
-5
-4 -2 0 2
|H|
A
-5
-4 -2 0 2
|H|
-5
-4 -2 0 2
|H|
-5
|H|
H
B
____________
5
[s]
-5
|H|
4.
0
ω
20
0
-20
Solution of ECE 315 Test 2 F09
1.
Find the numerical values of the constants in these Laplace transform pairs.
A
s−a
What is the ROC? ROC is ___________________________________
3
L
3e5t u ( t ) ←⎯
→
, σ >5
s−5
(a)
L
3e5t u ( t ) ←⎯
→
L
4δ ( 6t − 1) ←⎯
→ Ae− as
(b)
What is the ROC? ROC is ___________________________________
L
4δ ( 6t − 1) = 4δ ( 6 ( t − 1 / 6 )) = ( 2 / 3)δ ( t − 1 / 6 ) ←⎯
→ ( 2 / 3) e− s /6 , ROC is all s.
(c)
L
Aeat u ( bt ) + Cect u ( dt ) ←⎯
→
s−3
, σ >0
s ( s + 5)
s−3
−3 / 5 8 / 5
=
+
, σ >0
s ( s + 5)
s
s+5
L
⎡⎣ −3 / 5 + ( 8 / 5 ) e−5t ⎤⎦ u ( t ) ←⎯
→
−3 / 5 8 / 5
+
, σ >0
s
s+5
s−3
, −5<σ <0
s ( s + 5)
s−3
L
→
, −5<σ <0
( 3 / 5 ) u ( −t ) + ( 8 / 5 ) e−5t u (t ) ←⎯
s ( s + 5)
(d)
L
Aeat u ( bt ) + Cect u ( dt ) ←⎯
→
2.
Find the numerical values of the constants in these Fourier transform pairs.
(a)
F
−7 tri ( t − 3) ←⎯
→ Asinc 2 ( af ) ebf
F
−7 tri ( t − 3) ←⎯
→ −7sinc 2 ( f ) e− j 6 π f
(b)
F
Asin ( at ) sinc ( bt ) ←⎯
→ j ⎡⎣δ ( f − 8 ) − δ ( f + 8 ) ⎤⎦ ∗ 9 rect ( 5 f )
F
− (18 / 5 ) sin (16π t ) sinc ( t / 5 ) ←⎯
→ j ⎡⎣δ ( f − 8 ) − δ ( f + 8 ) ⎤⎦ ∗ 9 rect ( 5 f )
or
F
→ j ⎡⎣δ ( f − 8 ) − δ ( f + 8 ) ⎤⎦ ∗ 9 rect ( 5 f )
(18 / 5 ) sin ( −16π t ) sinc (t / 5 ) ←⎯
(c)
F
A rect ( at ) ∗ rect ( bt ) ←⎯
→ 9 sinc ( 2 f ) sinc ( 5 f )
F
→ 9 sinc ( 2 f ) sinc ( 5 f )
( 9 / 10 ) rect (t / 2 ) ∗ rect (t / 5 ) ←⎯
3.
If a CTFT in the f form is 30δ 4 ( f ) and in the ω form it is Aδ b ( cω ) , what are the
numerical values of A, b and c?
∞
⎡⎣ 30δ 4 ( f ) ⎤⎦ f →ω /2 π = 30δ 4 (ω / 2π ) = 30 ∑ δ (ω / 2π − 4k ) = 60π
k = −∞
⎡⎣ 30δ 4 ( f ) ⎤⎦ f →ω /2 π = 60πδ 8 π (ω )
∞
∑ δ (ω − 8π k )
k = −∞
Because of the way the problem was worded, I also accepted the solution
30δ 4 (ω / 2π )
even though that was not what I intended.
Below are some pole-zero diagrams and some magnitude frequency responses.
Match the frequency responses to the pole-zero diagrams by writing the letter
designation of the frequency response in the blank space provided above the polezero diagram. If there is no match, just write "None".
H
E
A
G
F
C
____________
5
[s]
____________
5
[s]
____________
5
[s]
0
0
0
0
-5
-4 -2 0 2
-5
-4 -2 0 2
-5
-4 -2 0 2
-4 -2 0 2
____________
5
[s]
____________
5
[s]
____________
5
[s]
____________
5
[s]
0
0
0
0
E
20
0
F
I
20
0
-20
0
J
|H|
0
ω
20
0
-20
0
-20
20
0
-20
20
20
0
K
0
20
0
ω
20
H
0
-20
20
L
1
1
0
-20
20
0.2
0.5
0.5
0
ω
0
0.4
1.5
1
0.05
G
0.2
2
0.1
0
|H|
|H|
0
0
-20
20
0.2
0.4
1
0.05
|H|
0.5
2
0.1
0
-20
0
-20
0.4
|H|
|H|
|H|
0
-4 -2 0 2
D
1
0.5
0.1
0
-20
C
1
0.2
0
-20
B
-5
-4 -2 0 2
|H|
A
-5
-4 -2 0 2
|H|
-5
-4 -2 0 2
|H|
-5
|H|
D
B
____________
5
[s]
-5
|H|
4.
0
ω
20
0
-20