HW1 Sol
HW1 Sol
HW1 Sol
Full Marks : 80
Q.1 (10)
∗
√ √
a) 𝑠 𝑒 √2 𝑒 𝑒 √2 𝑒 √2 𝑒 √2 𝑒 𝑗
√ √
𝑅𝑒 𝑠 𝐼𝑚 𝑠 |𝑠 | √2 ∠𝑠 𝑜𝑟
√
√
b) 𝑠 4√2 𝑒 4√2 𝑒
√ √
𝑅𝑒 𝑠 4 𝐼𝑚 𝑠 4 |𝑠 | 4√2 ∠𝑠 𝑜𝑟
Imaginary axis
Q.2 (14)
𝒙𝟏 𝒕 𝒙𝟏 𝒕 𝒙𝟏 𝒕 𝒙𝟏 𝒕
a) Even Odd
x1(t) - even
x1(t) - odd
b) Even part of 𝑥 𝑛 0, 2.5, 2, 3, 1, 4, 4, 4, 1, 3, 2, 2.5, 0
x2[n] - even
Odd part of 𝑥 𝑛 0, 2.5, 2, 1, 2, 2, 0, 2, 2, 1, 2, 2.5, 0
x2[n] - odd
c) Fundamental period T = 10 seconds
x3(t)
d) Fundamental period N = 7 points
x4[n]
e) Time‐shift (shift to the right by 3 seconds) first and then followed by time scaling (i.e. expansion) and time reversal.
4
x5(t)
-2
-4
-10 -8 -6 -4 -2 0 2 4 6
time (sec)
Q.3 (8)
.
a) 𝑅𝑒 𝑥 𝑡 𝑒 cos 𝑡 𝑢 𝑡 𝑢 𝑡 9
b)
Re[x(t)]
.
c) 𝐼𝑚 𝑥 𝑡 𝑒 sin 𝑡 𝑢 𝑡 𝑢 𝑡 9
d)
Im[x(t)]
Q.4 (8)
a) |𝑥 𝑛 | 0.95 𝑢𝑛 𝑢𝑛 9
b)
c) ∠ 𝑥 𝑛 𝑛 𝑢𝑛 𝑢 𝑛 9
d) = 3.1416 (The phase varies between and .)
Q.5 (10)
a) 𝜔 2000𝜋 rad/s sin 𝑛 sin 2𝜋𝑛 𝑛 sin 𝑛 𝜔
b) f = 1000 Hz f = 1/16
c) T = 1/1000 seconds = 1 ms N = 16 points
For DT signals, the highest fundamental angular frequency is while the highest fundamental ordinary frequency is .
/√
𝑒 √ cos 𝑡 𝑅𝑒 𝑒 √ 𝑅𝑒 𝑒 where 𝑠 √3 𝑗 √3 1 𝑒 2𝑒
|s| ∠
d) 𝑒 𝑠 𝑒 𝑒 𝑒
𝑥 𝑡 𝑅𝑒 2 𝑒 𝑒 √ 2 𝑒 √ cos 𝑡 13 2 𝑒 √ cos 𝑡
/
e) 𝑅𝑒 𝑒 𝑑𝜏 𝑅𝑒 𝑒
/
/
𝑒√ cos 𝜏 ∠𝑠 𝑒 √ / cos 𝑒 √ / cos
| | /
1 𝜋√3/6 1 𝜋√3/6
𝑒 𝑒
2 4
Q.6 (10)
Determine the properties for each of the following CT / DT systems. (True = Yes , False = F)
Causal Linear Time‐Invariant Stable Memoryless
a) (5) 𝑦 𝑡 𝑥 𝜏 𝑑𝜏 F Yes F F F
b) (5) 𝑦 𝑛 cos 𝑥 𝑛 1 1 Yes F Yes Yes F
Q.7 (10)
a) 𝑦 𝑡 𝑥 𝑡 ∗ℎ 𝑡 𝑥 𝜏 ℎ 𝑡 𝜏 𝑑𝜏 3 𝑥 𝜏 𝑑𝜏
1 𝑡 2 𝜏 𝑡 4
ℎ 𝑡 𝜏 ℎ 𝑡 3 𝑢 𝑡 4 𝑢 𝑡 2
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
b) 3 c) 𝑦 𝑡 𝑥 𝑡 ∗ℎ 𝑡 ℎ 𝑡 2
3
t
4 0 2 t
2 0 4
d) 𝑦 𝑡 𝑥 𝑡 ∗ℎ 𝑡
1 3 6
t *
t t
2 4 4 0 2 2 0 4 6
e) 𝑦 𝑡 ℎ 𝑡 ∗𝑥 𝑡 ℎ 𝑡 ∗ 𝑥 𝑡 𝑥 𝑡 𝑦 𝑡 𝑦 𝑡
Q.8 (10)
a) 𝑥 𝑛 𝛿𝑛 1 𝛿𝑛 2 𝛿𝑛 3 𝛿𝑛 4
𝑦 𝑛 𝑥 𝑛 ∗ℎ 𝑛 ℎ𝑛 1 ℎ𝑛 2 ℎ𝑛 3 ℎ𝑛 4 ∑ ℎ𝑛 𝑘
b)
1
0 1 2 3 n
c)
y[n]
d) 𝑦 𝑛 𝛿𝑛 1 2𝛿 𝑛 2 3𝛿 𝑛 3 3𝛿 𝑛 4 2𝛿 𝑛 5 𝛿𝑛 6
e) Window averager / smoother