TSE-M1
Digital Communication
Homework 1
1- Determine if the following signals are periodic. If yes, calculate the
fundamental period for the signals.
−5𝜋𝑡
𝜋
i)
𝑥(𝑡) = |sin (
ii)
𝑥(𝑡) = sin (
iii)
𝑥(𝑡) = exp (𝑗
iv)
v)
𝑥[𝑘] = 5 × (−1)𝑘
7𝜋𝑘
3𝑘
𝑥[𝑘] = exp (𝑗 4 ) + exp⁡(𝑗 4 )
vi)
𝑥[𝑘] = sin (
vii)
𝑥[𝑘] = exp (𝑗
8
6𝜋𝑡
+ 2 )|
3𝑡
) + 2cos⁡( 5 )
7
3𝜋𝑡
8
3𝜋𝑘
𝜋𝑡
) + exp⁡(86)
) + 𝑐𝑜𝑠⁡(
8
7𝜋𝑘
4
63𝜋𝑘
)
64
4𝜋𝑘
) + 𝑐𝑜𝑠⁡(
7
+ 𝜋)
2- Determine if the following signals are even, odd, or neither-even-nor-odd. In
the later case, evaluate and sketch the even and odd components of the CT
signals.
i)
ii)
iii)
iv)
v)
vi)
𝑥(𝑡) = 2sin⁡(2𝜋𝑡)[2 + cos⁡(4𝜋𝑡)]
𝑥(𝑡) = 𝑡 2 + cos⁡(3𝑡)
3𝑡
0≤𝑡≤2
6
2≤𝑡≤4
𝑥(𝑡) = {
3(−𝑡 + 6) 4 ≤ 𝑡 ≤ 6
⁡⁡⁡⁡⁡⁡⁡⁡0 ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
2𝜋𝑘
𝑥[𝑘] = sin(4𝑘) + cos⁡( 3 )
7𝜋𝑘
𝑥[𝑘] = exp⁡(𝑗 4 ) + cos⁡(
𝑘
𝑥[𝑘] = {(−1) 𝑘 ≥ 0
1
𝑘<0
4𝜋𝑘
7
+ 𝜋)
3- Determine if the following signals are energy or power or neither. Calculate
the energy and power of the signals in each case.
i)
𝑥(𝑡) = 𝑐𝑜𝑠⁡(2𝜋𝑡)sin⁡(3𝜋𝑡)⁡
ii)
𝑥(𝑡) = {
iii)
2014-2015
𝑐𝑜𝑠⁡(2𝜋𝑡) −3 ≤ 𝑡 ≤ 3
0
𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
𝑡
0≤𝑡≤2
𝑥(𝑡) = {4 − 𝑡 2 ≤ 𝑡 ≤ 4
0
𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
TSE-M1
iv)
𝑥[𝑘] = 𝑐𝑜𝑠⁡(𝜋𝑘)sin⁡(3𝜋𝑘)
v)
𝑥[𝑘] = (−1)𝑘
vi)
(−1)𝑘
𝑥[𝑘] = { 1
0
0≤𝑘≤2
2≤𝑘≤4
𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
4- Evaluate the following integrals
+∞
i)
∫−∞ (t − 1)δ(t − 5)dt
ii)
∫−∞ ( 3 − 5) δ ( 4 − 6) dt
iii)
∫−∞ exp(t − 1)sin⁡(
iv)
∫−∞ [sin (
v)
∫−∞ [u(t − 6) − u(t − 10)] sin (
+∞ 2t
3t
+∞
+∞
3πt
4
5
π(t+5)
4
)δ(1 − t)dt
) + exp(−2t + 1)]δ(−t − 5)dt
+∞
3πt
4
) δ(t − 5)dt
5- Find the Fourier transform of
i)
ii)
iii)
iv)
x(t) = te−t u(t)⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(a > 0)
x(t) = sin⁡(ω0 t)
δT (t) = ∑∞
n=−∞ δ(t − nT)
x(t) = cos⁡(ω0 t) + 𝑠𝑖𝑛2 (ω0 t)
6- Find the Fundamental period T0 and the Fourier coefficients cn of the signal:
i)
ii)
2014-2015
1
1
𝑥(t) = cos (3 t) + sin (4 t)
𝑥(t) = cos 4 (t)