Digital Communication Exercises.docx

ETC-M2

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)

2016-2017

𝑐𝑜𝑠⁡(2𝜋𝑡) −3 ≤ 𝑡 ≤ 3 0 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒 𝑡 0≤𝑡≤2 𝑥(𝑡) = {4 − 𝑡 2 ≤ 𝑡 ≤ 4 0 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒

ETC-M2

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)

2016-2017

1

1

𝑥(t) = cos (3 t) + sin (4 t) 𝑥(t) = cos 4 (t)