Telecommunication homework.

1. State whether y[n] = x² [n] is memoryless on not. Explain your answer.

2. Define a homogeneous system Is a system described by y[n] =x² [n]/x[n-1] homogeneous ?. Explain

3. Explain what is meant by a shift invariant system. State whether a system described by y[n] = x[n] + x[-n] is shift variant. Justify your answer

4. Define a causal system and classify system defined by y[n] = x[n] + x[n + 2] as causal or non-causal with a suitable justification.

5. Compute the discrete time convolution y(n) = x[n]*h[n] where x[n] = aⁿ u[n] and h[n]= u[n]

6. Prove the commutative property of convolution

7. Compute the DT convolution y(n) = x[n]*h[n]

given

x[n] = {1 10 n 20

{0 Otherwise

h[n]= = {n -5 n 5

{0 Otherwise

8. If the response of a linear shift-invariant system to a unit step is s[n] = n(1/2)ⁿ u[n] Find the unit sample response h[n]

9. Given two sequences x[n] = u[n] -u [n-6] and h[n] = u[n-2] -u[n-5]

i. Plot x[n] and h[n]

ii. Compute the correlation between the sequences

iii. Plot the correlation

10. Given two sequences x1[n] = {3,1,2} and x2[n] = [-1,2,2}

Compute the cross correlation Rxy and Ryx