Let n(t) be a white Gaussian noise process with noise (power) spectral density N₀/2. Define the following two random variables:

where the functions 1(t) and 2(t) satisfy the following properties:

Answer the following questions:

What kind of random variables are n1 and n2?

Hint: The answer can be found using the following facts:

For an input x(t), output y(t), and any function (t), the following is a linear time-invariant (LTI) operation:

If y(t) is a random process, then y(T) is a sample of y(t) at time t = T; hence, y(T) is a random variable.

(b) (6 Points) Determine the mean and variance of random variables n1 and of n2.

(c) (3 Points) Determine the covariance between random variables n1 and n2.

(d) (3 Points) Are n1 and n2 independent? Why?

ni = n1 01(t)n(t)dt 0 n2 = =1" 02(t)n(t)dt, 10:(t)/dt = 1, i=0,1. T 01(t)$2(t)dt = 0. y(t) = [ (1)x(1)ax. ni = n1 01(t)n(t)dt 0 n2 = =1" 02(t)n(t)dt, 10:(t)/dt = 1, i=0,1. T 01(t)$2(t)dt = 0. y(t) = [ (1)x(1)ax