Q1. Suppose the noise of a certain node of an electronic circuit can be modeled as a continuous-time, wide-sense stationary random process {X(t)} that is Gaussian with zero-mean and with the following autocorrelation function:

a) Compute the Pr {(X(5) +2X(6))² > 7}

b) For what values of t and s, if any, are X(t) and X(s) independent?

c) For what values of t and s, if any, are X(t) and X(s) statistically identical?

Q2. A signal S(t) is corrupted by additive noise N(t) and then filtered. The output of the filter is denoted as Y(t).

Assume that S(t) and N(t) are independent of each other and that they are wide-sense stationary random processes with zero-means and PSD functions:

a) Find the power of the noise process {N(t)}

b) Find the power in the output signal {Y(t)}

c) How much of the power in {Y(t)} is due to noise?

Q3. For a Poisson process, show that for s < t,

Q4. When a patient is screened for the presence of a disease in an organ, a section of tissue is viewed under a microscope and a count of abnormal cells is made. Even under healthy conditions, a small number of abnormal cells will be present. Presumably a much larger number will be present if the organ is diseased. Assume that the number L of abnormal cells in a section is geometrically distributed:

Pr[L = l] = (1 - α)α^l, I = 0, 1, . . .

The parameter α₁ of a diseased organ will be larger than that of a healthy one, α₀. The prior probability of a randomly selected organ being diseased is p.

a. Assuming that the values of the parameter α₀ and α₁ are known in each situation, and assuming uniform costs, find the Bayesian rule of deciding whether an organ is diseased.

b. What is the probability of correctly diagnosing a diseased organ?

Q5. Detecting a damped exponential. Suppose we have the following hypothesis test:

H₀: x(n) = w(n)

H₁: x(n) = Arⁿ + w(n)

where w(n) are iid ∼ N(0, σ²), r is known, and A is unknown. Based on x(n), n = 0,...,N-1, show that the GLRT decides H₁ if A² > γ where A is the MLE of A.

Q6. Suppose we observe

xi = θ + w_i, I = 1, 2

where θ is unknown and w_i are independent zero-mean Gaussian random variables with var(w_i) =1 and var(w₂) = 2

Based on the two observations x₁ and x₂,

a) Find the maximum likelihood estimate for θ.

b) Compute the CRLB for unbiased estimates for θ.