## Description

1. (a) Determine if the random process v[k] = A cos2

(2πf k + φ), where φ is a constant but A

is a random variable with zero mean and unit variance, is covariance stationary.

(b) The random walk process v[k] = v[k − 1] + e[k] is known to be variance non-stationary.

Assuming v[0] = 0, prove this result. Verify your finding numerically using Matlab.

2. A process evolves as y[k] = y

?

[k] + e[k], where y

?

[k] = b

0

2

q

−2

1 + f

0

1

q−1

u[k], u[k] is a known signal

and y[k] is the measured version of y

?

[k]. The measurement noise is e[k] ∼ WN(0, σ2

e

) and

u[k] ∼ WN(0, σ2

u

). Assume σeu[l] = 0, ∀l.

(a) Develop expressions for σ

2

y

, σyy[1], σyu[1], and σyu[2] in terms of the variances of u[k]

and the white-noise sequences, i.e., σ

2

u

and σ

2

e

respectively.

(b) Generate N = 500 observations of y[k] with σ

2

u = 2. Adjust σ

2

e

such that the SNR σ

2

y

? /σ2

e

is set to 10. Estimate the quantities (variance, auto-covariance and cross-covariance) in

(2a) and compare their closeness with the theoretical answers in (2a).

3. For the series given in a2_q3.mat,

(a) Determine the presence of any integrating effects.

(b) Fit a suitable ARIMA model. Report all the necessary steps and the final model.

4. (a) For a GWN process y[k] ∼ N (µ, σ2

), where 0 ≤ µ < ∞, derive the ML estimate and

Fisher information of µ given N observations and known σ

2

.

(b) Consider the linear regression problem Y = aX +b+ε. Determine the Fisher information

of parameters a and b contained in N observations {(y[k], x[k])}

N

k=1 assuming X is free of

randomness and ε ∼ N (0, σ2

e

).

Verify your analytical answer (for the ML estimate) with

simulation in Matlabby plotting the likelihood functions and locating the maximum.

Choose N = 100, σ

2

e = 1, a = 2, b = 3 and µ0 = 1 (true value).