# CH5115: Parameter and State Estimation Assignment 2 solution

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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
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
).