Description
Problem 1
A rather common problem that appears regularly in signal processing is that of estimating the
total phase ϕk = 2πf tk + θ. The frequency of the signal f is known and, thus, the dynamic model
is ˙ϕ = 2πf. The measurement model relates a measurement of its amplitude yk to ϕk as follows:
yk = sinϕk + vk = ŷk + vk, vk ∼ N(0, Rk)
This is a nonlinear measurement model. What we want to show in this problem is how a nonlinear
measurement model results in a predicted measurement ˆyk with a distribution that is very different
from that of a priori one or that of the measurement y. For what follows, ignore the measurement
noise and just focus on the random variable ϕk.
a) Derive an analytical expression for fŶ (y) when ϕ ∼ U(0, π)?
b) Using the pdf you derived in (a), determine E(Ŷ) and σ
2
Ŷ when ϕ ∼ U(0, π).
c) Using MATLAB draw 10,000 samples of ϕ and run a simple Monte Carlo experiment to
generate samples of ˆy. Using the generated data approximate E(Ŷ) and σ
2
Ŷ
(you can use the
MATLAB commands mean and var). Also plot a histogram-based approximation of fY (y).
To do this, use the histogram command in MATLAB (use at least 50 bins with histogram
and the option ‘Normalization’ set to ‘pdf’). How well do the Monte Carlo results for
E(Ŷ), σ
2
Ŷ
and fŶ (y) agree to the analytical results?
d) Use the analytical approach or a Monte Carlo simulation (your choice) to determine E(Ŷ),
σ
2
Ŷ
and fŶ (y) when ϕ ∼ N(0, 1). Which approach did you use (analytical or Monte Carlo)?
Why?
Problem 2
Do Problem 13.3 in your text.
Problem 3
Do Problem 14.4 in your text.
Problem 4
Do Problem 15.2 in your text.
Problem 5
Do Problem 15.1 in your text.
Problem 6
Do 15.15 in your text.
2
