Name:
Winter 2006 Quiz # 4 ECE S-510
You are given a periodic signal x(t) =
c
x
(t)+ n(t), where
c
x
(t)= a0,c + a1,c cos{2
π
t}, with
a0,1 = 2 and a1,1= 0, and a0,2 = 0 and a1,2= 1, and where n(t) is a zero mean Gaussian
white noise signal (i.e., for any t ≠
τ
, n(t) and n(
τ
) are i.i.d.) with variance 4.
(a) If a0* = < x(t), 1> and a1* = <x(t), cos{2
π
t}>, and a*= (a0*, a1*). Find the
p(a*|
ω
1) and p(a*|
ω
2).
(b) Find a Bayesian classifier in its simplest form that classifies the given signal
x(t) into either class
ω
1 or class
ω
2, given that we observe x(t) over the time
interval (0,1). Assume that the prior class probabilities are p(
ω
1) = p(
ω
2).
(c) If you want to use either the dc term or the first harmonic alone to decide on the
present class, which one would you use and why?
Fact: If X is a 2D vector which is normally distributed with mean
µ
and covariance
matrix
[Σ]
, then p(X) =
]det[2
1
Σ
π
exp{-
2
1
(X-
µ
)
t
[Σ]
-1(X-
µ
)}
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