Name:
Winter 2006 Quiz # 2 ECE S-510
You are given a periodic signal x(t) with period 2
π
and you want to approximate it by
x
(t)= a0 + a1 cos{4t} + a2 cos{6t} + b1 sin{6t}
such that the average error e2(t) = ||x(t) -
x
(t)|| 2 over one period is minimized.
(a) Find the least squares estimates for a0, a1, a2 and b1 that will minimize e2(t).
(b) What is the value of e2min(t) if the average power associated with x(t) is Pav.
If y(t) =
x
(t) + n(t), where n(t) is a zero mean Gaussian white noise signal (i.e., for any t
≠
τ
, n(t) and n(
τ
) are independent identically distributed) with variance
σ
2
.
(c) Find the MMSE estimates for a0, a1, a2 and b1.
(d) How are a0, a1, a2 and b1 statistically distributed (i.e., find the probability density
function associated with a0, a1, a2 and b1)?
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