EE 558 HW-1 (Spring 2012)
1-) Prove the following properties of the cumulative distribution functionlisted below:
2-) Consider the following question:
3-) Consider the following question:
4-) Consider the following question:
5-) Consider the following question:
6-) Consider the following question:
7-) Consider the problem below:
8-) Consider the problem below:
9-) Let , i = 1, 2 denote two estimates of the same scalar parameter θ. Assume, with N
denoting the number of data points, that
where var (·) is the abbreviation for variance (·). The mean square error (mse) is defined as
). Which one of , is the best estimate in terms of mse? Comment on the
result.
10-)For most consistent estimators of the parameters of stationary processes, the variance of the estimation error tends to zero as 1/N when N—> oo (N == the number of data points ). For nonstationary processes, faster convergence rates may be expected. To see this, derive the variance of the least squares estimate of α in
y(i) = αt + e(t), t = 1,2,..., N
where e(t) is white noise with zero mean and variance .
11-)Let be a sequence of independent and identically distributed Gaussian random
variables with mean μ and variance σ. Both μ and σ are unknown. Consider thefollowing estimate of μ:
Determine means and variances of the estimates , and . Discuss their
(un)biasedness and consistency properties. Compare and in terms of their mse’s.
12-)Let e(t) denote white noise of zero mean and variance . Consider the filtered white
noise process
where denotes the unit delay operator. Calculate the covariances
of y(k).
13-)Let denote the spectral density function of a stationary signal u(t):
Assume that
which guarantees the existence of . Show that has the following properties:
(a) is real valued and = .
(b) 0 for all ω.
Hint. Set . Then show that
and find out what happens when η tends to infinity.
14-)Let
be two stable matrix transfer functions, and let e(t) be a white noise of zero mean andcovariance matrix Λ (ηIη). Show that
The first equality above is called Parseval's formula. The second equality provides aninterpretation' of the terms occurring in the first equality.