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.