Lysario – by Panagiotis Chatzichrisafis

"ούτω γάρ ειδέναι το σύνθετον υπολαμβάνομεν, όταν ειδώμεν εκ τίνων και πόσων εστίν …"

Archive for the ‘Solved Problems’ Category

In [1, p. 94 exercise 4.2] we are asked to consider the estimator
\hat{P}_{AVPER}(0)=\frac{1}{N}\sum\limits_{m=0}^{N-1}\hat{P}_{PER}^{m}(0) (1)

\hat{P}_{PER}^{m}(0)= x^{2}[m] (2)

for the process of Problem 4.1. We are informed that this estimator may be viewed as an averaged periodogram. In this point of view the data record is sectioned into blocks (in this case, of length 1) and the periodograms for each block are averaged. We are asked to find the mean and variance of  \hat{P}_{AVPER}(0) and compare the result to that obtained in [2]. read the conclusion >
In problem [1, p. 94 exercise 4.1] we will show that the periodogram is an inconsistent estimator by examining the estimator at f=0, or
\hat{P}_{PER}(0)=\frac{1}{N}\left(\sum\limits_{n=0}^{N-1}x[n]\right)^{2}. (1)

If x[n] is real white Gaussian noise process with PSD
P_{xx}(f)=\sigma_{x}^{2} (2)

we are asked to find the mean an variance of hat{P}_{PER}(0). We are asked if the variance converge to zero as N \rightarrow \infty. The hint provided within the exercise is to note that
\hat{P}_{PER}(0)= \sigma_{x}^{2} \left(\sum\limits_{n=0}^{N-1}\frac{x[n]}{\sigma_{x}\sqrt{N}}\right)^{2} (3)

where the quantiiy inside the parenthesis is  \sim N(0,1) read the conclusion >
In [1, p. 62 exercise 3.19] we are asked to find for the multiple sinusoidal process
x[n]=\sum\limits_{i=1}^{P}A_{i}\cos(2\pi f_{i}n+\phi_{i})

the ensemble ACF and the temporal ACF as M\rightarrow \infty , where the \phi_{i}‘s are all uniformly distributed random variables on [0, 2 \pi) and independent of each other. We are also asked to determine if this random process is autocorrelation ergodic. read the conclusion >
In [1, p. 62 exercise 3.18] we are asked to find the temporal autocorrelation function for the real sinusoidal random process of Problem [1, p. 62 exercise 3.16]:

as M\rightarrow \infty. As a second step we are asked to determine if the random process autocorrelation ergodic. read the conclusion >
In [1, p. 62 exercise 3.17] we are asked to verify that the variance of the sample mean estimator for the mean of a real WSS random process

is given by [1, eq. (3.60), p. 58]. For the case when x[n] is real white noise we are asked to what the variance expression does reduce to. A hint that is given is to use the relationship from [1, eq. (3.64), p. 59]. read the conclusion >

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