Lysario – by Panagiotis Chatzichrisafis

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

Archive for Januar, 2012

In [1, p. 61 exercise 3.11] it is asked to repeat problem [1, p. 61 exercise 3.10] (see also the solution [2] ) for the general case when the predictor is given as
\hat{x}[n]=-\sum\limits_{k=1}^{p}\alpha_{k}x[n-k]. (1)

Furthermore we are asked to show that the optimal prediction coefficients \{\alpha_{1},\alpha_{2},..., \alpha_{p}\} are found by solving [1, p. 157, eq. 6.4 ] and the minimum prediction error power is given by [1, p. 157, eq. 6.5 ]. read the conclusion >
The desire to predict the complex WSS random process based on the sample x[n-1] by using a linear predictor
\hat{x}[n]=-\alpha_{1}x[n-1] (1)

is expressed in [1, p. 61 exercise 3.10]. It is asked to chose \alpha_{1} to minimize the MSE or prediction error power
MSE = \mathcal{E}\left\{\left| x[n] -\hat{x}[n] \right|^{2} \right\}. (2)

We are asked to find the optimal prediction parameter \alpha_{1} and the minimum prediction error power by using the orthogonality principle.
read the conclusion >
In [1, p. 61 exercise 3.9] we are asked to consider the real linear model
x[n]=\alpha + \beta n + z[n]  \; n=0,1,...,N-1 (1)

and find the MLE of the slope \beta and the intercept \alpha by assuming that z[n] is real white Gaussian noise with mean zero and variance \sigma_{z}^{2}. Furthermore it is requested to find the MLE of \alpha if in the linear model we set \beta=0. read the conclusion >

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