$($programming$)$ Construct a random, length$-1024,$ zero$-$mean, $1-$variance AR$-1$ sequence array with

$=0\mathrm{.}9.$ You may use "rand" function of Matlab $($with appropriate parameters$)$ in a for loop during

the generation of this random sequence with given statistics. Theoretically, the $88$ autocorrelation

matrix of this process is expected to be close to:

$R=\left[\begin{array}{ccc}1& \text{c}\text{d}\text{o}\text{t}\text{s}& {}^7\\ \text{v}\text{d}\text{o}\text{t}\text{s}& \text{d}\text{d}\text{o}\text{t}\text{s}& \text{v}\text{d}\text{o}\text{t}\text{s}\\ {}^7& \text{c}\text{d}\text{o}\text{t}\text{s}& 1\end{array}\right]$

You are supposed to construct the $88KLT$ matrix both from the above theoretical

autocorrelation, and from the numerical approximation of $R(0),R(1),$dots,$R(7)$ by multiplying

corresponding shifted samples and averaging: $R(k)=$average$\{x[n]*x[n-k]\}$ through your

generated data.

You need to find the eigenvectors $($corresponding to largest eigenvalues$)$ of the two cases of the

autocorrelation matrices to construct the basis matrix: $U,$ and then evaluate $T=U^{-1}.$

Plot the $64$ basis images $($with proper scaling for a nice gray$-$scale image output$)$ of both cases by

taking the outer products of columns of $U.$

Finally, plot the basis images for the $8$x$8$ DCT transform. Are the above basis images similar to

DCT basis images? Which would you prefer? Why?