Computer$-$Based HW#$2$

Submission Deadline: $12/2/2024$

Q$-1$: This exercise is to look at how accurately the sample autocorrelation is in estimating the autocorrelation and spectral density of white process.

a$.$ Generate $N=100000$ samples of the process $x(n),$ which takes on the values $+-1$ with equal probabilities.

b$.$ Estimate the first $10$ lags of the autocorrelation sequence as follows.

hat$(R{)}_{}(k)=\frac{1}{N}{\displaystyle \underset{n=0}{\overset{N-1}{}}}x(n)x(n-k),k=0,+-1,$dots,$+-9$

How close is your estimate, hat$(R{)}_{}(k),$ to the true autocorrelation sequence, $R_{}(k)=(k)?$ Plot both on the same figure.

c$.$ Estimate the power spectral density of the process generated in Part $($a$),$ in the range from $[0,2),$ and compare it with the true power spectral density. Plot both on the same figure.

Q$-2$: The process described in Q$-1$ is applied to an LTI communication channel to produce the output

$y(n)=h(0)x(n)-h(1)x(n-1)$

where $h(0)$ and $h(1)$ are the coefficients of channel impulse response. The files named 'Xin' and 'Yout' give the samples of the processes $x(n)$ and $y(n),$ respectively.

a$)$ Determine the two coefficients, $h(0)$ and $h(1),$ of impulse response of the channel.

b$)$ Estimate the power spectral density of the process $y(n),$ in the range from $[0,2),$ and compare it with the true power spectral density. Plot both on the same figure.

Remark: You may find the following Matlab commands useful: 'xcorr', $\text{'}$fft$\text{'},$ 'xlim', and 'ylim'.