0.10 Lab 7b - discrete-time random processes (part 2)  (Page 2/3)

Use Matlab to generate 1000 i.i.d. samples of $X$ , denoted as $X_1$ , $X_2$ , ..., $X_{1000}$ . Next, generate 1000 i.i.d. samples of $Y$ , denoted as $Y_1$ , $Y_2$ , ..., $Y_{1000}$ . For each of the four choices of $Z$ , perform the following tasks:

1. Use [link] to analytically calculate the correlation coefficient ${\rho }_{XZ}$ between $X$ and $Z$ . Show all of your work.Remember that independence between $X$ and $Y$ implies that $E\left[XY\right]=E\text{[}X\text{]}E\text{[}Y\text{]}$ . Also remember that $X$ and $Y$ are zero-mean and unit variance.
2. Create samples of $Z$ using your generated samples of $X$ and $Y$ .
3. Generate a scatter plot of the ordered pair of samples $(X_i,Z_i)$ . Do this by plotting points $(X_1,Z_1)$ , $(X_2,Z_2)$ , ..., $(X_{1000},Z_{1000})$ . To plot points without connecting them with lines, use the '.' format, as in plot(X,Z,'.') . Use the command subplot(2,2,n) (n=1,2,3,4) to plot the four cases for $Z$ in the same figure. Be sure to label each plot using the title command.
4. Empirically compute an estimate of the correlation coefficient using your samples $X_i$ and $Z_i$ and the following formula.
   $${\widehat{\rho }}_{XZ}=\frac{{\sum }_{i=1}^N(X_i-{\widehat{\mu }}_X)(Z_i-{\widehat{\mu }}_Z)}{\sqrt{{\sum }_{i=1}^N{(X_i-{\widehat{\mu }}_X)}^2{\sum }_{i=1}^N{(Z_i-{\widehat{\mu }}_Z)}^2}}$$

Inlab report

1. Hand in your derivations of the correlation coefficient ${\rho }_{XZ}$ along with your numerical estimates of the correlation coefficient ${\widehat{\rho }}_{XZ}$ .
2. Why are ${\rho }_{XZ}$ and ${\widehat{\rho }}_{XZ}$ not exactly equal?
3. Hand in your scatter plots of $(X_i,Z_i)$ for the four cases. Note the theoretical correlation coefficient ${\rho }_{XZ}$ on each plot.
4. Explain how the scatter plots are related to ${\rho }_{XZ}$ .

Autocorrelation for filtered random processes

In this section, we will generate discrete-time random processes and then analyze their behavior using the correlation measure introduced in the previous section.

Background

A discrete-time random process $X_n$ is simply a sequence of random variables. So for each $n$ , $X_n$ is a random variable.

The autocorrelation is an important function for characterizing the behavior of random processes.If $X$ is a wide-sense stationary (WSS) random process, the autocorrelation is defined by

Note that for a WSS random process, the autocorrelation does not vary with $n$ . Also, since $E\left[X_n{X}_{n+m}\right]=E\left[X_{n+m}{X}_n\right]$ , the autocorrelation is an even function of the “lag” value $m$ .

Intuitively, the autocorrelation determines how strong a relation there is between samples separated by a lag value of $m$ . For example, if $X$ is a sequence of independent identically distributed (i.i.d.) random variableseach with zero mean and variance ${\sigma }_X^2$ , then the autocorrelation is given by

We use the term white or white noise to describe this type of random process. More precisely, a random processis called white if its values $X_n$ and $X_{n+m}$ are uncorrelated for every $m\ne 0$ .

If we run a white random process $X_n$ through an LTI filter as in [link] , the output random variables $Y_n$ may become correlated. In fact, it can be shown that the output autocorrelation $r_{YY}\left(m\right)$ is related to the input autocorrelation $r_{XX}\left(m\right)$ through the filter's impulse response $h\left(m\right)$ .

Experiment

Consider a white Gaussian random process $X_n$ with mean 0 and variance 1 as input to the following filter.

Calculate the theoretical autocorrelation of $Y_n$ using [link] and [link] . Show all of your work.