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

Generate 1000 independent samples of a Gaussian random variable $X$ with mean 0 and variance 1. Filter the samples using [link] . We will denote the filtered signal $Y_i$ , $i=1,2,\cdots ,1000$ .

Draw 4 scatter plots using the form subplot(2,2,n) , $(n=1,2,3,4)$ . The first scatter plot should consist of points, $(Y_i,Y_{i+1})$ , $(i=1,2,\cdots ,900)$ . Notice that this correlates samples that are separated by a lag of “1”.The other 3 scatter plots should consist of the points $(Y_i,Y_{i+2})$ , $(Y_i,Y_{i+3})$ , $(Y_i,Y_{i+4})$ , $(i=1,2,\cdots ,900)$ , respectively. What can you deduce about the random process from these scatter plots?

For real applications, the theoretical autocorrelation may be unknown. Therefore, $r_{YY}\left(m\right)$ may be estimated by the sample autocorrelation , $r_{YY}^{\text{'}}\left(m\right)$ defined by

where $N$ is the number of samples of $Y$ .

Use Matlab to calculate the sample autocorrelation of $Y_n$ using [link] . Plot both the theoretical autocorrelation $r_{YY}\left(m\right)$ , and the sample autocorrelation $r_{YY}^{\text{'}}\left(m\right)$ versus $m$ for $-20\le m\le 20$ . Use subplot to place them in the same figure. Does [link] produce a reasonable approximation of the true autocorrelation?

Inlab report

1. Show your derivation of the theoretical output autocorrelation, $r_{YY}\left(m\right)$ .
2. Hand in the four scatter plots. Label each plot with the corresponding theoretical correlation, from $r_{YY}\left(m\right)$ . What can you conclude about theoutput random process from these plots?
3. Hand in your plots of $r_{YY}\left(m\right)$ and $r_{YY}^{\text{'}}\left(m\right)$ versus $m$ . Does [link] produce a reasonable approximation of the true autocorrelation? For what value of $m$ does $r_{YY}\left(m\right)$ reach its maximum? For what value of $m$ does $r_{YY}^{\text{'}}\left(m\right)$ reach its maximum?
4. Hand in your Matlab code.

Correlation of two random processes

Background

The cross-correlation is a function used to describe the correlation between two separate random processes.If $X$ and $Y$ are jointly WSS random processes, the cross-correlation is defined by

Similar to the definition of the sample autocorrelation introduced in the previous section,we can define the sample cross-correlation for a pair of data sets. The sample cross-correlation between two finite random sequences $X_n$ and $Y_n$ is defined as

where $N$ is the number of samples in each sequence. Notice that the cross-correlation is not an even function of $m$ . Hence a two-sided definition is required.

Cross-correlation of signals is often used in applications of sonar and radar, for exampleto estimate the distance to a target. In a basic radar set-up,a zero-mean signal $X\left(n\right)$ is transmitted, which then reflects off a target after traveling for $D/2$ seconds. The reflected signal is received, amplified, and then digitized to form $Y\left(n\right)$ . If we summarize the attenuation and amplification of the received signal by the constant $\alpha$ , then

where $W\left(n\right)$ is additive noise from the environment and receiver electronics.

In order to compute the distance to the target, we must estimate the delay $D$ . We can do this using the cross-correlation.The cross-correlation $c_{XY}$ can be calculated by substituting [link] into [link] .

Here we have used the assumptions that $X\left(n\right)$ and $W(n+m)$ are uncorrelated and zero-mean. By applying the definition of autocorrelation,we see that

Because $r_{XX}(m-D)$ reaches its maximum when $m=D$ , we can find the delay $D$ by searching for a peak in the cross correlation $c_{XY}\left(m\right)$ . Usually the transmitted signal $X\left(n\right)$ is designed so that $r_{XX}\left(m\right)$ has a large peak at $m=0$ .

Experiment

Download the file radar.mat for the following section.

Using [link] and [link] , write a Matlab function C=CorR(X,Y,m) to compute the sample cross-correlation between two discrete-time random processes, $X$ and $Y$ , for a single lag value $m$ .

To test your function, generate two length 1000 sequences of zero-mean Gaussian random variables, denoted as $X_n$ and $Z_n$ . Then compute the new sequence $Y_n=X_n+Z_n$ . Use CorR to calculate the sample cross-correlation between $X$ and $Y$ for lags $-10\le m\le 10$ . Plot your cross-correlation function.

Inlab report

1. Submit your plot for the cross-correlation between $X$ and $Y$ . Label the $m$ -axis with the corresponding lag values.
2. Which value of $m$ produces the largest cross-correlation? Why?
3. Is the cross-correlation function an even function of $m$ ? Why or why not?
4. Hand in the code for your CorR function.

Next we will do an experiment to illustrate how cross-correlation can be used to measure time delay in radar applications.Down load the MAT file radar.mat and load it using the command load radar . The vectors trans and received contain two signals corresponding to the transmitted and received signalsfor a radar system. First compute the autocorrelation of the signal trans for the lags $-100\le m\le 100$ . (Hint: Use your CorR function.)

Next, compute the sample cross-correlation between the signal trans and received for the range of lag values $-100\le m\le 100$ , using your $CorR$ function. Determine the delay $D$ .

Inlab report

1. Plot the transmitted signal and the received signal on a single figure using subplot . Can you estimate the delay $D$ by a visual inspection of the received signal?
2. Plot the sample autocorrelation of the transmitted signal, $r_{XX}^{\text{'}}\left(m\right)$ vs. $m$ for $-100\le m\le 100$ .
3. Plot the sample cross-correlation of the transmitted signal and the received signal, $c_{XY}^{\text{'}}\left(m\right)$ vs. $m$ for $-100\le m\le 100$ .
4. Determine the delay $D$ from the sample correlation. How did you determine this?