Convolution for Digital Communications Solution

In this lab, we will investigate convolution, starting with using unit impulses and convolution to implement time delay, and then learn how convolution can be used to decode transmitted messages in digital communications. This lab will also be used to review functions and introduce students to loops and decision functions.

Important Concepts From Lecture You Will Use In Lab 4

Describing LTI systems using the impulse response h(t)

Computing the output y(t) of an LTI system using the graphical method of convolution

What is expected from you in Lab 4

Completion of 3 pre-lab exercises (3 points total)

Completion of 2 in-lab check offs with TA (4 points) Completion of a lab report (3 points)

Note: all pre-lab exercises must be completed by each student individually before walking in to the lab. Each student will be checked off by the TA at the beginning of lab section.

PRE-LAB:

Read the Matlab Lab 4 tutorial, particularly the sections on scaling the convolution output, for loops and decision functions.

In this lab, we will see how we can use convolution to measure the similarity between two signals. This important measurement is called correlation. The correlation between two signals can be computed from the convolution between signal x1(t) and signal h(t), and sampling the convolution output at a particular time point. Recall the convolution formula:
+

 =  1 ∗ℎ =      1ℎ  − 
,+

For this lab, we will consider the case where the signals are both of the same duration and perfectly symmetric around their midpoint in time:











In this case, the correlation is computed from the convolution by sampling the y(t) at time t when there is full overlap between x1(τ) and h(t - τ):
- =   ./0122 456728- 98:6

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The correlation measurement will be maximum when the two signals are identical, h(t)=x1(t), so this system is referred to as a matched filter. If the two signals are different, the correlation measurement will be lower, and it can be negative, as when h(t)=-x1(t).










You can use this in digital communications, where messages consisting of binary values 0 and 1 are sent by using a signal x(t) that can take two different forms (x0(t) or x1(t)) depending on which value is being sent. After the signal is transmitted over the air, the signal received r(t) will have added noise, so it will not be perfectly matched.









If the signals are chosen to be sufficiently different and the noise is not too high, the correlation measurement will still be higher for the matching signal. In this lab we will use the signals x1(t)
    • u(t) – u(t – 1) and x0(t) = -x1(t) to represent 1 and 0, respectively.
    1) Of the two plots below, which one is the correct plot for y1(t) = x1(t) ∗ x1(t)? Using the graphical method of convolution, find the peak amplitude and label critical time points for y1(t). Repeat for y0(t) = x1(t) ∗ x0(t). At what time point would you take the correlation measurement?













    2) In digital communications, we can use can correlate r(t) separately with x1(t) and then with x0(t) to decode which signal was transmitted.











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        a) Write the header for the following desired function: a function called decode that takes in the input signal x, the width d of signal x, the sampling rate Fs, the low signal x0, and the high signal x1, and then outputs the decoded symbol to the variable s.
        b) Write the Matlab decision statement for the following scenario:

            ▪ If the variable y1_corr is greater than the variable y0_corr, assign the value 1 to s

            ▪ Otherwise, assign the value 0 to s

    3) A useful digital message typically has multiple bits of information. In this lab, you will decode a message with 16 bits, where the received signal has been reshaped in to a matrix rm to separate the binary signals. Each row of rm contains one of the 16 binary high or low signals in the message:










You will need to extract each binary signal from the matrix and perform a separate correlation to decipher each bit. We will employ a for-loop to aid us in this task.

    a) What is the line of code to get the number of rows in matrix rm, the value of which will be stored in the variable rm_rows?

    b) What is the line of code to create a row vector of zeros called message_bits with the same number of elements as rm_rows?

    c) Using rm_rows and the loop index variable i, write the for-loop statement to loop through all the rows in matrix rm, starting from i = 1 to the last row of matrix rm. In other words, fill in the following line of code:
for FILL IN CODE

    d) Assuming your loop variable is i, what is the line of code to extract all the columns in the ith row of matrix rm, the values of which will be stored in the variable signal?






 EXERCISE #1: Convolution of Two Signals

In this experiment, we will investigate what happens when you convolve a signal with itself and also with a different signal. The signals we will use involve unit step functions and serve as the basis for the digital communications problem we will explore later in this lab.

Lab Exercise: Let us now perform these convolutions you analyzed using Matlab.

    a) Create a new working directory in your U Drive called Lab 4. In this directory, open up a script file and call it Ex1.m. Make sure to clear all variables and close all figures.

    b) We will first implement the high and low signals x1(t) = u(t) – u(t-1) and x0(t) = -x1(t):

        i) Using the ones function, create a vector of ones called x1 that lasts for 1 second with a sampling rate of Fs = 8000.

        ii) Using the vector x1, create the vector x0.

    c) Using the conv function, compute the convolution of the two high signals y1(t) = x1(t) ∗ x1(t), as well as the convolution between the high and the low signal y0(t) = x1(t) ∗ x0(t). Store your results in vectors y1 and y0, respectively. For each convolution, make sure to apply the correct amplitude scaling factor.

    d) Using y1, compute the corresponding time samples vector t_y. Since y1 and y0 are the same length, we will use t_y for both vectors.

    e) Using a 2 x 1 subplot, plot the following:

            ▪ y1 vs. t_y

            ▪ y0 vs. t_y
You do not need to adjust the axes, but title and label each subplot.

    f) Comment your script, and then run it. Verify that your plots and time boundaries match your answers from pre-lab. Also, verify your answer as to whether convolving a signal with itself produces the same plot as the convolution of two different signals.

Lab Check-Off #1 of 2: Show your script Ex1.m to a lab TA and demonstrate you know how to perform convolution in Matlab.

Lab Report Question #1 of 3: An inquisitive student decides to perform the convolution for

y0(t) in reverse order and computes: y0(t) = x0(t) ∗ x1(t). The student claims s/he produced a completely different plot. Explain why this CANNOT be true.

EXERCISE #2: Convolution and Matched Filtering

Lab Exercise:

a)    Open up a script file and call it Ex2.m. Clear all variables and close all figures.



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    b) Your communication high and low signals are stored in a Matlab data file CommsSignals.mat. Download the file from the class website, and then load the file. The file will contain the following three variables: x1, x0, and Fs.

    c) We will first compute the correlation between identical signals x1(t) and x1(t):

        i) Compute the convolution y1(t) = x1(t) ∗ x1(t) and store the result in vector y1. Make sure to scale the output of the convolution appropriately.

        ii) To get the correlation measurement, extract the value of the convolution output y1 at the index that corresponds to correlation measurement time that you found in the prelab. Store the correlation measurement in the variable y1_corr. Display the value of y1_corr on the COMMAND window.

    d) Repeat the steps in part (c) for the correlation between two different signals x1(t) and x0(t), storing the resulting signal in vector y0 and the correlation measurement in variable y0_corr.

    e) Comment your script, and then run your script to view your correlation measurements on the COMMAND window. Verify that your measurements are close to your answers from pre-lab. Note: we are using discrete-time signals in Matlab, so the correlation measurements will not be the same as your pre-lab answers (where continuous-time signals were analyzed).

    f) Let us now analyze the digital communication scenario from pre-lab. We will create r(t) by adding the high signal x1(t) with a random noise signal n(t):

        i) Download Noise.mat from the class website and load the file. The file will contain two variables: n and Fs. The sampling rate Fs is identical to x1 and x0.

        ii) Using the high signal x1 and the noise signal n, produce the received signal r(t) = x1(t) + n(t). Store the result in variable r.

        iii) Create the corresponding time samples vector t_r.

        iv) On a 2 x 1 figure window, plot r vs. t_r as the 1st subplot. Adjust your x-axis to be between 0 and 2. Leave y-axis unchanged. Title and label your plot

    g) Let us now compute the correlation measurement between r and x1, as well as r and x0:

        i) Compute the convolution between r and x1, and store the output in yr1. Make sure to scale the output of the convolution.

        ii) Compute the convolution between r and x0, and store the output in yr0. Make sure to scale the output of the convolution.

        iii) Using variable yr1, compute the time samples vector t_yr. We will use this vector of time samples for yr0 as well.


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        iv) On the 2nd subplot, plot both yr1 vs. t_yr (using the color magenta) and yr0 vs. t_yr (using the color red) on the same plot. No need to adjust your axes. Title and label your plots. Also, provide a legend to label each signal.

        v) Compute the correlation yr1_corr between r and x1 by sampling yr1 at the correct time. Display the output of yr1_corr on the COMMAND window.

        vi) Compute the correlation yr0_corr between r and x0 by sampling yr0. Display the output of yr0_corr on the COMMAND window.

    h) Write a function to decode a received signal.

        i) Open a new script and call it decode.m. As the first line, write your function header.

        ii) Using your code above, compute the correlation between an input signal x and both the signals x0 and x1, and sample to get the correlation measurement.

        iii) Using your decision statement from pre-lab, write the if-else statement to determine the output decoded symbol s from the correlation measurements y0_corr and y1_corr.

    i) Let us now test your decode function on a separate Matlab script. Open a new script and call it TestDecode.m. Make sure to clear all variables and close all figures.

        i) Load CommsSignals.mat to get access to the variables: x0, x1, and Fs. The signals x0 and x1 will again have the same duration of 1 second.

        ii) Decode the signal x1 by calling the function decode and passing x1 as the input signal, a duration of 1 second, and the low and the high signals x0 and x1. Store the output in the variable s1. Display the output of s1 on the COMMAND window.

        iii) Repeat (ii) for input signal x0 and store the output in the variable s0. Display the output of s0 on the COMMAND window.

        iv) Run your script, and verify you get the correct values for s1 and s0. If your function works as expected, passing the input signal x1 should produce the symbol 1, while passing the signal x0 should produce the symbol 0.


Lab Check-Off #2 of 2: Show your plots to a lab TA and demonstrate you know how to compute correlation measurements in Matlab.

Lab Report Question #2 of 3: In this exercise, we assumed signals x1(t) and x0(t) were both box signals but with different amplitudes. Suppose x0(t) had a different shape, say a triangle like the one below:





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Using your understanding of correlation, do you expect the correlation measurement now between x1(t) and x0(t) to be greater or less than the correlation measurement for the case when both signals were box signals? Explain your answer.

EXERCISE #3: Deciphering Received Message in Digital Communication System

In this exercise, we will take what you learned in the previous exercises and apply it to a digital communication system application. As shown already, we can use correlation measurements to determine whether each part of a received signal is a 0 or 1.








For this exercise, you will decipher an entire message consisting of 16 bits, where each bit is represented as the same high and low signals we have already been using, x1(t) and x0(t), both of duration 1 second for a total duration of 16 seconds for the whole message.


Lab Exercise:

    a) Open up a new script and call it Ex3.m. Clear all variables and close all figures.

    b) Load the data file CommsSignals.mat. This will again contain: x0, x1, and Fs.

    c) Download L4Ex3signal.mat from the class website and load it. This file will contain:

        ◦ r: This is the entire received messaged vector

        ◦ rm: This is the formatted received message as a matrix
        ◦ Fs: Sampling rate of received signal (same as x0 and x1)

    d) Compute the time samples vector t_r for the received signal vector r.

    e) To see what kind of signal you will be working with, plot r vs. t_r. No need to adjust the axes, but title and label your plot.

    f) Get the number of rows in matrix rm and store that value in variable rm_rows.

    g) Create a row vector of zeros called message_bits with the dimensions 1 x rm_rows. You will use this vector to store every bit you decipher.



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        h) We will now decipher each bit in the received message. Here is the outline of the loop:

            ▪ LOOP through all rows in matrix rm using variable index i


                • Extract all the columns in the ith row of matrix rm and store

                • the values in the variable signal


                • Call the decode function to decipher the binary value of signal and

                • store the output in the variable symbol


                • Store symbol in the ith element of message_bits

            ▪ END of loop

        i) Display the values of message_bits on the COMMAND window.

        j) Comment your script, and then run it. Make sure the values of message_bits are displayed on the COMMAND window.

    2) Lab Report Question #3 of 3: Provide the decoded message in your lab report. What is the

hidden message? You can use this link: http://www.binaryhexconverter.com/binary-to-ascii-text-converter to translate the bits to ASCII symbols.


FOR NEXT WEEK: LAB REPORT DUE

Turn in a PDF of your lab report (one per team) online via the link on the class website. A sample format for the lab report is on the class website. You may vary from this format but please include the basic sections. The report is due 24 hours prior to the start of your next lab section. Lab turn-in times will be checked, so no late lab reports will be accepted unless arranged in advance with the instructor.


























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