EEE203 Signals and Systems I
Lab: What for Lunch?
Bob is very appreciative of your help throughout the semester. He wants to treat you for lunch. Since he is an amateur radio fan, he transmits the lunch menu to you through ham radio to see if you like it. The radio transmission uses amplitude modulation and the carrier frequency is 1.885 MHz. You receive the modulated signal. Now you need to demodulate the signal to find out the lunch menu.
To understand the theory of amplitude modulating and demodulating, it is crucial to understand the Fourier transform. of a signal multiplied by a sinusoid. Suppose we have a signal m(t) and its Fourier transform is M(jw), what is the Fourier transform. of m(t)cos(wct)? From the multiplication property, we know
Due to the sifting property of delta function, we have
In other words, if any signal is multiplied by a cosine signal in the time domain, in the frequency domain, it is equivalent to split the spectrum of the signal into two halves 2/1 M(jw), with one half shifted to the left by the frequency of the cosine signal wc and the other half shifted to the right by wc.
For example, assume m(t) = cos(2π100t) and wc = 2π1000. Figure 1 shows the Fourier transforms (the FFT magnitude) of m(t) = cos(2π100t) and m(t) cos(2π1000t) respectively. As shown in the figure, the Fourier transform. of m(t) is two impulses at ±100Hz. After multiplied by cos(2π1000t), the spectrum of m(t) is split into two halves, one shifted to the left by 1000Hz, and the other shifted to the right by 1000Hz, and they end up centered around at ±1000Hz respectively.
Lab Objectives:
1. Apply the multiplication property of Fourier transform, i.e., time domain multiplication of two
signals is equivalent to frequency domain convolution. See Table 4.1 for the property. Review Week
4 practice and homework problems 4.28 (b) (ii) and (iii) for an example. (This is a key concept of communication system.)
2. Understand the procedure to amplitude modulate a signal and be able to derive the Fourier transform. of the modulated signal.
3. Understand the procedure to coherently demodulate a signal and be able to derive its Fourier transform.
4. Resample the signal when a signal is modulated and demodulated based on Nyquist Theorem.
5. (Optional) Understand the impact of modulation index on coherent demodulation.
Lab Tasks:
Submit your derivation, answer to questions, MATLAB script and plot in a signal PDF file.
1. In this lab, you will need to use the “amdemod” (and possibly “ammod”) commands from MATLAB Communications Toolbox. If you haven’t installed the toolbox, you can go to “Home->Add-Ons->Manage Add-Ons” and go to “Get Add-Ons” and choose Communications Toolbox to install.
2. Read the tutorial on amplitude modulation/demodulation.
3. Derive the Fourier transform expression of the amplitude modulated signal ΦAM(jw) in the tutorial. (Hint: All you need is Table 4.1 and 4.2.)
4. Derive the Fourier transform expression R(ju) of r(t) in the tutorial. (Hint: Expand the expression and again all you need is Table 4.1 and 4.2.)
5. Given the R(ju) you derived, what needs to be done to restore the original signal? In the MATLAB
command window, type “type amdemod” . It will display the script. for the demodulation function.
Read the code and describe the demodulation process. How is the signal restored to its original
form? Does the procedure make sense based on the R(jω) you derived? The script uses a
butterworth low pass filter. What is the default cutoff frequency? Why is it a good choice? Refer to the MATLAB documentation (you can type “help butter” at the command line) or noise removal lab for the syntax of “butter” command.
6. Download the modulated signal “lunch_modulated.txt” . Due to its large size, the file is compressed. You need to unzip the file first.
7. Download the MATLAB script. “lab_mod_demod_mesg.m” . The part of MATLAB code to modulate the message is commented out and included for your reference. Don’t uncomment this part until you decode the message and if you want to complete Task 9. Complete the script, and submit the completed script. Your task is to
a. Demodulate the message using “amdemod”, which is the inverse of “ammod” (example on how to use “ammod” is in the commented out code section).
b. Down sample the demodulated message so that it can be played by a speaker. It is the inverse of up-sampling (example on how to use “resample” to do up-sampling is in the commented out code section).
c. Plot and play the recovered message.
8. Submit the MATLAB plot of the recovered message. What is the lunch menu Bob sends?
9. (Optional) Once you successfully demodulate the message, you can save the message as
“lunch.wav” using the “audiowrite” command. Now you can uncomment the modulation part of the code (also comment out the two dlmwrite and dlmread lines, so you don’t repeatedly writing and reading file), and experiment with different modulation indices (adjust the mu variable), can you recover message correctly when the modulation index is greater than 1?