University of South Australia EEET 3028 2017 SP2 Communication Systems
School of Engineering
Assignment 1: Signals & Spectra, Filtering, and Amplitude Modulation
Due date: 5 pm, Friday, 5th May 2017
General Information:
Each task consists of:
1. A set of tutorial/exam type problems, which you will need to solve with pen and paper.
2. A set of simulation tasks in MATLAB related to the specific questions.
This document and all required files are available for downloading from the learn online course web page. Important Information: Kindly use the function ftB.m provided the course web page for computing the Fourier transform (MATLAB tasks only). You can find it under the section about MATLAB.
Final Report and Evaluation
Your report should consist of:
• Solutions to the problems (including mathematical derivations).
• A working copy of the MATLAB code.
• All the plots generated in MATLAB, with appropriate title description. (You can add the titles to the plots in the report document after copying them to your document, if that is easier).
Evaluation will be based on the components above. Hand-written solutions to the problems are to be scanned into a pdf file, to which you will attach the plots and the MATLAB code.
Note: Hand-written answers/mathematical derivations will be accepted only if done neatly and clearly. You may scan them and submit electronically.
Assignment 1: Signals & Spectra, Filtering, and Amplitude Modulation Page 1 of 5
University of South Australia EEET 3028 2017 SP2 Communication Systems
School of Engineering
Task 1: Fourier Series
Let fc > 0. Consider the half-wave-rectified cosine wave, a periodic signal defined as: v(t) = Ac maxcos(2πfct),0. 1. Sketch this signal. What is the fundamental period Tv of this signal? (4 marks)
2. The Fourier coefficients of the above signal v(t) are given by:
cn =
Ac π
(−1)
n 2 −
1 n2−1 ,n even
Ac 4 ,n = ±1 0 ,n odd and n 6= ±1.
Using the above information, sketch the amplitude spectrum of v(t) for the frequency range −9fc to 9fc. (4 marks)
3. Using Parseval’s Theorem, write the power of the signal as an infinite series. (4 marks)
Now consider the fully rectified cosine, w(t) = Ac maxcos(2πfct),−cos(2πfct). 4. Sketch this signal. What is the fundamental period Tw of this signal as a function of Tv? (4 marks)
5. Find the Fourier coefficients of w(t). (Hint: express w(t) as a function of two v(t)s, then use Fourier series properties. Be wary that Tw 6= Tv.) Using the above information, sketch the amplitude spectrum of w(t) for the same frequency range. (10 marks)
6. What is the power of w(t) in terms of the power of v(t)? (you shouldn’t need Parseval’s theorem!) (2 marks)
Now, write a script in MATLAB to perform the following. Use a time resolution of 128 samples per period. Plots of signals in the time domain must be for −10/fc ≤ t ≤ 10/fc. Plots of spectra must be for the frequency range −500 kHz to 500 kHz and must be “stem” plots:
7. Generate and plot v(t) and w(t), with Ac = 1 and fc = 150 kHz. Plot the signals in the same axis, using different colours/linestyles. (2 marks)
8. Plot the magnitude spectra of the two signals, using the MATLAB function ‘ftB.m’ provided in the course web page. How does the observation from your plot correspond with the plot you sketched in Questions 2 and 5 above? (4 marks)
Assignment 1: Signals & Spectra, Filtering, and Amplitude Modulation Page 2 of 5
University of South Australia EEET 3028 2017 SP2 Communication Systems
School of Engineering
Task 2: Envelope Detection
Consider standard AM modulation, with the message signal x(t) of bandwidth W and satisfying |x(t)| < 1. This message signal modulates a carrier of amplitude Ac and frequency fc ≫ W so that the modulated signal carrying the message is given by: xc(t) = Ac1 + µx(t)cos(2πfct). 9. Explain why xc(t) can be considered as a bandpass signal. Find the envelope A(t) and phase φ(t) of this bandpass signal, without assuming µ < 1. (8 marks)
10. Now assume µ < 1. Comment on the shape of the envelope and identify whether the envelope corresponds to the original message. (4 marks)
Assume an ideal envelope detector with the structure shown in the figure below:
full−wave
rectifier
lowpass filter
DC block outputx c(t) x′ c(t) H(f)
The input-output relationship for the full-wave rectifier stage is given by:
x′ c(t) = maxxc(t),−xc(t). 11. Does this demodulator need carrier synchronization when µ < 1? (2 marks)
12. Assuming µ < 1, show that the output of the rectifier stage is given by:
x′ c(t) =1 + µx(t)w(t), where w(t) is the fully-rectified cosine signal discussed in Task 1. (4 marks)
13. Using the Fourier series expansion for w(t) found in Task 1, show that the spectrum of x′ c(t) is given by
X′ c(f) = Ac ∞
X n=−∞
c′nδ(f − n2fc+ µX(f − n2fc),
where c′n are the Fourier coefficients of w(t) computed in task 1 part 5 and X(f) is the spectrum of the message x(t). (8 marks)
Hint: Use the superposition and the complex modulation (frequency shift) properties of the Fourier transform.
14. Assume that the message signal is given by x(t) = sinc2(Wt), where W ≪ fc. Compute X(f) and use this to sketch the spectrum of the signal at the output of the rectifier stage for the frequency range −9fc to 9fc. (8 marks)
Assignment 1: Signals & Spectra, Filtering, and Amplitude Modulation Page 3 of 5
University of South Australia EEET 3028 2017 SP2 Communication Systems
School of Engineering
15. Referring to the above sketch of spectra, explain in a few sentences how the low pass filter in combination with the DC block recovers the message signal. Without full derivation, instead referring to the spectrum of the half-wave rectified cosine computed in task 1 part 2, describe how the spectrum would change if half-wave rectification had been used. Would there be any implications for the lowpass filter design? (8 marks)
Task 3: Simulating Envelope Detection in MATLAB
Use MATLAB to generate the following signals and spectra corresponding to the envelope detector in Task 2. Use a time resolution of 128 samples per period. All plots of signals in the time domain must be for −10/fc ≤ t ≤ 10/fc. All plots of spectra must be for the frequency range −500 kHz to 500 kHz and must be “stem plots”: 16. Assume fc = 150 kHz, Ac = 1, µ = 0.75, and the message signal x(t) = sinc2(Wt), with W = 40 kHz. Plot the modulated signal xc(t) and its spectrum. (4 marks)
17. Plot the signal at the output of the rectifier stage and the corresponding amplitude spectrum. Consider both half-wave and full rectifiers. (4 marks)
18. Assume that the low-pass filter block in the above demodulator has the following transfer function: H(f) = 1 1 + j(f/f0)4 . Plot the magnitude response of this filter in MATLAB for the following cases, using different colours in the same plot:
(a) f0 = 20 kHz (b) f0 = 55 kHz (c) f0 = 80 kHz
For each case, explain how the performance of the demodulator varies. Identify the case(s) that recover the best replica of the original message. (4 marks)
19. For each of the low-pass filters listed in the previous question, and for both half-wave and full rectification, generate the plots of the signals and spectra (amplitude only) at the output of the low-pass filter. What do you observe, in each case? (12 marks)
Hint: To obtain the signal and spectrum after filtering, it is easy to operate in the frequency domain first. You first need to multiply the spectrum of the signal with the transfer function. This corresponds to convolution in the time domain. Plot the resulting spectrum, and then find and plot the time-domain signal by performing the inverse Fourier transform using the function ft.m, using the duality and conjugation properties of Fourier Transform. Remember to use the ‘real’ function to discard any tiny imaginary parts after the inverse Fourier transform, which should not exist in theory, but occurring due to MATLAB’s numerical precision issues. Use the following MATLAB code segment as a guide:
Assignment 1: Signals & Spectra, Filtering, and Amplitude Modulation Page 4 of 5
University of South Australia EEET 3028 2017 SP2 Communication Systems
School of Engineering
N = length(t); [f,X_ft] = ft(t,x); % The Fourier transform of the signal to be filtered figure; stem(f,abs(X_ft)); % Spectrum of the signal to be filtered
f_0 = 20e3; % Filter parameters m = 4;
X_filt = X_ft./(1+1i*(freq/f_0).^m); % Fourier transform of the filtered signal
figure; stem(freq,abs(X_filt)); % Plot of the spectrum of the real signal [t,x_filt] = ft(f,conj(X_filt)); % Inverse Fourier transform operation
figure; plot(t,x_filt); % Plot of the filtered signal in time domain
Assignment 1: Signals & Spectra, Filtering, and Amplitude Modulation Page 5 of 5