Signals and Systems Final Project Academic Year 2018/19
Goals of the Final Project
The Final Project has the following goals:
1. develop critical thinking: the student should be able to understand if the obtained result (either theoretical or measured through a suitable computer program) is realistic/reasonable or not
2. improve the ability of writing reports: the student should practice this ability
3. develop the ability of the student to correctly/reasonably set unknown parameters
Therefore, the project is not described with all the details (you have to make your choices and justify them). On the other side, it is required that you get your results in two or more different ways in order to check that they are correct and find potential errors (critical thinking). The report must include the description of all the required activity, in many case it is expressly said what the report must include, in other cases it is left to you to decide what to write and how to show your results. Take your responsibility.
Who has to complete the project
The project must be carried out by students who are attending Signals and Systems with 10 credits. They have to write the Matlab code and the report that comments the results; the maximum grade of the report is 6 points (20% of 30), the oral exam gives the remaining part of the final grade. Each student shall write his/her own report. Plagiarism will result into suspension.
Deadlines
The report must be uploaded in folder ”elaborati” on the course web-page by
January 20th 2018;
intermediate reports are not required. The report must be a unique pdf file with filename yoursurname sxxxxx SaS 2018 19.pdf,
(where xxxxx is your matricola number) that also includes the Matlab scripts in the appendix. The expected length of the document is 5 pages (not including the Matlab scripts).
It is not possible to sit the exam without the report.
YOU MUST UPLOAD THE FILE BY JANUARY 20th, whatever is its content.
1 Minimum requirements for writing the report
• Thereportmusthaveafirstpagewiththetitle(SignalsandSystemFinalProjectReport,Academic Year 2018/19), the author, the “numero di matricola”.
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• All the pages (apart from the very first) of the report must be numbered, you must justify the text.
• The report must be organized in sections and subsections.
• Youmustdescribethegoaloftheproject:thereportmustbeself-standing,itshouldbecompletely understood by a professor of Signals and Systems who teaches in another university (you must explain the notation you used, you don’t need to explain what the FFT is)
• Usethespellchecker,readyourreportatleast3timestoremovethegrammarerrors(missing“s”, wrong use of articles, etc); if you are not sure about the use of a word, check it.
• Avoid using “I” in the report, it is not professional.
• The figures in the report must be numbered and must have a caption (note: the title of the figure you specify in the Matlab script is NOT the caption). The caption of figure must be below the figure.
• In the report you must refer in the text to all the figures included in the report, and you must use the number of the figure (i.e. don’t write “In the figure below” but “In figure 3.2”).
• The plots must include the labels on the axes and the grid.
• The scale in the plots must be such that the plots are clearly readable.
• You are responsible of choosing if the x and/or y data are represented in a logarithmic scale or in a linear scale, and you must choose the most appropriate one.
The quality of the report must be that of a technical book written by a professor.
We suggest that you use Latex for writing the report, but it is not mandatory. In any case you must give in a pdf file, the Matlab scripts must be included in the appendix. Once you obtain the final Matlab script and you get the correct results, immediately write the report, don’t wait till the last useful day. Writing the report takes time. Once you have your report, check again the list of point above, in case
you forgot one.
2 Task 1 of the project – 4 points
Task 1 aims at the simulation and analysis of a type 1 Chebyshev analog filter of order 4, using the impulse invariance technique.
The transfer function of the type 1 Chebyshev analog filter of order n is
n −sm Ha(s)= ∏s−sm
m=1
sm0 = −sinh narsinh ε
θm = π 2m−1 2n
sm = 2π f0sm0, f0 = 1kHz ε=10(RdB/10)−1, RdB=0.1
Set n = 4.
1 1 narsinh ε
1 1
sin(θm)+ jcosh
cos(θm)
2
(1)
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1. Write a Matlab script that plots in the complex plane the poles s1,s2,s3,s4. Is the system stable? Write your comments in the report and include the figure together with the formula of the transfer function Ha(s) and of the poles (you cannot cut and paste the above text, this is plagiarism).
2. Plot the transfer function Ha( f ) = Ha(s = j2π f ) using Matlab and include the figure in your report. Which kind of filter is this one? low-pass? band-pass? high-pass? How many dB/dec?
3. Theoretically (pen and paper) derive the impulse response ha(t) of the filter and include this derivation in the report (don’t substitute the numerical values of sm). Plot the impulse response with Matlab using a correct sampling frequency fs (explain how you chose it in the report). To check that the impulse response is correct, evaluate its approximate Fourier transform (through the FFT) and compare it with Ha(f). If the parameters are correctly set, the two functions are practically equal. In the report discuss why the approximate Fourier transform is not exactly equal to Ha(f).
4. Theoretically derive the transfer function H(z) of the digital filter, designed according to the im- pulse invariance technique. Assume that the sampling frequency is the same fs you used at point 3. Don’t substitute the numerical values of the poles, but write them as pm in your formulas (but write the formula that allows to obtain pm from sm). Sketch the block diagram of the digital filter (there are several, use the one that you want to implement in Matlab). Include in the report the derivation of H(z) and the block diagram.
5. Implement the block diagram in Matlab (don’t use the Matlab built-in functions conv or filter or anything like that, but implement the finite difference equation!). Obtain the impulse response of the implemented system by using signal δ [n] as input. Compare the measured impulse response and the theoretical impulse response you derived in point 3. If they are not equal, write the reason. NOTE: the Matlab implementation must use real coefficients, complex coefficients are not allowed.
6. Plot in the complex plane the position of the poles of H(z) (include the figure and the comments in the report).
7. Plot H(z) for z = ej2π f∆t for the correct range of f and parameter ∆t. Compare this plot with Ha ( f ), include this figure in the report and write your comments.
Task 2 of the project – 2 points
Use the Matlab script you wrote for task 1, and modify it as follows (you have to add a few lines).
1. Use the Matlab function randi to generate a random sequence of 50 bits
2. Use these bits to generate the sampled version x[n] of signal
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x(t) = ∑biΠ(t −iTb) i=0
where Π(t) = 1 for t ∈ [0,Tb] and Π(t) = 0 elsewhere. Set Tb so that the bit rate Rb = 1/Tb is 500 bit/s. Signal x(t) is sampled at the same sampling frequency you used in Task 1.
3. Usex[n]asinputofthediscrete-timefilteryoudesignedintask1,plottheoutputy[n]andcomment the result.
4. Measure the mean square value, the mean and the variance of x[n] and y[n] and write your com- ments.
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