Description
Work on these questions as a homework first: answer the analytical parts of the questions and write the answers on a paper, write a MATLAB program (or many such programs) to perform the tasks that need computation, print your MATLAB code(s), print your computer outputs (numerical and graphic) whenever needed; the collection of all those will be your lab report. Bring your code (in a computer readable form) to the lab; transfer your code to one of the lab computers; run and show your TA the results. Answer all the questions your TAs may ask. Modify your lab report, including any modifications needed in your MATLAB codes, during the lab hours in the lab. Finally, hand your TAs the lab report prepared as described above. Your report will get a grade based on your preparedness when you come to the lab, performance of your codes in the lab (any modifications needed and conducted during the lab hours included), your answers to the oral questions during your demo(s) in the lab, and the entire content of the submitted report at the end of the lab.
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In this work, you will learn the Fourier series expansion and some related approximations.
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1- ya(t) is a rectangular waveform defined as: |
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t ∈ [0, 4)s. |
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ya(t) = |
5 |
t ∈ [4, 6)s. |
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0 |
t ∈ [6, 12)s. |
and ya(t) is periodic with a period of 12 seconds.
Any signal processing in a digital environment involves sampling; that includes your MATLAB environement, as well.
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Discretize ya(t), using a sampling period Ts = 1/5 s. Plot (using MATLAB) y[n] = ya(nTs), for n ∈ [−40, 215].
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Analytically find the Fourier series expansion of ya(t).
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Plot (approximately by hand) the spectrum of ya(t).
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Write a MATLAB code that computes the discrete function zN [n] using your results in (b), where,
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N
∑
zN [n] =
akejω0knTs
k=−N
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and ak’s are the FSE coefficients found in (b). Plot the result for n ∈ [−40, 215], and for N = 70.
Comment on the results: does the plot look like ya(t)?
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Repeat (e) for N = 25.
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Repeat (e) for N = 7.
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Repeat (e) for N = 3.
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Repeat (e) for N = 1.
Consider your plots in (d) through (h) and comment on the quality of approximations zN [n], of ya(t), by taking only some of the FSE components during the synthesis; pay attention also to the Gibbs phenomenon.
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Plot the zeroth, first, second, and third harmonics of ya(t), using the same scale for all these plots. (Hint: zeroth harmonic is also the DC component of ya(t) that is a constant function whose amplitude (value) is equal to a0. k′th harmonic of this real valued ya(t) is a−ke−jωokt + akejωokt which is a real valued sinusoidal. Hint: Though you cannot plot a continuous function using MATLAB, you can get a very good approximation if the sampling rate is high enough.)
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2- Replace ya(t) in (1), as, |
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π |
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ya(t) = |
4 cos |
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6 t) |
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Repeat (1). (Full-wave rectifier.) (You need to find the FSE analytically, first; to do that you need to find the fundamental period.)
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3- Replace ya(t) in (1), as, |
∈ |
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4 cos |
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π |
t |
t |
[ 3, 3) s . |
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ya(t) = |
. |
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{ 0 |
t |
[3,)9)s |
∈ − |
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6 |
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and it is periodic with period T = 12s. Repeat (1). (Half-wave rectifier.) (You need to find the FSE analytically, first; to do that you need to find the fundamental period.)
(⃝cLevent Onural, 2018. Prepared only for the students registered to EEE321 course at Bilkent University in 2018-2019
fall semester. Not to be distributed to others by any means.)
