Methods of Applied Mathematics 1— Assignment 1 SP2, 2019
Due Date: 11 April.
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Please consult the Course Outline on the web page for this course for details of how to submit your assignment, and penalties for late presentation. Note: The due date has changed! You need to submit your assignment via the online submission tool on the web site. The system will automatically put a coversheet on your assignment.
Make sure to relate on the front page of the assignment any information about help given/received for any of the questions you attempted, who received/gave help, and to reference any sources other than the notes and exercises for the course.
You will submit only one file to the learnonline website. This file must contain all of your work, including graphs, MATLAB code, etc.
Cheers, Jorge.
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1. This exercise is a typical final exam question (minus the MATLAB content). Consider the signal below, periodic with period T = 2.
f(t) = (1−t2)cos(3πt), −1 ≤ t ≤ 1.
(a) What is ω0 for this signal?
(b) Draw a picture of this signal for −3 ≤ t ≤ 3. Label your axes correctly! (You should use MATLAB. Include your code.)
(c) For which values of t will the Fourier series provide a good approximation for this signal? Justify, based on your answer to the previous item.
(d) Find the complex Fourier coefficients cn from the definition (that is, integrate, either by hand or using a table of integrals, to find the answer).
cn = T
−T /2
1 ∫ T/2
f(t)e
−jnωt
dt, n = 0,±1,±2,···
(e) Find the real Fourier coefficients an, bn. You may use your knowledge of cn to do this, without integrating. Alternatively, you may choose instead to find an and bn by integration, and use this knowledge to find cn.
(f) Find the average energy of f over one period; that is, compute
1 ∫ T/2
T
(g) Use Parseval’s identity to answer the following question: How many terms do we need to add in the Fourier series in order to approximate f, with at least 99% of the (average) energy of f being present in the approximation? Justify. (This can, and should, be done using MATLAB. Include your code in the answer.)
(h) Use MATLAB to plot, on the same graph, both f and the approximation obtained in the last item.
(i) Draw a picture of the (complex) amplitude spectrum of f, containing at least the first 5 frequencies. For full marks you will need to label all axes correctly, making clear which angular frequencies are present in this signal.
|f(t)|2 dt.
−T /2
2. This is a MATLAB exercise. You are not expected to perform any computations by hand, but you do need to deliver your own MATLAB code for this question.
Consider the signal
{ − t , −π < t < 0; π
f(t) =
(a) Find the average energy of f. (Use MATLAB.)
tcos(3t), 0 ≤ t < π.
(b) Obtain the complex Fourier coefficients c0, c1,. . . , c10. (Use MATLAB, display your
answers in a printout.)
(c) Plot f and the approximation
jnω0t
(d) Find a value for N (a number) for which the Fourier approximation
|f(t)−SN(t)|< 1. 10
Justify your answer by plotting a graph.
(e) In the last item, is it possible to obtain an approximation that works for all values of t? Why, or why not?
S10(t) =
overlayed on the same graph. Use different colours.
cnejnω0t of f, only for t between −1 and 1, is such that
SN (t) =
10
∑
n=−10
∑N n=−N
cne
3. This is an exercise about convolution. Consider the signals f and g below, both periodic with T = 2.
f(t) =
t2, 1,
0 ≤ t ≤ 1; 2
g(t) =
tsin(2t), −1≤t<0;
{ 1(t+1), −1≤t<0; 2
|t−0.5|, 0≤t≤1.
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