讲解Mathematics留学生、辅导Matlab程序设计、辅导amplitude spectrum、讲解Matlab编程 解析R语言编程|调试
Methods of Applied Mathematics 1— Assignment 2
SP2, 2018
Due Date: 4:00 pm, 31 May, 2018.
oooOooo
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.
Make sure to relate on the front page of your 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 need to turn in your Matlab code (but you do
not need to turn in the code in a disc). Cheers, Jorge.
oooOooo1. Consider the signal
f(t) = {
0, t < 0;
e
t
sin t, t ≥ 0.
(a) Draw a picture of this signal over a suitable range for t, and classify the signal as
odd, even, or neither. (You may use MATLAB. If so, include your code.)
(b) Find the Fourier Transform fb(α) for this signal. Simplify your answer.
(c) Use MATLAB to draw the amplitude spectrum of f over an appropriate spectral
range. (That is, draw |fb(α)|.) [20 marks]
2. Consider the signal
f(t) = 1
3
cos(3t) + 2 sin(6t) ? sin(8t).
(a) What is the highest angular frequency present in this signal? What is the highest
numerical frequency present in this signal?
(b) What is the Nyquist rate for this signal? Did you use the angular or the numerical
frequency?
(c) If you sample this signal with sampling period T, which values of T may you choose
to be in accordance with the Nyquist rate? Choose and fix one such T.
(d) After sampling you pass the sampled signal through a low-pass filter. Which threshold
M0 can be used in the low-pass filter?
[3 marks each = 12 marks]
3. Consider the time-limited signal
f(t) =
0, t < π;
et
sin t, π ≤ t ≤ π;
0, t > π.
(a) Use MATLAB to plot the signal for ?4 ≤ t ≤ 4.
(b) Use MATLAB to plot the amplitude spectrum |fb(α)| over the range ?5 ≤ α ≤ 5.
(c) We will arbitrarily pick M = 4, and try to use Shannon’s formula to sample and
reconstruct f. By looking at the graph of |fb(α)| (or by magnifying that graph), what
is the amplitude of the frequencies we are cutting out?
(d) Fix M = 4, and treat it as the highest frequency in the signal f (although it really
isn’t). What would be the Nyquist rate in this case? Which sampling period T would
then be appropriate? Choose a sampling period T, and fix it.
(e) Choose an appropriate value M0 > 0 to serve as the low-pass filter threshold for
the sampled signal. What is the minimum value that M0 can have? What is the
maximum value to avoid aliasing (in theory)? Choose your M0 and fix it.
(f) Using the values of M0 and T chosen/obtained above, try to use Shannon’s Sampling
Formula to reconstruct the signal f. (This is a MATLAB exercise.) Plot the reconstructed
signal by itself, then plot it superimposed over the original signal f. Include
your code in your answer.
[25 marks]4. Use the Laplace Transform method to solve the following differential equation problem:
y
′′(t) y(t) = sin(2t), y(0) = 0, y′
(0) = 1.
[15 marks]
5. Use the z-transform method to solve the following difference equation:
y[n + 2] = 2y[n + 1] + 8y[n], y[0] = 0, y[1] = 1.
[15 marks]
6. Suppose Φ is a uniform random variable on the interval [0, 2π].
(a) What is the density f of Φ?
(b) Use the formula
E[g(Φ)] = ∫ ∞
∞
g(x)f(x) dx
to compute E[sin(3 + Φ)].
(c) Compute E[e
jΦ].
[13 marks]
Bonus Question. This is an Internet search bonus question. Let’s look at the stars. The
chemical composition of our sun (the one we see during the day) is given in this table:
Element Hydrogen Helium Oxygen Carbon Nitrogen Silicon Magnesium Neon Iron Sulphur
Mass % 71.0 27.1 0.97 0.40 0.096 0.099 0.076 0.058 0.014 0.040
(Source: https://www.thoughtco.com/element-composition-of-sun-607581)
How do we know this? Search the web and find an explanation based on the observed
properties of light emitted by our sun. Reference your sources carefully.
This is a subjective question to mark. For full marks (5 marks at most), try to make your
explanation as clear as possible.