代做Mathematics留学生、代写Matlab程序设计、代写amplitude spectrum、代做Matlab编程

日期：2018-12-11 10:37

Methods of Applied Mathematics 1— Assignment 2

SP2, 2018

Due Date: 4:00 pm, 31 May, 2018.

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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.

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.

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1. 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.