Engineering computations - draw the following function in

Question

Question 1: The following function/equation is given:

            1: for 2 < x ≤ 5

f(x) =  -1: for - 2 < x < 2  for - 5 < x ≤ 5

            1: for - 5 < x ≤ -2 

with f(x + 10) = f(x)

a) Sketch the function!

b) Calculate the Fourier series for that function!

c) Write down the fifth partial sum!

Question 2: The following function/equation is given where f(x) is from question 1:

g(x) = f(x) - 1 for - 5 ≤ x ≤ 5

with g(x + 10) = g(x)

a) Sketch the function

b) Without calculation explain how does the Fourier series of f(x)(as derived in Question 1) differ from the Fourier series of g(x)? Explain why?

c) Calculate the minimal N (number of components of the Fourier series), so that the Square Error of.the Fourier series of g(x)is E* < 10^-2 (Hint: Use Excel, Matlab or any other tool to calculate N.)

d) Sketch the development of E*(n)! (Hint: Use Excel, Matlab or any other tool to calculate N.)

Question 3: The following integral is given:

-2Π∫2Π(∑_n=1 ⁿ⁼¹⁰∑_m=1 ^m=10 [sin(ω₀mt)])dt with ω₀ = 5

TIP: A trigonometric system such as sin(at)cos(bt) has one important characteristic that will help you simplifying the given integral. What is that characteristic and how does it simplify the integral?

Calculate the integral.

Question 4: Draw the following function in frequency domain (amplitude spectrum) and time domain (for t=0 until 4Π):

y = 4/Π² + 1/Π²sin(0.5t) - 3/Π²sin(t) + 2/Π²sin(1.5t) - 1/Π²sin(2t)

Hint: Use Matlab, or Excel to draw the function in time domain.