Reference no: EM132928898

Question 1. a) Find the Fourier coefficients and the Fourier series for the follwing periodic function. For full credit, show your work when integrating to compute the coefficients. Write at least three cosine terms and at least three sine terms, in addition to the DC term (if any).

f (t) = (-t if -2 ≤ t ≤ 0)
f (t) = 0, if 0 ≤ t ≤ 2                      and f (t + 4) = f (t)

b) Use Desmos to graph both the function and Fourier series and to confirm your result. Attach a snapshot of the graph.

Question 2. Consider the function:

x(t) = 1    for -Π ≤ t ≤ 0

x(t) = 0    otherwise

a) Using the definition of the Fourier Transform, find X(ω).

b) Write x(t) as the sum of two unit step functions.

c) Use the Fourier Transform Tables to confirm the answer you calculated above:

Question 3. Use the functional and operational rules to find the Fourier Transform for the given functions:

a. u(t)e-2t.
b. δ(t + 5) - 2
c. 4rect(t/3)
d. sin(6Πt)
e. 3e^-2iΠt

Question 4. Circle the best response:
a. f(t) = e^-3t is an even function.
b. cos(nΠ) = (-1)ⁿ.
c. The Fourier transform take a non-periodical signal from the time domain into the frequency domain.
d. (e^iθ - e^-iθ)= 2sinθ
e. The Fourier series of an odd function contains only sine terms.