Answer following questions accordingly. a) Based on your understanding, discuss the use of Fourier series and Fourier transform in linear time invariant analysis and their relation. b) Determine the complex exponential Fourier series of x(t) given as follow Figure 1 c) Determine the Fourier transform of following continuous signals using analysis equation. i. x(t) = t e−2tu(t) ii. x(t) = e−t cost u(t)
![Problem 4: Fourier Series [24 points] (a) (6 points) Obtain the Fourier series of the signal x(t) given in Figure 3. Using numerical solvers (like Wolfram alpha), sketch the synthesis equation of the Fourier series and see if it gives the same sketch as in Figure 3. Figure 3: Figure for Question 4 part (a) (b) (6 points) Consider a signal g(t) = t2 over the interval (−1, 1) which repeats its self with a period T = 4. Find the trigonometric Fourier Series to represent g(t). Sketch the Fourier Series over frequency. (c)(6 points) Verify Parseval's Theorem for this case given that ∑n = 1∞1 n4 = π490 (d) (6 points) Determine the periodic signal x(t) of period T = 4 whose Fourier series coefficients are given below cn = {jn|n| < 30 otherwise](https://www.doubtrix.com/uploads/editor/0258520695NeIiMMsiju.png)