( 9 marks total) Consider the Fourier series of the function f(t) = t(u(t+1) − u(t−1)) = {t if −1 < t < 10 otherwise defined on the interval [−2, 2]. (a) Sketch the function f(t) and find the Fourier frequencies ωn. (2 marks) (b) Calculate the Fourier coefficients a0, an, and bn. (5 marks) (c) Write down the resulting Fourier series, substituting in the expressions for the Fourier coefficients. (2 marks)
![( 9 marks total) Consider the Fourier series of the function f(t) = t(u(t+1) − u(t−1)) = {t if −1 < t < 10 otherwise defined on the interval [−2, 2]. (a) Sketch the function f(t) and find the Fourier frequencies ωn. (2 marks) (b) Calculate the Fourier coefficients a0, an, and bn. (5 marks) (c) Write down the resulting Fourier series, substituting in the expressions for the Fourier coefficients. (2 marks)](https://www.doubtrix.com/uploads/editor/3150152742hBuJyCNiUv.jpg)
![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)
