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Inverse Fourier Series (CE Form)

The questions below are due on Monday March 01, 2021; 10:00:00 PM.
 
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Note that this exercise is intended to be solved by hand, without the use of computation.

Determine the CT signals with the following Fourier series coefficients. Assume that the signals are periodic in T = 4. Enter an expression that is valid for 0\leq t < 4.

For each, enter your expression in terms of trig functions (sin and cos) rather than entering complex exponentials, and recall the following:

  • e^{j\theta} + e^{-j\theta} = 2\cos \theta

  • e^{j\theta} - e^{-j\theta} = 2j\sin\theta

1) Part 1

X_1[k] = \begin{cases} -jk, & |k| = 2\\ 3, & |k| = 1\\ 0.5 & k = 0\\ 0 & \text{otherwise} \end{cases}

x_1(t) =

2) Part 2

X_2[k] = \begin{cases} jk, & |k| \lt 3\\ 0, & \text{otherwise} \end{cases}

x_2(t)=

3) Part 3

X_3[k] = \begin{cases} \frac{1}{2} + \frac{1}{2}j, & k = -1\\ \frac{1}{2} - \frac{1}{2}j, & k = 1\\ 0, & \text{otherwise} \end{cases}

x_3(t)=