Fourier Transforms

The questions below are due on Thursday March 13, 2025; 02:00:00 PM.
 
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For all subparts of this question, you should enter your answers as Python expressions. Your answers may contain any of the following:

  • omega represents \omega
  • OMEGA represents \Omega
  • sin and cos represent \sin(\cdot) and \cos(\cdot), respectively
  • sqrt represents the square root function
  • e, j, and pi represent e, j, and \pi, respectively

Part 1

The continuous-time signal x_1(t) is defined by the following plot

and is zero outside the indicated range of t.

Find X_1(\omega), the Fourier Transform of x_1(t).
X_1(\omega) =~

Part 2

Determine the Fourier transform of x_2(t) given by the following expression

x_2(t) = \begin{cases} e^{|t|}&\text{if}~-1\lt t\lt1\\ 0&\text{otherwise} \end{cases}

and illustrated below.

Enter a closed-form expression for the Fourier transform in the box below, in terms of sines and cosines.
X_2(\omega)=~

Part 3

Consider a DT signal x_3[\cdot] described by the following diagram:

Determine a closed-form expression for the Fourier transform of x_3[\cdot].
X_3(\Omega) =~