Delayed Pulse

The questions below are due on Thursday March 13, 2025; 02:00:00 PM.

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Please answer the following questions by entering Python expressions in the boxes provided. 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, 1j, and pi represent e, j, and \pi, respectively
Re and Im represent {\rm Re}(\cdot) and {\rm Im}(\cdot), respectively

Consider the CT signal x(\cdot) defined by the following plot (you can assume that x(t) = 0 for all t not shown in the plot below):

This function has a Fourier transform given by X(\cdot). A portion of the magnitude of the Fourier transform is shown below:

Determine a simple expression for b in terms of the parameters of x(\cdot).
When entering your answer, you may use t_1 to refer to t_1, t_2 to refer to t_2, and/or a to refer to a.
b =~

Determine a simple expression for \omega_1 in terms of the parameters of x(\cdot).
When entering your answer, you may use t_1 to refer to t_1, t_2 to refer to t_2, and/or a to refer to a.
\omega_1 =~