(Solved): Figure 1 shows the Fourier Transform pair of a rectangular pulse (Heaviside \( \mathrm{Pi} \) ) in ...

Figure 1 shows the Fourier Transform pair of a rectangular pulse (Heaviside \( \mathrm{Pi} \) ) in time domain and a Sinc function in Frequency domain. The duality property makes this pair reversible in shape. We want to observe the symptoms of the impact of time duration of rectangular pulse on the effective bandwidth and peak point in the frequency domain. \( \operatorname{lsin} \) Figure 1: Rectangular Pulse and its Fourier Iranstorm A. Mathematically derive the Fourier Transform of the rectangular pulse described in Figure 1. Note that \( g(t)=\operatorname{rect}\left(\frac{t}{\tau}\right)=\Pi\left(\frac{t}{\tau}\right) \). Plot the magnitude and phase of the theoretical Fourier Transform function (Sinc) in MATLAB for \( \tau= \) \( 10,20,40 \) [seconds]. Overlap the three Sinc graphs in one figure. B. Using \( \mathrm{fft} \) () and its routine to plot spectrum, plot the magnitude and phase of the Numerical Fourier Transform of the rectangular pulse for \( \tau=10,20,40 \) [seconds]. Overlap the three Sincs graph in one figure and compare the shape to the theoretical result in \( \mathrm{A} \).