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Find the Fourier transform of the rectangular pulse function shown in the figure,

Find the Fourier transform of the rectangular pulse function shown in the figure,

enter image description here

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Solution:

$ x(t)=\pi(t)=A \quad ; \frac{-T}{2} \leq t \leq \frac{T}{2}\\ $

$ F[\pi(t)]=\int_{-\frac{T}{2}}^{\frac{T}{2}} A e^{-j \Omega t} d t\\ $

$ =A\left[\frac{e^{-j \Omega t}}{-j \Omega}\right]_{-\frac{T}{2}}^{\frac{T}{2}}\\ $

$ =\frac{A}{-j \Omega}\left[e^{-j \Omega \frac{T}{2}}-e^{j \Omega \frac{T}{2}}\right]\\ $

$ =\frac{2 A}{j \Omega}\left[\frac{e^{j \Omega \frac{T}{2}}-e^{-j \Omega \frac{T}{2}}}{2}\right]=\frac{2 A}{\Omega} \sin \Omega \frac{T}{2}\\ $

$ =\frac{2 A}{\Omega T} T \sin \Omega \frac{T}{2}\\ $

$ =A T \frac{\sin \Omega \frac{T}{2}}{\Omega \frac{T}{2}} A T \sin c \Omega \frac{T}{2}\\ $

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