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Q#1 Fourier Transform GATE EC 2001 MSQ +2 marks -0 marks

The Fourier transform G(ω) of the signal g(t) in Figure (a) is given as . Using the information and the time-shifting and time-scaling properties, determine the Fourier transform of signals in Figures (b), (c) and (d).


Figure (a)

Figure (b)

Figure (c)

Figure (d)

\( \scriptstyle G_1(\omega) = \frac{1}{\omega^2}\left[ e^{-j\omega} + j\omega e^{-j\omega} + 1 \right] \)

\( \scriptstyle G_1(\omega) = \frac{1}{\omega^2}\left[ e^{-j\omega} + j\omega e^{-j\omega} - 1 \right] \)

\( \scriptstyle G_2(\omega) = \frac{1}{\omega^2}\left[ 1 + j\omega - e^{j\omega} \right] \)

\( \scriptstyle G_3(\omega) = \frac{2 \sin\left( \frac{\omega}{2} \right)}{\omega} \)

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