Question

Find the fourier transform of the gate function.

(a) \(\frac{1}{ω} sin⁡(\frac{ωτ}{2})\)

(b) \(\frac{1}{ω} cos⁡⁡(\frac{ωτ}{2})\)

(c) \(\frac{2}{ω} sin⁡(\frac{ωτ}{2})\)

(d) \(\frac{2}{ω} cos⁡⁡(\frac{ωτ}{2})\)

This question was addressed to me in an online quiz.

My question is based upon Fourier Transforms in portion Fourier Transform of Signals and Systems

Answer 1

The correct answer is (c) \(\frac{2}{ω} sin⁡(\frac{ωτ}{2})\)

Easy explanation: Gate function is defined as

\(

G(t)=\begin{cases}

1 &\text{\(|t|<\frac{τ}{2}\)} \\

0 &\text{elsewhere} \\

\end{cases} \)

The fourier transform is  \(F(ω) = \int_{-∞}^∞ f(t)e^{-jωt} \,dt = \int_{-τ/2}^{τ/2} e^{-jωt} \,dt = \frac{2}{ω}  sin⁡(\frac{ωτ}{2})\).