Question

The Fourier transform of a function x(t) is X(ω). What will be the Fourier transform of \(\frac{dX(t)}{dt}\)?

(a) \(\frac{X(f)}{jf}\)

(b) j2πfX(f)

(c) \(\frac{dX(f)}{dt}\)

(d) jfX(f)

This question was posed to me in an interview for job.

Origin of the question is Properties of Fourier Transforms topic in portion Fourier Transform of Signals and Systems

Answer 1

Right option is (b) j2πfX(f)

Explanation: We know that  x(t) = \(\frac{1}{2π} \int_{-∞}^∞ X(ω) e^{jωt} \,dω\)

\( \frac{d}{dt} \,x(t) =  \frac{1}{2π} \int_{-∞}^∞ X(ω) \frac{d}{dt} e^{jωt} \,dω = \frac{1}{2π} jω X(ω) \int_{-∞}^∞ e^{jωt} \,dω\)

= jω X(ω) = j2πfX(f).