0913 Fourier Transform of Derivative, Time Shift
ph
derivative
$$ f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) \, e^{i \omega t} d\omega \\ f'(t) = \frac{d}{dt}\!\left( \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) \, e^{i \omega t} d\omega \right)= \frac{1}{2\pi} \int_{-\infty}^{\infty} i \omega \, F(\omega) \, e^{i \omega t} d\omega $$
따라서 \( f'(t)\)의 변환은 \(i \omega \, F(\omega)\) [1]
time shift
Let \(t’ = t \pm t_0\), .i.e. \(t = t’ \mp t_0 \) $$ \int_{-∞}^∞ f(t ± t_0) e^{-jωt} dt = \int_{-∞}^∞ f(t’)e^{-jω(t’ \mp t_0)} dt’ \\ e^{±jωt_0} \int_{-∞}^∞ f(t’)e^{-jωt’} dt’ = F(ω)e^{±jωt_0} $$
따라서 \(f(t±t_0)\)의 변환은 \(e^{±jωt_0}F(ω)\). [2]