[SS] Fourier Transform of Real Exponential Function

다음과 같은 Real Exponential Function이 있다고 하자.

$$b e^{-at} u(t), a>0$$

해당 Real Exponential Function의 Fourier Transform은 다음과 같음.

$$\begin {align} \int^\infty_{-\infty}b e^{-at} u(t)e^{-j\Omega t} \text{d}t &= b \int^{\infty}_{-\infty}u(t)e ^{-(a+j\Omega)t}\text{d}t\\ \quad &= b \int^{\infty}_{0}e ^{-(a+j\Omega)t}\text{d}t\\ \quad &= b \left [ \frac{e ^{-(a+j\Omega)t}}{-(a+j\Omega)}\right]^\infty_0\\ \quad &= b \left [ 0-\frac{1}{-(a+j\Omega)} \right ] \\ \quad &= \frac{b}{(a+j\Omega)} \\ \\ \therefore \mathcal{FT}\left[b e^{-at} u(t)\right] &= \frac{b}{(a+j\Omega)} \end {align}$$