Calculation the Fourier transform of $f(t)=e^{-|t+1|} \, u(t+3)$

Question

The transform of $f(t)=e^{-|t+1|} \, u(t+3)$ seems to me like mixing the Fourier transform of two functions $f(t)=e^{-a|t|}$ and $f(t)=u(t)$. I am not sure if I can use convolution or I if I can separate the absolute value.

Answer 1

Consider it as follows:

$$f(t)=f_1(t)+f_2(t)\Rightarrow F(\omega)=F_1(\omega)+F_2(\omega)$$

• $f_1(t)=f(t)\vert_{t>-1}$: $$f_1(t)=e^{-(t+1)}u(t+1)$$
• $f_2(t)=f(t)\vert_{t\le-1}$: $$f_2(t)=e^{(t+1)}(u(-t-1)-u(-t-3))$$

To calculate $F_1(\omega)$ and $F_2(\omega)$, use $\mathcal{F}\{f(t-t0)\}=e^{-j\omega t_0}\mathcal{F}\{f(t\}$.