I wanted to calculated the inverse fourier transform of the folowing function : $\frac{\frac{1}{1+2i\pi f}+\frac{1}{3+2i\pi f}}{\frac{2}{1+2i\pi f}-\frac{2}{4+2i\pi f}}$ ($f$ is the variable)
I tried the exponential form, Unifying the denominators but in vain, the premitive is still impossible to be found !
can I get some help ? much appreciated.
After simplifying, we have $$H(f) =\frac{\frac{1}{1+2i\pi f}+\frac{1}{3+2i\pi f}}{\frac{2}{1+2i\pi f}-\frac{2}{4+2i\pi f}} = 1 + \frac{2 i f \pi}{3} + \frac{i}{-9 i + 6 f π}$$ Now use these facts $$\mathcal{F}\{g'(x)\} = 2\pi i f G(f)$$ $$\mathcal{F}\{e^{-ax}u(x)\} = \frac{1}{a + 2\pi i f}$$ $$\mathcal{F}\{\delta(x)\} = 1$$ You can check your result(Be careful about the scaling factors which are result of how the Fourier transform is defined).