Problem 4.5
Find the inverse Fourier transform of:
- $ X(j\omega) = |X(j\omega)|e^{j \sphericalangle X(j\omega)} $
Given that:
- $ \big|X(j\omega)| = 2\lbrace u(\omega +3) - u(\omega - 3)\rbrace $
- $ \sphericalangle X(j \omega) = -\frac{3}{2} \omega + \pi $
The entire integral:
- $ \frac{2}{2\pi}\int_{-\infty}^{\infty}\Bigg(e^{-\frac{3}{2} j \omega + \pi j + j\omega t} u(\omega + 3) - e^{-\frac{3}{2} j \omega + \pi j + j\omega t} u(\omega - 3)\Bigg)\,d\omega $
Change the limits:
- $ \frac{1}{\pi}e^{\pi j}\Bigg\{ \int_{3}^{\infty}\Bigg(e^{j\omega(t-\frac{3}{2})}\Bigg)\,d\omega - \int_{-3}^{\infty}\Bigg(e^{j\omega(t-\frac{3}{2})} \Bigg)\,d\omega \Bigg\} $
Integrate:
- $ \frac{1}{\pi}e^{\pi j}\Big\{ \Big(\frac{e^{j\omega(t-\frac{3}{2})}}{ jt- j\frac{3}{2} }\Big)\Bigg|_{3}^{\infty} - \Big(\frac{e^{j\omega(t-\frac{3}{2})}}{ jt- j\frac{3}{2} } \Big)\Bigg|_{-3}^{\infty} \Big\} $
The infinite terms cancel out:
- $ \frac{1}{\pi}e^{\pi j}\Big\{ \Big( \frac{e^{3j(t-\frac{3}{2})}}{ jt- j\frac{3}{2} } \Big) - \Big( \frac{e^{-3j(t-\frac{3}{2})}}{jt - j\frac{3}{2}} \Big) \Big\} $
Combine terms:
- $ \frac{1}{\pi}e^{\pi j}\Big\{ \frac{e^{3j(t-\frac{3}{2})} - e^{-3j(t-\frac{3}{2})}} {j(t-\frac{3}{2})} \Big\} $
Simplify using euler's crap:
- $ -\frac{2}{\pi}\Bigg\{ \frac{sin(3(t-\frac{3}{2}))} {t-\frac{3}{2}} \Bigg\} $
