Example of Computation of inverse Fourier transform (CT signals)
A practice problem on CT Fourier transform
$ X(w) = \pi \delta (w - 2 \pi)(3j - 7) + \pi \delta (w + 2 \pi) (3j + 7) $
$ x(t) = \frac{1}{2 \pi} \int^{\infty}_{- \infty} X(w) e^{jwt} dw $
$ = \frac{1}{2 \pi} \int^{\infty}_{- \infty} [ \pi \delta (w - 2 \pi)(3j - 7) + \pi \delta (w + 2 \pi) (3j + 7)] e^{jwt} dw $
$ = \frac{3j - 7}{2}\int^{\infty}_{- \infty}\delta (w -2\pi) e^{jwt} dw + \frac {3j + 7}{2}\int^{\infty}_{- \infty}\delta (w + 2\pi) e^{jwt} dw $
$ = \frac{3j - 7}{2} e^{j2\pi t} + \frac{3j + 7}{2} e^{-j2\pi t} $
$ = \frac{3j}{2} e^{j 2\pi t} - \frac{7}{2} e^{j 2\pi t} + \frac{3j}{2} e^{-j 2\pi t} + \frac{7}{2} e^{-j 2\pi t} $
$ = \frac{3j(e^{j 2\pi t} + e^{-j 2\pi t})}{2} + \frac{7(- e^{j 2\pi t} + e^{-j 2\pi t})}{2} $
$ = 3sin(2\pi) + 7cos(2\pi) $
