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| − | <math>\,\mathcal{X}( | + | <math>\,\mathcal{X}(\omega)=\int_{-\infty}^{+\infty}x(t)e^{-j\omega t}\,dt\,</math> |
Revision as of 17:31, 8 October 2008
$ X(\omega)=\cos{(6\omega + \pi/6)} $
Start by guessing the solution:
$ X(t)=(1/2)e^{-j(\pi/6)}\delta(t-6)+(1/2)e^{j(\pi/6)}\delta(t+6) $
Then take the fourier transform of the guessed solution to make sure it's right...
$ \,\mathcal{X}(\omega)=\int_{-\infty}^{+\infty}x(t)e^{-j\omega t}\,dt\, $
$ \,\mathcal{X}(\omega)=\int_{-\infty}^{+\infty}[(1/2)e^{-j(\pi/6)}\delta(t-6)+(1/2)e^{j(\pi/6)}\delta(t+6)]e^{-j\omega t}\,dt, $
$ \,\mathcal{X}(\omega)=\int_{-\infty}^{+\infty}(1/2)e^{-j(\pi/6)}\delta(t-6)e^{-j\omega t}\,dt + \int_{-\infty}^{+\infty}(1/2)e^{j(\pi/6)}\delta(t+6)e^{-j\omega t}\,dt\, $
$ \,\mathcal{X}(\omega)=(1/2)e^{-j(\pi/6)}\int_{-\infty}^{+\infty}\delta(t-6)e^{-j\omega t}\,dt + (1/2)e^{j(\pi/6)}\int_{-\infty}^{+\infty}\delta(t+6)e^{-j\omega t}\,dt\, $
$ \,\mathcal{X}(\omega)=(1/2)e^{-j(\pi/6)}e^{-6jt} + (1/2)e^{j(\pi/6)}e^{6jt} $
$ \,\mathcal{X}(\omega)=(1/2)e^{-j(6t + \pi/6)} + (1/2)e^{j(6t + \pi/6)} $
$ \,\mathcal{X}(\omega)=\cos{(6\omega + \pi/6)} $
