(→Computing the Inverse Fourier Transform) |
(→Computing the Inverse Fourier Transform) |
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The inverse Fourier transform is defined as: | The inverse Fourier transform is defined as: | ||
| − | <math> x(t) = \int_{-infty}^{infty} \frac{X(w)}{2 \pi} e^{jwt} dw </math> | + | <math> x(t) = \int_{-\infty}^{\infty} \frac{X(w)}{2 \pi} e^{jwt} dw </math> |
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| + | Using this formula to determine the signal: | ||
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| + | <math>\ x(t) = \frac{8 \pi}{2 \pi} \int_{-\infty}^{\infty} w e^{jwt} \delta(w-9) dw + \frac{2}{2 \pi} \int_{-\infty}^{\infty}w^{3} \delta(w-4 \pi) e^{jwt} dw </math> | ||
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| + | Now using the sifting property of the delta function we find that the signal is | ||
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| + | <math>\ x(t) = 36 e^{j9t} + 64 \pi^{2} e^{j4\pi t} </math> | ||
Revision as of 17:31, 8 October 2008
Computing the Inverse Fourier Transform
$ \ X(\omega)= 8 \pi w \delta(w-9) + 2 \pi w^{3} \delta(w-4 \pi) $
The inverse Fourier transform is defined as:
$ x(t) = \int_{-\infty}^{\infty} \frac{X(w)}{2 \pi} e^{jwt} dw $
Using this formula to determine the signal:
$ \ x(t) = \frac{8 \pi}{2 \pi} \int_{-\infty}^{\infty} w e^{jwt} \delta(w-9) dw + \frac{2}{2 \pi} \int_{-\infty}^{\infty}w^{3} \delta(w-4 \pi) e^{jwt} dw $
Now using the sifting property of the delta function we find that the signal is
$ \ x(t) = 36 e^{j9t} + 64 \pi^{2} e^{j4\pi t} $
