(→Computing the Fourier Transform) |
(→Computing the Fourier Transform) |
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Compute the Fourier Transform of the signal | Compute the Fourier Transform of the signal | ||
| − | <math>\ x(t)= \sin(2 \pi t+ \pi/4) </math> | + | <math>\ x(t)= t \sin(2 \pi t+ \pi/4) </math> |
By definition the Fourier Transform of a signal is defined as: | By definition the Fourier Transform of a signal is defined as: | ||
| Line 9: | Line 9: | ||
<math>X(\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt</math> | <math>X(\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt</math> | ||
| − | First | + | First expressing the signal in as a Fourier series: |
| − | <math>\ | + | However before finding the transform we note that multiplication in the time domain is just differentiation in the frequency domain. So the game plan is to find the Fourier series of x(t)/t then differentiate it with respect to w in the frequency space. |
| + | |||
| + | <math>\ x1(t)=\sin(2\pi t+ \pi/4) = \frac{e^{2 \pi jt + \pi/4}{2j}</math> | ||
Revision as of 16:49, 8 October 2008
Computing the Fourier Transform
Compute the Fourier Transform of the signal
$ \ x(t)= t \sin(2 \pi t+ \pi/4) $
By definition the Fourier Transform of a signal is defined as:
$ X(\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt $
First expressing the signal in as a Fourier series:
However before finding the transform we note that multiplication in the time domain is just differentiation in the frequency domain. So the game plan is to find the Fourier series of x(t)/t then differentiate it with respect to w in the frequency space.
$ \ x1(t)=\sin(2\pi t+ \pi/4) = \frac{e^{2 \pi jt + \pi/4}{2j} $
