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now we let <math>\omega</math> = 2<math>\pi</math><math>f</math> | now we let <math>\omega</math> = 2<math>\pi</math><math>f</math> | ||
| − | <math>X(2\pi f)=\int\limits_{-\infty}^{\infty} x(t)e^{-j2\pi ft} dt </math> making X(<math>2\pi f</math> = X(<math>f</math> | + | <math>X(2\pi f)=\int\limits_{-\infty}^{\infty} x(t)e^{-j2\pi ft} dt </math> \qquad \qquad making X(<math>2\pi f</math> = X(<math>f</math> |
Revision as of 11:41, 18 September 2014
Fourier Transform as a Function of Frequency w Versus Frequency f (in Hertz)
A slecture by ECE student Randall Cochran
Partly based on the ECE438 Fall 2014 lecture material of Prof. Mireille Boutin.
To show the relationship between the Fourier Transform of frequency $ \omega $ versus frequency $ f $ (in hertz) we start with the definitions: $ X(w)=\int\limits_{-\infty}^{\infty} x(t)e^{-jwt} dt \qquad \qquad \qquad \qquad X(f)=\int\limits_{-\infty}^{\infty}x(t)e^{-j2\pi ft} dt $
now we let $ \omega $ = 2$ \pi $$ f $
$ X(2\pi f)=\int\limits_{-\infty}^{\infty} x(t)e^{-j2\pi ft} dt $ \qquad \qquad making X($ 2\pi f $ = X($ f $
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