Fourier Transform as a FUnction of Frequency w versus Frequency f (in Hertz) - Rhea

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 $

making X($ 2\pi f $) = X($ f $)

Examples of the relationship can be shown by starting with known CTFT pairs:

Example 1. $ x(t)= e^{j\omega_o t} \qquad \qquad X(f) = 2\pi \delta (\omega - \omega_o ) $

Again we will let $ \omega $ = 2$ \pi $$ f $ in our Fourier Transform $ X(f) $

$ X(f) = 2\pi \delta (2\pi f - 2\pi f_o ) $

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