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 $\omega = 2\pi f$

$X(2\pi f)=\int\limits_{-\infty}^{\infty} x(t)e^{-j2\pi ft} dt$

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

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

Example 1.

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


Again we will let $\omega = 2\pi f$ in our Fourier Transform $X(f)$ , and we will use the scaling property of the Dirac$\delta$ Function: $c\delta (ct) = \delta (t)$

\begin{align} \\ X({\color{red}2\pi f}) & = 2\pi \delta ({\color{red}2\pi f} - ({\color{red}2\pi f_o}))\\ & = {\color{red}2\pi} \delta ({\color{red}2\pi}(f - f_o )\\ & = \delta (f - f_o ) \end{align}

And previously it was shown that $X(2\pi f) = X(f)$ completing the change of variables.

Example 2.

$x(t) = sin(\omega t) \qquad \qquad X(\omega ) = \frac{\pi }{j}\ [\delta (\omega - \omega_o ) - \delta (\omega + \omega_o )]$

As earlier we will let $\omega = 2\pi f$ in our Fourier Transform $X(f)$ , and we will use the scaling property of the Dirac$\delta$ Function: $c\delta (ct) = \delta (t)$

\begin{align} \\ X({\color{red}2\pi f}) & = \frac{\pi }{j}\ [\delta ({\color{red}2\pi f} - {\color{red}2\pi f_o } ) - \delta ({\color{red}2\pi f} + {\color{red}2\pi f_o })]\\ & = \frac{\pi }{j}\ ( [ \delta ({\color{red}2\pi}(f - f_o ) - \delta({\color{red}2\pi} (f + f_o ))\\ & = \frac{\pi }{j}\ {\color{red}\frac{1}{2\pi }} ({\color{red}2\pi } [ \delta ( {\color{red}2\pi }(f - f_o ) - \delta({\color{red}2\pi} (f + f_o ))\\ & = \frac{1}{2j}\ ( \delta (f - f_o ) - \delta(f + f_o ))\\ \end{align}

References

[1].Mireille Boutin, "ECE438 Digital Signal Processing with Applications," Purdue University August 26,2009