# Difference between 2$\pi f$ and $\omega$ in Fourier transform

What is the difference when we use $e^{-j2\pi f}$ and $e^{-j\omega n}$ for Fourier transformation?

• $\omega = 2\pi f$ – endolith Nov 10 '16 at 17:44
• @endolith Thanks :) I know that, but I don't know how the meaning of Fourier transform and Fourier transform table pairs change if we use $f$ or $\omega$, Can you help me with that? – user137927 Nov 10 '16 at 17:46
• It's exactly the same thing, just expressed in different units. You should really be comparing $e^{-j\omega}$ with $e^{-j\omega n}$, for instance. – endolith Nov 10 '16 at 19:00
• An interesting side note: white noise has (two-sided) power spectral density $\frac{N_0}{2}$ regardless of whether you are denoting it $S_N(\omega)$ or as $S_n(f)$ !! In either case, the noise power in a bandwidth of $B$ Hz works out to be $N_0B$ watts or volts$^2$. – Dilip Sarwate Nov 10 '16 at 20:57
• Thank @endolith, What about normalization coefficient? As I know it is different in use of $f$ and $\omega$? Is there any pattern for this difference? – user137927 Nov 11 '16 at 6:30

You might be mixing two concepts, pertaining to (following Robert Bristow-Johnson)

• the "analog context" (more formally, the "continuous-time case"),
• the "digital context" (more formally the "discrete-time case").

The first concept corresponds to the continuous Fourier transform, for which you can use a form of normalized frequency cycles per second or Hertz ($$e^{-j2\pi f}$$), while the second is the angular frequency in radians per second ($$e^{-j\omega}$$).

The second one, with integer index $$n$$ relates first instance to discrete summations, e.g. the discrete-time Fourier transform (DTFT), where the Fourier kernel writes $$e^{-j2\pi f n}$$ with $$f\in \mathbb{R}$$ (for instance) or $$e^{-j\omega n}$$ with $$\omega\in \mathbb{R}$$ (for instance).

The first concept is in use in the inverse DTFT, since it involves integrals.

• i would say, in the "analog case" (more formally, the "continuous-time case") it's $$\Omega = 2\pi f$$ and in the "digital case" (more formally the "discrete-time case") it's $$\omega = 2\pi \frac{f}{f_\text{s}}$$ in my book it's $$s=j \Omega$$ and $$z = e^{j \omega}$$ – robert bristow-johnson Nov 10 '16 at 23:57
• "in my book" is meant as a figure of speech. – robert bristow-johnson Nov 11 '16 at 0:02