# DFT independent variable as fraction of sampling rate

When the frequency domain's independent variable is labeled as a fraction of the sampling rate, the values along the horizontal axis always run between $0$ and $0.5$, since discrete data can only contain frequencies between DC ($0$) and half the sampling rate.

But why is the highest frequency pegged at exactly $0.5$ of the sampling rate? Isn't it possible to have a scenario where the highest frequency is less than half the sampling rate?

E.g., consider a speech signal that has been filtered to remove all frequencies above $3.3$ kHz and has been sampled at $10$ kHz. The highest frequency ($3.3$ kHz) is less than $0.5$ of the sampling rate ($5$ kHz). How would the frequency domain of this signal (labeled as a fraction of the sampling rate) look like?

The DFT calculates spectral components up to $f_\mathrm{s}/2$, no matter what the input signal is. If the input signal to the DFT contains (absolutely) no spectral components between $0.33 f_\mathrm{s}$ and $0.5 f_\mathrm{s}$ the corresponding DFT outputs are zero (assuming that there is no spectral leakage).