# Uncertainty Relation between time and frequency

Typically, humans can hear sound waves in the frequency range 20 Hz to 20 kHz. If one wants to make digital record of sound such that no audible information is lost, what is the longest time interval between samples that could be used?

• Can you please clarify the question a bit? Off the shelf cheap interfaces can record at 192kHz / 24bit is that "long enough"? – A_A Apr 21 '18 at 12:59
• looks like homework about sampling theorem, doesn't it? – AlexTP Apr 21 '18 at 13:20
• My brother is taking a course on Fourier Analysis and wanted me to help on this particular problem. It seems like to have something to with some sort of inequality relation with f and t but I don't know how to proceed. – Zeki Zeybek Apr 21 '18 at 13:57
• Do you think it is about delta(t) * delta(f) ~ 1 ? – Zeki Zeybek Apr 21 '18 at 14:01
• @ZekiZeybek it seems that you want to talk about Heisenberg's uncertainty principle. However, for me the question is about the sampling theorem. A simple search gives you lots of references. – AlexTP Apr 21 '18 at 21:44

Sampling a signal creates its spectrum replica as Nyquist Shannon theorem stated. As a consequence, to avoid aliasing, one must sample at a rate greater than twice the max frequency $B$ of the signal spectrum. Otherwise, you have aliasing as illustrated in the Figure below. In your case, $B = 20$kHz hence sampling rate $f_s > 2B$ and the interval between samples $T_s = 1/f_s < 1/(2B)$. You can see that $T_s$ is upperbounded by $1/(2B) = 50$ microseconds.
Note that the longest $T_s$ is not 50 microseconds but $(50-\epsilon)$ microseconds with $\epsilon$ is arbitrarily small positive quantity.