# Translations:The Nyquist frequency/12/en

According to our definition, ${\displaystyle \Delta t}$ is the sampling interval. The reciprocal of ${\displaystyle \Delta t}$, namely ${\displaystyle {\rm {1/}}\Delta t}$, is the sampling frequency (or, equivalently, the sampling rate). For example, if ${\displaystyle \Delta t{\rm {=0.004s}}}$, then the sampling frequency is ${\displaystyle f_{s}{\rm {=}}{\rm {1/}}\Delta t{\rm {=}}{\rm {1/0.004=250}}}$ samples/s. The sampling angular frequency is ${\displaystyle {\omega }_{s}{\ =2}\pi f_{s}{\ =2}\pi /\Delta t}$ radians per second. Thus, we always must subtract some integer multiple of the sampling angular frequency to bring an actual angular frequency within the Nyquist range. The sampling frequency ${\displaystyle f_{s}}$ is one sample per ${\displaystyle \Delta t}$, that is, ${\displaystyle f_{s}{\rm {=}}{\rm {1/}}\Delta t}$. Thus, we see that the Nyquist frequency ${\displaystyle f_{\rm {n}}{\rm {=}}{\rm {1/}}\left({\rm {2}}\Delta t\right)}$ is one-half the sampling frequency. In the case when ${\displaystyle \Delta t{\ =0.004}{\rm {s}}}$, the Nyquist frequency is ${\displaystyle f_{n}{=}{\ 1/}\left({\rm {2}}\cdot {\rm {0.004}}\right){\rm {=\ }}{\rm {125}}}$, and the sampling frequency is ${\displaystyle f_{s}{\rm {=2}}f_{n}{\rm {=}}{\rm {1/}}\left({\rm {0.004}}\right){\rm {=250Hz}}}$. If a frequency f is outside the Nyquist range, its alias is found by subtracting from f an integral multiple of the sampling frequency; that is, the aliased frequency is ${\displaystyle f_{a}{\ =\ }f-kf_{s}}$, where k is the integer that makes the aliased frequency lie in the Nyquist range.