# The Nyquist frequency

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Series Geophysical References Series Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing Enders A. Robinson and Sven Treitel 4 http://dx.doi.org/10.1190/1.9781560801610 9781560801481 SEG Online Store

The upper critical angular frequency, namely ${\displaystyle {\omega }_{n}{\ =}\pi {\rm {/}}\Delta t}$ is called the Nyquist angular frequency. The lower critical angular frequency, namely ${\displaystyle -{\omega }_{n}{\ =}-\pi {\rm {/}}\Delta t}$, is the negative of the Nyquist angular frequency. The interval between the two Nyquist angular frequencies, namely

 {\displaystyle {\begin{aligned}-{\frac {\pi }{\Delta t}}\leq \omega \leq {\frac {\pi }{\Delta t}},\end{aligned}}} (14)

is called the Nyquist range, which is of total length ${\displaystyle {\rm {2}}\pi {\rm {/}}\Delta t}$. As we have seen, any actual rotation always appears as an apparent rotation within the Nyquist range. For example, the actual angular frequency ${\displaystyle \omega {\ =}\pi {\ /}\left({\rm {6}}\Delta t\right)}$ is already in the Nyquist range, so in this case, the apparent angular frequency is the same as the actual angular frequency. Because the cyclic frequency f is ${\displaystyle \omega {\ /2}\pi }$, the cyclic Nyquist frequency is

 {\displaystyle {\begin{aligned}f_{n}{\rm {=}}{\frac {{\omega }_{n}}{{\rm {2}}\pi }}{\ =\ }{\frac {\left(\pi {\rm {/}}\Delta t\right)}{{\rm {2}}\pi }}{\ =\ }{\frac {\rm {1}}{{\rm {2}}\Delta t}}\end{aligned}}} (15)

and the Nyquist range is

 {\displaystyle {\begin{aligned}-{\frac {\rm {l}}{{\rm {2}}\Delta t}}\leq f\leq {\frac {\rm {l}}{{\rm {2}}\Delta t}},\end{aligned}}} (16)

which is of total length ${\displaystyle {\rm {1/}}\Delta t}$.

In contrast, the actual angular frequency ${\displaystyle \omega {\ =}{\ 11}\pi {\rm {/}}\left({\rm {6\ }}\Delta t\right)}$ is outside the Nyquist range, so the apparent angular frequency is the aliased version, namely

 {\displaystyle {\begin{aligned}{\frac {{\rm {11}}\pi }{{\rm {6}}\Delta t}}-{\frac {{\rm {2}}\pi }{\Delta t}}{\rm {=}}-{\frac {\pi }{{\rm {6}}\Delta t}},\end{aligned}}} (17)

where we recall that ${\displaystyle {\rm {2}}\pi {\rm {/}}\Delta t}$ is the length of the Nyquist range.

Because one complete rotation is ${\displaystyle {\rm {36}}0^{\rm {o}}}$, or ${\displaystyle {\rm {2}}\pi }$, we always must add or subtract some integer multiple of ${\displaystyle {\rm {2}}\pi {\rm {/}}\Delta t}$ from the actual frequency to obtain the apparent or aliased frequency, which lies within the Nyquist range.

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.

Our results to this point imply that at the sample points, any sinusoid of arbitrary frequency is equivalent to a sinusoid with a frequency lying within the Nyquist range. Here we mean “equivalent” in the sense that the two sinusoids have the same numerical values at the sample points.

Let us pick a sampling increment of 0.004 s, so that the sampling frequency is 250 Hz (the Nyquist frequency is thus 125 Hz). Figures 6 and 7 illustrate the impossibility of distinguishing sampled signals outside of the Nyquist range — that is, within the range from –125 Hz to 125 Hz. Let the actual frequency be f and the sampling frequency be ${\displaystyle f_{s}}$. In each of these figures, the aliased frequency is given by ${\displaystyle f_{a}{\ =\ }f-f_{s}}$. If the resulting ${\displaystyle f_{a}}$ is negative, replace it by ${\displaystyle -f_{a}{\ =\ }f_{s}-f}$. We can do this because (for real signals) a positive frequency cannot be distinguished from its negative counterpart. For example, in Figure 7a, the actual frequency is 175 Hz. Thus, the aliased frequency is 250 –175 = 75 Hz.

Figure 6.  (a) Actual signal of frequency 250 Hz, with the same sample points as its aliased signal of frequency 250 - 250 = 0 Hz. (b) Actual signal of frequency 225 Hz, with the same sample points as its aliased signal of frequency 250 - 225 = 25 Hz. (c) Actual signal of frequency 200 Hz, with the same sample points as its aliased signal of frequency 250 - 200 = 50 Hz.
Figure 7.  (a) Actual signal of frequency 175 Hz, with the same sample points as its aliased signal of frequency 250 - 175 = 75 Hz. (b) Actual signal of frequency 150 Hz, with the same sample points as its aliased signal of frequency 250 - 150 = 100 Hz. (c) Actual signal of frequency 125 Hz, with the same sample points as its aliased signal of frequency 250 - 125 = 125 Hz.

The sampling limitation is one of bandwidth as given by the Nyquist range. In seismic exploration work, we are interested in the lower frequencies, so we always choose the Nyquist range to center around zero frequency. Other disciplines might be concerned with frequencies in some higher band. If there are no frequencies outside a selected band, sampling economies can result.