Consider the sinusoid in Figure 1.1-7. This signal is resampled as before to 4 and 8 ms. The amplitude spectra indicate that all three have the same frequency, 25 Hz. Nothing happened to the signal after resampling it to a coarser sampling interval. Now examine the higher frequency sinusoid (75 Hz) in Figure 1.1-8. It appears the same at both 2- and 4-ms sampling. However, resampling to 8 ms changed the signal and made it appear to be a lower frequency sinusoid. The resampled signal has a frequency of 50 Hz as seen in the amplitude spectrum. The Nyquist frequency for an 8-ms sampling interval is 62.5 Hz. The true signal frequency is 75 Hz. As a result of resampling, the signal with 75-Hz frequency folded back onto the spectrum and appeared at its alias frequency of 50 Hz. Finally, a 150-Hz sinusoid resampled to 4 and 8 ms is shown in Figure 1.1-9. This time, the 4-ms sampling made the signal appear as a 100-Hz signal, while the 8-ms sampling made it appear as a 25-Hz signal. By using a single-frequency sinusoid, we see that frequencies above the Nyquist really are not lost after sampling, but appear at frequencies below the Nyquist.
Now consider the superposition of two sinusoids with frequencies of 12.5- and 75-Hz as shown in Figure 1.1-10. Digitization of this signal at 2- and 4-ms sampling intervals does not alter the original signal, since its frequency components are below the Nyquist frequencies associated with 2- and 4-ms sampling intervals — 250 and 125 Hz, respectively. However, when the signal is digitized at a coarser sampling interval, such as 8 ms, the amplitude spectrum changes. The 12.5-Hz component is not affected, because 8-ms sampling still is sufficient to sample this low-frequency component. On the other hand, the 75-Hz component is seen as a lower-frequency component (50 Hz). Once again, note that those frequencies in the original signal above the Nyquist frequency corresponding to the chosen sampling interval are folded back onto the amplitude spectrum of the digitized version of the signal.
This analysis can be extended to many sinusoids of different frequencies. In particular, the discrete time series derived from a too coarse sampling (undersampling) of a continuous signal actually contains contributions from high-frequency components of that continuous signal. Those high frequencies fold back onto the spectrum of the discrete time series and appear as lower frequencies. The phenomenon that is caused by undersampling the continuous signal is termed frequency aliasing.
To compute the alias frequency fa, use the following relation
where fN is the folding frequency, fs is the signal frequency, and m is an integer such that fa < fN. For example, suppose that fs = 65 Hz, fN = 62.5 Hz, which corresponds to 8-ms sampling rate. The alias frequency then is fa = |2 × 62.5 − 65| = 60 Hz.
In conclusion, undersampling has two effects:
- band limiting the spectrum of the continuous signal, with the maximum frequency being the Nyquist, and
- contamination of the digital signal spectrum by high frequencies beyond the Nyquist, which may have been present in the continuous signal.
Nothing can be done about the first problem. The second problem is of practical importance. To keep the recoverable frequency band between zero and the Nyquist frequency free from aliased frequencies, a high-cut antialiasing filter is applied in the field before analog-to-digital conversion of seismic signals. This filter eliminates those frequency components that would have been aliased during sampling. Typically, the high-cut antialiasing filter has a cutoff frequency that is either three-quarters or half of the Nyquist frequency. This filter rolls off steeply so that aliases of frequencies above the Nyquist are highly attenuated.
- Analog versus digital signal
- Phase considerations
- Time-domain operations
- Crosscorrelation and autocorrelation
- Vibroseis correlation
- Frequency filtering
- Practical aspects of frequency filtering
- Bandwidth and vertical resolution
- Time-variant filtering