Aliasing
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| Series | Geophysical References Series |
|---|---|
| Title | Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing |
| Author | Enders A. Robinson and Sven Treitel |
| Chapter | 4 |
| DOI | http://dx.doi.org/10.1190/1.9781560801610 |
| ISBN | 9781560801481 |
| Store | SEG Online Store |
The time t is a continuous real variable; that is, time t can take on any real value. However, in digital processing, the signals are sampled at equally spaced time instants. The time increment between adjacent sampling points is called the sampling interval. If we denote continuous time by t, the sampling interval by $ \Delta t $, and the discrete time index by the integer n, then we have the fundamental relation $ t{\ =}n\Delta t $.
Let us now apply the sampling concept to simple harmonic motion. We will see that the most important consequence of sampling a function at equally spaced time points is the phenomenon called aliasing. For example, in seismic recording, the sampling interval of 4 ms often is used. This causes frequencies greater than 1/(2)(0.004), namely 125 Hz (called the Nyquist frequency), to be aliased (see below). Figure 3a shows the amplitude spectrum of a wavelet with a 4-ms sampling interval. If the wavelet is decimated by resampling it at 8 ms (so that one-half of the values are thrown away), then the new Nyquist frequency becomes 1/(2)(0.008), which is 62.5 Hz. In Figure 3b, the portion of the spectrum beyond 62.5 Hz is folded back, as we see in Figure 3a. For this reason, the Nyquist frequency also is known as the folding frequency. The amplitude spectrum of the decimated wavelet is the sum of the parts shown in Figure 3b. This result is shown in Figure 3c (Robinson and Clark, 1991[1]).
Aliasing is commonplace, but we are so accustomed to it that we are only vaguely aware of its existence. Suppose we see an object at point A and then at point B, but our eye does not see the actual motion between A and B. Our mind interpolates this motion and makes the apparent path the shortest distance between A and B. Many of the magician’s sleight-of-hand tricks, such as the pea under the walnut shell, are based on this phenomenon.
Many of the old western movies had scenes with wagons and stagecoaches with wheels containing spokes. The spokes let us see the motion of the wheel. But what did we see in our mind? As the stagecoach started up and increased in speed, we saw the wheels turn faster and faster in the forward direction. Then suddenly they would reverse direction and slow to a stop, and the process would be repeated. The wheels would turn faster, reverse direction, slow to a stop, and so on. Our mind saw the wheels go backward as the coach went forward, but our eyes saw only a sequence of still pictures making up the moving-picture film. In fact, a moving-picture film represents a sampling of the actual physical motion being photographed. Any actual high rate of rotation of the wheel appears, as a result of the sampling process, to be aliased into a lower rotational frequency.
Consider the case of a single spoke, which we represent with the vector $ e^{i\omega t} $. We sample this exponential function at times $ t{=}n\Delta t $ so our sampled function is $ e^{i\omega n\Delta t} $. At low speeds, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \omega is small; for example, suppose Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \omega \Delta t{ =}\pi { /6} radians, which is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \omega \Delta t{ =}{ 3}0^{{ o}} . Then our sample function is the sequence of vectors Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): e^{i\left(\pi {\rm /6}\right)n} . We show these vectors for n = 0, 1, 2 in Figure 4a.
The mind interprets this motion as the slowest motion that can account for the actual observations. In the above case, the apparent rotation is the same as the actual rotation. At a higher speed, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \omega is larger. For example, suppose Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \omega \Delta t{\ =}{\ 11}\pi {\ /6,} , which is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \omega \Delta t{\ =33}0^{{\ o}} . Then our sample function is the sequence of vectors Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): e^{i\left({\rm 1l}\pi {\rm /6}\right)n} . We show these vectors for n = 0, 1, 2 in Figure 4b.
Again, the mind sees the smallest rotation velocity that accounts for the observations. We thus see an apparent Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \omega \Delta t{\rm \ of}-\pi {\rm /6\ =}-{\rm 3}0^{{\rm o}} instead of the actual Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \omega \Delta t of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\rm 11}\pi /{\rm 6=33}0^{{\rm o}} . That is, we see an apparent angular frequency of
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} &\omega {\ =\ }-\frac{\pi }{{\rm 6}\Delta t} \end{align} ()
instead of the actual angular frequency of
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \omega {\ =\ }\frac{{\rm ll}\pi }{{\rm 6}\Delta t}. \end{align} ()
We see that the actual angular frequency has been aliased to the apparent angular frequency. In this case, we have to subtract the Nyquist range Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 2\ \pi {\rm /}\Delta t from the actual angular frequency to obtain the apparent frequency; that is,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \frac{{\rm 11}\pi }{{\ 6}\Delta t}-\frac{{\rm l2}\pi }{{\ 6}\Delta t}{\rm =}-\frac{\pi }{{\rm 6}\Delta t}. \end{align} ()
The critical points occur when either Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \omega {\ =}-\pi {\rm /}\Delta t or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \omega {\ =}\pi {\rm /}\Delta t . When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \omega {\ =}\pi {\rm /}\Delta t, , for example, the sample function is the sequence of vectors
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} e^{i\omega {\rm t}} {\rm =}e^{i\frac{\pi }{\Delta t}\Delta tn}{\rm =}e^{i\pi n}{\rm =}{\left(e^{j\pi }\right)}^n{\rm =}{\left(-{\rm l}\right)}^n. \end{align} ()
We show these vectors in Figure 5. The vector just flops back and forth between +1 and –1, so the apparent motion could be either way.
References
- ↑ Robinson, E., and D. Clark, 1991, Sampling and the Nyquist frequency: The Leading Edge, 10, no. 3, 51-53.
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| Sinusoidal motion | The Nyquist frequency |
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| Visualization | Filtering |
Also in this chapter
- Time series
- The wavelet
- Digitization
- Frequency
- Sinusoidal motion
- The Nyquist frequency
- Sampling geophysical data
- Appendix D: Exercises
External links
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