Sinusoidal motion
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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 |
At this point, we introduce an important relation that is familiar to us from elementary calculus — Euler’s equation:
$ {\begin{aligned}&e^{i\theta }{\ =\ cos\ }\theta {\ +\ }i{\rm {\ sin\ }}\theta .\end{aligned}} $ ()
For negative $ \theta $, this is
$ {\begin{aligned}&e^{-i\theta }{\rm {=\ cos\ }}\theta -i{\rm {\ sin\ }}\theta .\end{aligned}} $ ()
Here, $ i{\rm {=}}{\sqrt {-{\rm {l}}}} $. The two versions of Euler’s equation above give the following expressions for the cosine and the sine:
$ {\begin{aligned}&{\rm {\ cos\ }}\theta {\ =\ }{\frac {e^{i\theta }{\rm {+}}e^{-i\theta }}{\rm {2}}}{\rm {,\ \ sin\ }}\theta {\ =\ }{\frac {e^{i\theta }-e^{-i\theta }}{{\rm {2}}i}}.\end{aligned}} $ ()
We now introduce the continuous time variable t, and we let $ \theta {\ =\ }\omega \ t $ in Euler’s equation. The result is
$ {\begin{aligned}&e^{i\omega t}{\ =\ cos\ }\omega t{\ +\ }i{\rm {\ sin\ }}\omega t.\end{aligned}} $ ()
The function $ e^{i\omega t} $ represents the rotating vector shown in Figure 2a. The angular frequency is $ \omega $, which for this discussion, we take to be an intrinsically positive number. The cyclical frequency is $ f{\rm {=}}\omega {\rm {/2}}\pi $. As t increases, the vector $ e^{i\omega t} $ rotates in the counterclockwise direction. In Cartesian coordinates, we let the x-axis represent the real axis and the y-axis represent the imaginary axis (Figure 2b). Then the quantity $ e^{i\omega t} $ for fixed $ \omega $ and t represents a vector whose projection on the x-axis is $ {\rm {\ cos\ }}\omega t $ and whose projection on the y-axis is $ {\rm {\ sin\ }}\omega t $. The angle of this vector is $ \omega t $, and the length of this vector is one. The (x,y)-plane is called the complex z-plane, where $ z{\rm {=}}x{\rm {+}}iy $. As time t increases, this vector rotates in a counterclockwise direction, and the tip of the vector traces out a circle. Because this circle has unit radius, it is called the unit circle in this complex z-plane.
As the vector rotates, its projection $ {\rm {\ cos\ }}\omega t $ on the x-axis traces out a cosine curve, and its projection $ {\rm {\ sin\ }}\omega t $ on the y-axis traces out a sine curve. Both $ {\rm {\ cos\ }}\omega $ and $ {\rm {\ sin\ }}\omega t $ represent sinusoidal motion at a fixed frequency $ \omega $. Such motion is called either sinusoidal motion or simple harmonic motion. Instead of using a continuous time scale t for the signal, digital processing requires first choosing a time increment $ \Delta t $ and then defining the time index n so that time is given by $ t{\rm {=}}n\Delta t $. A typical time increment is 0.004 s.
The two vectors in Figure 2a show that an angle of $ \omega n\Delta t $ radians is swept out in $ t{\rm {=}}n\Delta t $ seconds. The lower vector corresponds to time index n = 0 and the upper vector to time index n. Instead of considering a rotating wheel, we simply can think of a single vector that rotates at a constant angular velocity $ \omega $ (Figure 2b).
At time n = 0, the vector lies in the positive direction along the horizontal coordinate axis. Then at some arbitrary time index n, the vector will make an angle of $ \omega n\Delta t $ radians with the horizontal axis. The projections of this vector on the x- and y-axes give the horizontal and vertical components, respectively: $ \left({\rm {\ cos\ }}\omega n\Delta t{\rm {,\ \ sin\ }}\omega n\Delta t\right) $. Let the vertical axis be the imaginary axis (i.e., a unit distance on the vertical axis is $ i{\rm {=}}{\sqrt {-{\rm {1}}}} $). We can represent the vector at time $ n\Delta t $ in terms of its components $ {\rm {\ cos\ }}\omega n\Delta t $ and $ {\rm {\ sin\ }}\omega n\Delta t $ as follows:
$ {\begin{aligned}{\rm {vector}}\ ({\rm {at}}\ {\rm {time}}\ n\Delta t)\ {\rm {=\ cos\ }}\omega n\Delta t{\ +\ }i{\rm {\ sin\ }}\omega n\Delta t{\ =\ }e^{i\omega n\Delta t}\end{aligned}} $ ()
This equation, which is a form of Euler’s equation, shows that the exponential $ e^{i\omega n\Delta t} $ represents a unit vector (or wheel) rotating at constant angular velocity $ \omega $. The components of this vector represent simple harmonic motion.
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Also in this chapter
- Time series
- The wavelet
- Digitization
- Frequency
- Aliasing
- The Nyquist frequency
- Sampling geophysical data
- Appendix D: Exercises
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