Sinusoidal motion

Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing

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:

${\begin{aligned}&e^{i\theta }{\ =\ cos\ }\theta {\ +\ }i{\rm {\ sin\ }}\theta .\end{aligned}}$

(5)

For negative $\theta$, this is

${\begin{aligned}&e^{-i\theta }{\rm {=\ cos\ }}\theta -i{\rm {\ sin\ }}\theta .\end{aligned}}$

(6)

Here, $i{\rm {=}}{\sqrt {-{\rm {l}}}}$. The two versions of Euler’s equation above give the following expressions for the cosine and the sine:

${\begin{aligned}&{\rm {\ cos\ }}\theta {\ =\ }{\frac {e^{i\theta }{\rm {+}}e^{-i\theta }}{\rm {2}}}{\rm {,\ \ sin\ }}\theta {\ =\ }{\frac {e^{i\theta }-e^{-i\theta }}{{\rm {2}}i}}.\end{aligned}}$

(7)

We now introduce the continuous time variable t, and we let $\theta {\ =\ }\omega \ t$ in Euler’s equation. The result is

${\begin{aligned}&e^{i\omega t}{\ =\ cos\ }\omega t{\ +\ }i{\rm {\ sin\ }}\omega t.\end{aligned}}$

(8)

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).

Figure 2.  (a) Spoke of wheel rotating at angular velocity $\omega$. (b) Vector rotating at angular velocity $\omega$.

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:

${\begin{aligned}{\rm {vector}}\ ({\rm {at}}\ {\rm {time}}\ n\Delta t)\ {\rm {=\ cos\ }}\omega n\Delta t{\ +\ }i{\rm {\ sin\ }}\omega n\Delta t{\ =\ }e^{i\omega n\Delta t}\end{aligned}}$

(9)

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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Table of Contents (book)

Also in this chapter

• Time series
• The wavelet
• Digitization
• Frequency
• Aliasing
• The Nyquist frequency
• Sampling geophysical data
• Appendix D: Exercises

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