At this point, we introduce an important relation that is familiar to us from elementary calculus — Euler’s equation:
For negative , this is
Here, . The two versions of Euler’s equation above give the following expressions for the cosine and the sine:
We now introduce the continuous time variable t, and we let in Euler’s equation. The result is
The function represents the rotating vector shown in Figure 2a. The angular frequency is , which for this discussion, we take to be an intrinsically positive number. The cyclical frequency is . As t increases, the vector 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 for fixed and t represents a vector whose projection on the x-axis is and whose projection on the y-axis is . The angle of this vector is , and the length of this vector is one. The (x,y)-plane is called the complex z-plane, where . 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 on the x-axis traces out a cosine curve, and its projection on the y-axis traces out a sine curve. Both and represent sinusoidal motion at a fixed frequency . 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 and then defining the time index n so that time is given by . A typical time increment is 0.004 s.
The two vectors in Figure 2a show that an angle of radians is swept out in 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 (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 radians with the horizontal axis. The projections of this vector on the x- and y-axes give the horizontal and vertical components, respectively: . Let the vertical axis be the imaginary axis (i.e., a unit distance on the vertical axis is ). We can represent the vector at time in terms of its components and as follows:
This equation, which is a form of Euler’s equation, shows that the exponential represents a unit vector (or wheel) rotating at constant angular velocity . The components of this vector represent simple harmonic motion.
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Also in this chapter
- Time series
- The wavelet
- The Nyquist frequency
- Sampling geophysical data
- Appendix D: Exercises