Why is the use of sines and cosines of the time variable t relevant to the theory of digital filtering? The reason is that sines and cosines let us define frequency, and the concept of frequency is basic to what we mean by a filter, be it digital or analog. Let us think about a rotating wheel, as on an old-fashioned wagon or stagecoach. Consider, for example, a wheel that is rotating at a rate of 10 times a second. If we consider one spoke on this wheel to be a reference vector, this vector makes 10 complete rotations in every second. Each rotation through (which is radians) represents one cycle, so we say that the vector has a cyclic frequency of 10 cycles per second. The word hertz (Hz) stands for cycles per second, so alternatively, we say that the vector has a frequency of 10 Hz. Up to this point, we have not specified in which direction the vector is rotating. As a matter of mathematical convention, we say that the vector has a frequency of +10 Hz if it is rotating in the counterclockwise direction, whereas it has a frequency of –10 Hz if it is rotating in the clockwise direction. The frequency customarily is denoted by the symbol f.
We talk about negative frequencies, but do we encounter them in seismic work? In the seismic case, only real-valued signals are recorded. Therefore, a positive frequency cannot be distinguished from its negative counterpart. For example, a frequency of 10 Hz is indistinguishable from a frequency of –10 Hz. Hence, only positive frequencies need be considered in ordinary circumstances.
Period is a general term used to denote an interval of time, but the word period has special meaning in frequency analysis. By the period of a repeating process, we mean the shortest time interval for which the process exactly repeats itself. In our example of the rotating vector, the motion exactly repeats itself 10 times a second, so we say that it has a period . What does this have to do with the subject of frequency?
The period T and the frequency f are reciprocals of each other, that is, and . As we have seen, hertz is a measure of cyclic frequency, that is, a measure of the number of cycles per second. However, there is yet another type of frequency: the angular frequency , which is measured in radians per second. Angular frequency is related to cyclic frequency f by the fundamental relation
In general, the word frequency can be used for either or f, so one often must determine from the context which frequency is meant. Usually, if the word frequency occurs by itself, cyclic frequency in units of hertz is implied. The factor in the above formula comes from the fact that there are radians (or ) in each complete rotation.
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Also in this chapter
- Time series
- The wavelet
- Sinusoidal motion
- The Nyquist frequency
- Sampling geophysical data
- Appendix D: Exercises