The 1-D Fourier transform
Consider the following experiment. Hold a spring at one end and attach a weight to the other end. Pull the weight down a certain amount, say 0.8 units of distance. Release the weight. Assume that the spring is elastic — it bounces up and down ad infinitum. Set the time to zero at the onset of motion. Displacement of the weight as a function of time should vary between the peak amplitudes (+0.8, −0.8). If you had a device that could trace the amplitude of the displacement as a function of time, it would produce a sinusoidal curve as shown in Figure 1.1-1 (frame 1). Measure the time interval between two consecutive peaks; you will find that it is 0.080 s (80 ms). This time interval is called the period of the spring and it depends on the spring constant — a measure of spring stiffness. We say that the spring has completed one cycle of motion in a single period of time. Count the number of cycles within one second. This should be 12.5 cycles, which is called the frequency associated with the spring motion. One cycle per second (cps) is one hertz (Hz). Note that 1/0.080 s = 12.5 Hz; that is, frequency is the inverse of the period.
To continue, repeat the above experiment using a spring with a higher stiffness. Give the second spring a peak displacement amplitude of 0.4 units. The motion of the spring is traced as another sinusoid in Figure 1.1-1 (frame 2). The period and frequency of the spring are 0.040 s and 25 Hz, respectively. To keep track of these measurements, plot the peak amplitude of each spring as a function of frequency. These are the amplitude spectra shown in Figure 1.1-1.
Working with two identical springs, release spring 1 from a peak amplitude displacement of 0.8 units and set the time to zero at the onset of the motion. When spring 1 passes through the zero amplitude position, set spring 2 in motion from the same peak amplitude displacement (0.8). The motion of spring 1 is plotted in frame 1, while the motion of spring 2 is plotted in frame 3 of Figure 1.1-1. Because the springs were set to motion with the same peak amplitude displacement, the amplitude spectra of the two sinusoidal time functions should be identical. However, a difference is noted between the time functions in frames 1 and 3. In particular, when the sinusoid in frame 1 takes the peak amplitude value, the sinusoid in frame 3 takes the zero amplitude value. There was a time delay (20 ms) equivalent to one-quarter of a full cycle in setting spring 2 in motion relative to spring 1. This time delay is the difference between the two sinusoids shown in frames 1 and 3. A full cycle is equivalent to 360 degrees or 2π radians. Therefore, a time delay of one-quarter of a cycle is equivalent to a +90-degree phase-lag.
Phase is defined as the negative of phase-lag . Thus, a negative time shift corresponds to a positive phase value. Note that in Figure 1.1-1, if we apply a time shift of one-quarter of a full cycle (20 ms) to the sinusoid in frame 3 in the negative time direction, we obtain the sinusoid in frame 1. Although their amplitude spectra are identical, these two sinusoids can be distinguished based on their phase spectra as seen in Figure 1.1-1.
The experiment is completed. What is learned? First, the motion of an elastic spring can be described by a sinusoidal time function. Second, and more important, a complete description of a sinusoidal motion is given by its frequency, peak amplitude, and phase. This experiment teaches us how to describe spring motion as a function of time and frequency.
Now imagine an ensemble of many springs, each with a sinusoidal motion with a specific frequency, peak amplitude, and phase. The sinusoidal responses of all the members are shown in Figure 1.1-2. Suppose the motions of the individual springs are superimposed by adding all the traces. The result is a time-dependent signal that is represented by the first trace in Figure 1.1-2 (as indicated by the asterisk). The superposition (synthesis) allows us to transform the motion from frequency to time domain. This transformation is reversible; that is, the time-domain signal can be broken down (analyzed) into its sinusoidal components in the frequency domain.
Mathematically, this two-way process is achieved by the Fourier transform. In practice, the standard algorithm used on digital computers is the fast Fourier transform (FFT). Analysis of a time-dependent signal into its frequency components is done by forward Fourier transform, while synthesis of all the frequency components to the time-domain signal is done by inverse Fourier transform.
Figure 1.1-3 is a display of the Fourier transform of the time-dependent signal from Figure 1.1-2. The amplitude and phase spectra constitute a more condensed frequency-domain representation of the sinusoids in Figure 1.1-2. We can clearly see the parallelism between the two types of displays. In particular, the amplitude spectrum in Figure 1.1-3 has a large and a relatively small peak at about 20 and 40 Hz, respectively. Darker bands corresponding to larger peak amplitudes occur in Figure 1.1-2 at about the same frequencies. On the other hand, zones of weak amplitudes at about 30 Hz and at the low- and high-frequency ends of the spectrum also are apparent in both types of representations. Remember that the amplitude spectrum curve represents the peak amplitudes of the individual sinusoidal components as a function of frequency.
Figure 1.1-1 Tracing the motion of a spring in time yields a sinusoidal curve where positive amplitude corresponds to spring motion in the upward direction. The peak amplitude represents the maximum displacement of the weight at the end of the spring from the unstretched position. The time between the two consecutive peaks is the period of the sinusoid, the inverse of which is called frequency. Amplitude spectra distinguish sinusoids 1 and 2, which have different peak amplitudes and frequencies. The time delay of the onset of one spring relative to another is defined as phase-lag. Phase spectra (the negative of phase-lag spectra) distinguish sinusoids 1 and 3.
Now examine the phase spectrum. From the spring experiment, recall that the time delay of a particular frequency component also was expressed as a phase-lag. To better trace phase-lag as a function of frequency, a part of Figure 1.1-2 is magnified in Figure 1.1-4. Follow the positive-peak trend denoted by P. Note that the peaks fall above the zero timing line on the negative side of time axis at the low-frequency end of the spectrum. They then cross over to the positive side of the time axis at about 20 Hz and stay on that side over the rest of the frequency axis. The path that the peaks follow in Figure 1.1-4 can be plotted as the phase spectrum of Figure 1.1-3. If all the peaks were aligned along the zero timing line in Figure 1.1-4, then the corresponding time-domain signal would have a zero-phase spectrum. In this case, all the sinusoids would reinforce each other, causing a maximum peak value at zero time (Figure 1.1-11).
Figure 1.1-3 The information from Figure 1.1-2 can be condensed into amplitude and phase spectra. Each point along the amplitude spectrum curve corresponds to the peak amplitude of the sinusoid at that frequency plotted as a trace in Figure 1.1-2. Note the equivalence of the two peaks in the amplitude spectrum with the two high-amplitude zones in Figure 1.1-2. Each point along the phase spectrum corresponds to the time delay of a peak or trough along the sinusoid at that frequency with respect to the timing line at t = 0. Note the equivalence of the phase curve with the trend of a positive peak from trace to trace in Figure 1.1-4.
The physical significance of the amplitude spectrum is easier to understand than that of the phase spectrum (basic mathematical details of the Fourier transform).
- Robinson and Treitel, 1980, Robinson, E. A. and Treitel, S., 1980, Geophysical signal analysis: Prentice-Hall, Inc.
- Analog versus digital signal
- Frequency aliasing
- Phase considerations
- Time-domain operations
- Crosscorrelation and autocorrelation
- Vibroseis correlation
- Frequency filtering
- Practical aspects of frequency filtering
- Bandwidth and vertical resolution
- Time-variant filtering
- A mathematical review of the Fourier transform