Digital Analysis of Geophysical Signals and Waves

3.1 Fourier Analysis

In the previous section we said that most geophysical signals can be expressed as a decomposition of the signal into sine and cosine functions of different frequencies (also referred to as harmonics). This is called Fourier analysis. We are usually first exposed to this concept in a calculus or physics course where sine and cosine functions expressed as a Fourier series are used to represent a periodic function of time. (In 1822, French mathematician Joseph Fourier was the first person who attempted to prove the convergence of such a series.) There are the usual conditions placed on the signal, i.e.: 1) it can't be multivalued at any given time, 2) it can't have an infinite number of discontinuities, or maxima or minima, and 3) it must be bounded within its period. The frequencies of the trigonometric functions are the spectral components of the Fourier series. These frequencies are predetermined by the periodicity, T of the function and are equal to n/T, n = +1, +2, ... Therefore, the frequency spectrum is composed of discrete line spectra.

When a signal is not periodic, the spectrum is not discrete and the Fourier series must be generalized into the Fourier integral or Fourier transform. As long as the integral of the absolute value of the signal converges, the continuous signal s(t) can be expressed as the Fourier integral

 

where

Equation 3.1.2 defines the Fourier transform of s(t); equation 3.1.1 is the inverse Fourier transform that recovers s(t) back from S(f). These equations are at the heart of spectral analysis and they are so tightly connected that they are usually called the Fourier transform pair. It is customary to use a lower case symbol for the time (or space) domain function and an upper case symbol for the corresponding function of frequency. S(f) and s(t) are referred to as the frequency domain and time domain representations of the signal, respectively. In general transform language, the terms in the integrands exclusive of the s(t) and S(f) are called the kernels of the transforms. In the Fourier transform pair the kernels differ only slightly; the sign of the exponent in the Fourier transform is - and in the inverse transform it is +. To save space in writing the Fourier transform pair, the common shorthand expressions S(f) = [s(t)] and s(t) = ^-1[S(f)] will be used for the remainder of this web site for the Fourier transform and inverse Fourier transform, respectively, unless the explicit expressions are required. Another common notation which we shall use is to indicate a Fourier transform pair using a bold, double-ended arrow, accordingly s(t)S(f).

The 2p appearing in the transform kernels can be included with the frequency f to express the Fourier transform pair in the angular frequency, w domain (in radians/s) as

and

 

 

We seemed to have made a gigantic jump from considering the digitizing of s(t) to now expressing it in a Fourier transform pair. Furthermore, it is obvious from the equations that the transform pair are complex functions with the inclusion of i = (-1)^1/2 in the transform kernels. Let's soften the blow by simply remembering a couple of things from basic mathematics. For one, the transform kernels, e.g., exp(i2pf) are of the general form of Euler's identity,

so,

From Euler's relationship we clearly see that the Fourier transform pair have sine and cosine terms just like a Fourier series does. And, since we know that integration is the limiting expression of a summation that becomes continuous, we realize that the Fourier transform is really the expression of a infinite, continuous "summation" of sine and cosine functions. In fact, the Fourier transform can be expressed using separate sine and cosine transforms. So Fourier analysis expressed by the Fourier transform is simply the decomposition of a signal into its composite frequency (sine and cosine) components.

Using Euler’s relation, a Fourier transform (equation 3.1.2) can be rewritten into it’s sin and cosine transform components as

From this we can visualize the Fourier transform of a real function, s(t) by realizing that:

1. To calculate the real part of the Fourier transform, at a frequency f = f₀, we multiply s(t) by cos(2pf₀t) and integrate (find the area under the resulting curve).
2. To calculate the imaginary part of the Fourier transform, at a frequency f = f₀, we multiply s(t) by sin(2pf₀t) and integrate (find the area under the resulting curve).
3. The Fourier transform at f = 0 is simply the integral (the area under the curve) of s(t).

...a rainbow is really nature's Fourier transform.

Rather than the discrete spectral lines (frequencies) appearing in a Fourier series, the Fourier transform has a continuous (as used in Figure 2.5) spectrum to represent a nonperiodic process. The transform of a signal into its continuous frequency components is familiar to us all in nature when white light passing through a glass prism exposes its color spectrum (Figure 3.1a). When this happens with rain drops it's called a rainbow. So a rainbow is really nature's Fourier transform (Figure 3.1b) although we've never heard anyone call a rainbow a Fourier transform.