Examples of the Fourier Transform

This article is referenced from "Digital Geophysical Analysis" by Sage.

Figure 3.4 contains a simple composite signal in the time domain made up of three separate cosinusoids. Assuming that these cosinusoids extend in time from - infinity to + infinity, the Fourier transform of this composite signal yields the result depicted in Figure 3.5.

Figure 3.4. Three cosine waves with amplitudes A1, A2, and A3 combine to form a composite signal with amplitude A1 + A2 + A3.

Figure 3.5. Fourier transform of three-cosine composite signal in Figure 3.4 yields three pairs of real, even delta functions with corresponding amplitudes A1/2, A2/2, and A3/2.

Since the original signal is real and even (cosine functions are clearly even functions), the Fourier transform must be real and even. Three pure cosine oscillations summate to make up s(t) so only three spectral lines are present in the Fourier transform, S(f). These spikes can be represented by Dirac delta functions that are functions of frequency, not of time as we defined in Section 2.3. For example, the Fourier transform of A₁cos 2pf₁t is

(3.10)

This reveals an interesting aspect of the Fourier transform that we avoided talking about earlier, namely that there are values (spectra lines) at both positive and negative frequencies. In this case they appear where the delta functions are non-zero, i.e., where their arguments are zero, at f = +f₁ and f = -f₁.

The concept of negative frequencies is not widely understood, even though the proper handling of this concept is critical for practical applications of digital processing in the frequency domain. Therefore, we are compelled to convince you of the validity of both positive and negative frequencies so you will appreciate the subtleties when working with them. This we will do at the end of this section. First let’s see what the Fourier transforms are of several of the functions that we’ve encountered so far. The Fourier transform pairs appearing in Figure 3.6 are as important as they are famous. Select a Fourier transform pair and guess what will happen in the frequency domain before your actually drag the button to change the spacing in the time domain function. We will not derive the equations given in Figure 3.6 that dictate the graphed results. Consult the general references at the end of the web site if you really want to see the derivations of the Fourier transform pairs.

Figure 3.6 Equations and graphs of several important, famous Fourier transform pairs. Select a pair and guess what will happen in the frequency domain before you move the drag button to vary the spacing in the time domain function..

Negative Frequencies

Virtually every text book on Fourier analysis treats the introduction of negative frequencies as a natural occurrence, one that is merely a convention, not worthy of any justification. Yet, in our experience, this concept is one of the least understood basic tenants of Fourier analysis and, consequently, it is often ill-applied by students. Actually, as shown by the symmetry properties of a real-valued function in the time or space domain (Section 3.3), there is no new information in the negative frequency spectrum.

A common way to describe the idea of negative frequencies is to visualize a wheel rotating in one direction and then reversing the direction. Rotating in say the counterclockwise (CCW) direction illustrates positive frequency and clockwise (CW) rotation describes negative frequency. The rotating wheel view is a perfectly correct way of interpreting the + and - frequencies of the complex Fourier spectrum as we will now show.We will justify this statement by providing a detailed understanding of what the Fourier transform of A₁ cos (2pf₁t) in equation 3.10 actually means.

First, we recall Euler’s relation (equation 3.5b) that decomposes e^if into real and imaginary parts, i.e.,

(3.11)

from which we can write,

(3.12)

The inverse of Euler’s relation allows us to express the trigonometric functions as

(3.13)

and

(3.14)

Concentrating on the cosine relationship, the term A₁/2 e^i2pf₁t can be mapped as a vector with real and imaginary parts A₁/2 cos (2pf₁t) and A₁/2 sin (2pf₁t), respectively, as in Figure 3.8a. Similarly, A₁/2 e^-i2pf₁t is plotted in Figure 3.8b.

Figure 3.8. Mapping of complex exponentials as vectors with real and imaginary parts: a) A1/2 ei2pf1t, and b) A1/2 e-i2pf1t.

The key observation is that since f = +2pf₁t is an angle that varies linearly with time, the vectors e^{+i2p f₁t} also vary with time. For example, at t = 0 the e^{+i2p f₁t} vectors lie along the positive, horizontal axis; at a time t = 1 s later the e^{+i2p f₁t} vector has rotated through a CCW angle of 2pf₁. At an arbitrary time t, the rotating e^{+i2p f₁t} vector has an angle of 2pf₁t CCW from the positive real axis. The e^{-i2p f₁t} vector, with the negative exponent, rotates similarly but in a CW direction. In Figure 3.9a the two counter rotating vectors from Figures 3.8a and b are summed as they rotate CCW and CW each with angular velocity w₁ = 2pf₁ radians/s. In other words, Figure 9a graphically performs

(3.15)

Figure 3.9. Development of positive and negative frequencies: a) A1 cos (2pf1t) is sum of two counter rotation vectors from Figure 3.8, and b) A1 sin (2pf1t) is difference of these counter rotating vectors.

This clearly shows that the cosine function can be viewed as being composed of both positive (CCW) and negative (CW) frequency components. These are expressed in the frequency domain after Fourier transform as d-functions at f = +f₁ in equation 3.10. It is also clear from equation 3.15 and the vector summation in Figure 3.9a that the amplitudes of the d-functions must be A₁/2, not A₁. Equation 3.14 tells us that the picture for the sine function is that of a difference between counter rotating vectors both of which have imaginary values (Figure 3.9b).

A very convenient way to graph general complex frequency domain quantities at one instant of time was presented by Bracewell (1965) and extended by harris (1977). Figures 3.10a and b illustrate this method for A₁ cos (2pf₁t) and A₁ sin (2pf₁t), respectively. The positions of the vectors on the + frequency axis are the frequencies of rotation. This plotting scheme allows the presentation of both real and imaginary parts (or amplitude and phase values) on a single plot.