3.4. Examples of the Fourier Transform
At this point let us pause to make sure we understand what is produced when the frequency domain representation of s(t) is obtained by Fourier transform (equation 3.1.2 or 3.1.4). Figure 3.5 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 3.6.
Figure
3.5. Three cosine waves
with amplitudes A1, A2, and A3 combine
to form a composite signal with amplitude
A1 + A2 + A3. 
Figure
3.6. Fourier transform of
three-cosine composite signal in
Figure 3.5 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 2πf1t is
(3.4.1)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 in Appendix B. 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 3.7 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 3.7 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.
