TABLE OF CONTENTS The Fourier series is a frequency-domain representation of a continuous-time periodic signal. What about aperiodic signals? The concept of Fourier series can be extended to the Fourier transform, which applies to many aperiodic signals. For instance, all signals of finite energy have a Fourier transform, also called a spectrum.
As mentioned earlier, engineers often like to think of a signal as a spectrum whose energy is mostly contained in certain frequency bands. For instance, the Fourier transform of a human voice signal would show that most of the spectral contents of the signal are contained between 300 Hz and 3400 Hz. This is why the old telephone lines, seen here as linear time-invariant (LTI) systems that transmit continuous-time voice signals as voltage signals, were designed to have a flat frequency response over that band so as to maintain signal integrity.
In order to introduce the Fourier transform, we will use the specific example of a periodic rectangular signal, and we will let the period tend to infinity so as to have only the base period effectively remaining, that is, the part of the waveform around time t = 0. We will see that the Fourier transform in this particular case is nothing but the sinc envelope of the Fourier series coefficients.
Consider the Fourier series representation of the rectangular wave 
shown in Figure 5.1.
