Most of the signals and waveforms that we discuss and analyze are expressed in the time domain. That is, they are variations of voltage, current, or power with respect to time.

All the signals shown in the previous illustrations are examples of time-domain waveforms. Their mathematical expressions contain the variable time t, indicating that they are a time-variant quantity. Fourier theory gives us a new and different way to express and illustrate complex signals. Here, complex signals containing many sine and/or cosine components are expressed as sine or cosine wave amplitudes at different frequencies. In other words, a graph of a particular signal is a plot of sine and/or cosine component amplitudes with respect to frequency. A typical frequency-domain plot of the square wave is shown in Fig. 2-61(a). Note that the straight lines represent the sine wave amplitudes of the fundamental and harmonics, and these are plotted on a horizontal frequency axis. Such a frequency domain plot can be made directly from the Fourier expression by simply using the frequencies of the fundamentals and harmonics and their amplitudes. Frequency-domain plots for some of the other common nonsinusoidal waves are also shown in Fig. 2-61.

Note that the triangle wave in Fig. 2-61(c) is made up of the fundamental and odd harmonics. The third harmonic is shown as a line below the axis, which indicates a 180° phase shift in the cosine wave making it up. Fig. 2-62 shows how the time and frequency domains are related. The square wave discussed earlier is used as an example. The result is a three-axis threedimensional view.

Signals and waveforms in communication applications are expressed by using both time-domain and frequency-domain plots, but in many cases, the frequency-domain plot is far more useful. This is particularly true in the analysis of complex signal waveforms as well as the many modulations and multiplexing methods used in communication. Test instruments for displaying signals in both time and frequency domains are readily available. You are already familiar with the oscilloscope, which displays the voltage amplitude of a signal with respect to a horizontal time axis. The test instrument for producing a frequency-domain display is the spectrum analyzer. Like the oscilloscope, the spectrum analyzer uses a cathode-ray tube for display, but the horizontal sweep axis is calibrated in hertz and the vertical axis is calibrated in volts or power units or decibels.

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