- 2.1 The Time Domain
- 2.2 Sine Waves in the Frequency Domain
- 2.3 Shorter Time to a Solution in the Frequency Domain
- 2.4 Sine Wave Features
- 2.5 The Fourier Transform
- 2.6 The Spectrum of a Repetitive Signal
- 2.7 The Spectrum of an Ideal Square Wave
- 2.8 From the Frequency Domain to the Time Domain
- 2.9 Effect of Bandwidth on Rise Time
- 2.10 Bandwidth and Rise Time
- 2.11 What Does Significant Mean?
- 2.12 Bandwidth of Real Signals
- 2.13 Bandwidth and Clock Frequency
- 2.14 Bandwidth of a Measurement
- 2.15 Bandwidth of a Model
- 2.16 Bandwidth of an Interconnect
- 2.17 The Bottom Line

## 2.4 Sine Wave Features

As we now know, by definition, the only waveforms that exist in the frequency domain are sine waves. We should also be familiar with the description of a sine wave in the time domain. It is a well-defined mathematical curve that has three terms that fully characterize absolutely everything you could ever ask about it. An example is shown in Figure 2-3.

Figure 2-3 Top: Description of a sine wave in the time domain. It is composed of over one thousand voltage-versus-time data points. Bottom: Description of a sine wave in the frequency domain. Only three terms define a sine wave, which is a single point in the frequency domain.

The following three terms fully describe a sine wave:

- Frequency
- Amplitude
- Phase

The frequency, usually identified using a small f, is the number of complete cycles per second made by the sine wave, in Hertz. Angular frequency is measured in radians per second. A radian is like degrees, describing a fraction of a cycle. There are 2 x p radians in one complete cycle. The Greek letter w is often used to refer to the angular frequency, measured in radians per second. The sine-wave frequency and the angular frequency are related by:

#### Equation 2-2

where:

- w = angular frequency, in radians/sec
- p = constant, 3.14159...
- f = sine-wave frequency, in Hz

For example, if the frequency of a sine wave is 100 MHz, the angular frequency is 2 x 3.14159 x 100 MHz ~ 6.3 x 10^{8} radians/sec.

The amplitude is the maximum value of the peak height above the center value. The wave peak goes below the horizontal just as much as it goes above.

The phase is more complicated and identifies where the wave is in its cycle at the beginning of the time axis. The units of phase are in cycles, radians, or degrees, with 360 degrees in one cycle. While phase is important in mathematical analysis, we will minimize the use of phase in most of our discussion to concentrate on the more important aspects of sine waves.

In the time domain, describing a sine wave requires plotting a lot of voltage-versus-time data points to draw the complete sine-wave curve. However, in the frequency domain, describing a sine wave is much simpler.

In the frequency domain, we already know that the only waveforms we can talk about are sine waves, so all we have to identify are the amplitude, frequency, and phase. If there is only one sine wave we are describing, all we need are these three values and we have identified a complete description of the sine wave.

Since we are going to ignore phase for right now, we really only need two terms to completely describe a sine wave: its amplitude and its frequency. These two values are plotted with the frequency as one axis and the amplitude as the other axis, as shown in Figure 2-3. Of course, if we were including phase, we'd have a third axis. A sine wave, plotted in the frequency domain, is just one single data point. This is the key reason why we will go into the frequency domain. What might have been a thousand voltage-versus-time data points in the time domain is converted to a single amplitude-versus-frequency data point in the frequency domain.

When we have multiple frequency values, the collection of amplitudes is called the spectrum. As we will see, every time-domain waveform has a particular pattern to its spectrum. The only way to calculate the spectrum of a waveform in the time domain is with the Fourier Transform.