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2.2 Sine Waves in the Frequency Domain

We hear the term frequency domain quite a bit, especially when it involves radio frequency (rf) or communications systems. We will also encounter the frequency domain in high-speed digital applications. There are few engineers who have not heard of and used the term multiple times. Yet, what do we really mean by the frequency domain? What is the frequency domain and what makes it special and useful?

The most important rule in the frequency domain is that the only kind of waveforms that exist are sine waves. Sine waves are the language of the frequency domain.

There are other domains that use other special functions. For example, the JPEG picture-compression algorithm takes advantage of special waveforms that are called wavelets. The wavelet transform takes the space domain, with a lot of x-y amplitude information content, and translates it into a different mathematical description that is able to use less than 10% of the memory to describe the same information. It is an approximation, but a very good one.

It's common for engineers to think that we use sine waves in the frequency domain because we can build any time-domain waveform from combinations of sine waves. This is a very important property of sine waves. However, there are many other waveforms with this property. It is not a property that is unique to sine waves.

In fact, there are four properties that make sine waves very useful for describing any other waveform. These properties are as follows:

  1. Any waveform in the time domain can be completely and uniquely described by combinations of sine wave.
  2. Any two sine waves with different frequencies are orthogonal to each other. If you multiply them together and integrate over all time, they integrate to zero. This means you can separate each component from every other.
  3. They are well defined mathematically.
  4. They have a value everywhere with no infinities and they have derivatives that have no infinities anywhere. This means they can be used to describe real world waveforms, since there are no infinities in the real world.

All of these properties are vitally important, but are not unique to sine waves. There is a whole class of functions called orthonormal functions, or sometimes called eigenfunctions or basis functions, which could be used to describe any time-domain waveform. Other orthonormal functions are Hermite Polynomials, Legendre Polynomials, Laguerre Polynomials, and Bessel Functions.

Why did we choose sine waves as our functions in the frequency domain? What's so special about sine waves? The real answer is that by using sine waves, some problems related to the electrical effects of interconnects will be easier to understand and solve using sine waves. If we switch to the frequency domain and use sine-wave descriptions, we can sometimes get to an answer faster than staying in the time domain.

Sine waves can sometimes provide a faster path to an acceptable answer because of the types of electrical problems we often encounter in signal integrity. If we look at the circuits that describe interconnects, we find that they will often include combinations of resistors (R), inductors (L), and capacitors (C). These elements in a circuit can be described by a second-order linear differential equation. The solution to this type of differential equation is a sine wave. In these circuits, the naturally occurring waveforms will be combinations of the waveforms that are solutions to the differential equation.

We find that in the real world, if we build circuits that contain Rs, Ls, and Cs and send any arbitrary waveform in, more often than not, we get waveforms out that look like sine waves and can more simply be described by a combination of a few sine waves. An example of this is shown in Figure 2-2.

Figure 2-2

Figure 2-2 Time-domain behavior of a fast edge interacting with an ideal RLC circuit. Sine waves are naturally occurring when digital signals interact with interconnects, which can often be described as combinations of ideal RLC circuit elements.

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