# Linear does not mean no distortion

#### Article By : Bob Witte It is tempting to conclude that linear circuits are always distortionless, but that is not correct.

When we think about signals and systems, we often do so with an emphasis on creating and transmitting signals with specific characteristics. That is, we want the signal to have some desired amplitude, frequency content, or waveform shape. And we don’t want our electronics messing around inappropriately with the waveform. Another way to say this is that we want to keep the signal free from distortion.

In some cases, we want the signal to have a specified amplitude. For example, digital signals must be below or above a specific voltage level to be considered a valid low or high logic value. In other cases, we expect or tolerate changes in amplitude, but we do want to maintain a waveform’s shape. For example, signals that propagate through a communication channel are normally attenuated. Time delay is also usually accepted as the signal passes through various points in the system.

I looked around the web for a good definition of distortion in the context of electrical engineering and found this one [Ref 1].

Distortion: An undesired change in the signal’s waveform after it passes through a device or a system.

The same source also provides a more elaborate definition:

Distortion can occur because the device characteristic is not linear or because the circuit elements and devices respond to the input signal differently at various frequencies. When distortion occurs, the output will not be an exact duplicate (except for magnitude) of the input signal.

Looking closely at this second definition reveals that distortion can be caused by two mechanisms: nonlinearities and frequency response.

No Distortion
A system or network (Figure 1) is called distortionless if its output is an exact replica of its input, except for amplitude scaling and time delay [Ref 2]. Put mathematically,

y(t) = kx(tt0)

where

y(t) = output signal

x(t) = input signal

k = amplitude scale factor

t0 = time delay in the system

Note that k and t0 are constants and are not a function of frequency. In other words, k must be a constant for all frequencies of interest. This could imply that the system must have unlimited bandwidth but engineers usually accept a more practical bandwidth requirement which may be expressed as “sufficient bandwidth to support frequencies of interest.” Figure 1. Simple block diagram showing input and output signals of a system.

Figure 2 shows a plot of output (y) versus the input (x) of the system. A system with a constant k factor has a straight line or linear characteristic. A plot that isn’t a straight line is termed nonlinear and the system will introduce distortion. That is, k is not a constant. The nonlinear function will impact the shape of the waveform and produce distortion. In the time domain, the distortion appears as a change in waveform shape. In the frequency domain, the frequency content of waveform changes, typically introducing harmonic distortion or intermodulation distortion. Figure 2. Linear (blue) and nonlinear (black) relationships of input and output show differences in response.

Linear does not mean no distortion
From Fig. 2, it seems intuitive that a linear function won’t produce distortion, but a nonlinear function will. It is tempting to conclude that linear circuits are always distortionless systems, but that is not correct. Linear circuits that introduce variation in frequency response (either magnitude or phase) can distort the signal.

Let’s look at a simple linear circuit, the RC lowpass circuit as shown in Figure 3. (We will consider the R’s and C’s to be ideal.) Figure 3. A low pass RC circuit is a simple linear network.

The step response of this RC circuit is a rising exponential with a time constant equal to RC (Figure 4). In the frequency domain, the circuit is a low pass filter with a –3 dB frequency of 1/(2πRC).

Suppose the input to the RC circuit is a square wave (Figure 5). The output waveform will exhibit the same exponential response on each edge of the square wave. The output waveform is a distorted version of the input, so we’d probably conclude that the RC circuit introduces distortion into the waveform. What’s happening is that the high frequency content of the square wave is being removed by the circuit, causing the sharp edge of the waveform to be slowed down.

The amount of waveform distortion is, however, going to depend on the RC time constant of the circuit relative to the period of the square wave. If the RC time constant is fast enough, corresponding to a high cutoff frequency, we might consider the output waveform to be close enough to the input to call the system distortionless. This judgment depends on the application, determined by asking how good does the waveform need to be for this particular use case?

Sometimes this type of distortion is called linear distortion, indicating that the circuit is operating in a linear fashion while still distorting the signal. Linearity alone is not sufficient to avoid distortion. The system must also pass the frequency content of the waveform such that the waveform shape is maintained. (The phase response is also important. We’ll look at that in my next post.)

It turns out that a square wave is a common test signal for wideband systems because it is rich in harmonic content. Being able to pass a square wave of suitable frequency is a quick way to verify system performance, including distortion and bandwidth. It is equivalent to inputting multiple sinusoids simultaneously into a system and seeing what shows up at the output.

Distortion on purpose
Distortion is not always a bad thing. There are plenty of applications where distortion is intentionally introduced to accomplish some purpose. A great example of this is the wide range of distortion boxes and other effects used with electric guitars [Ref 3]. Radio frequency design also takes advantage of nonlinear devices to generate harmonics and to mix frequencies together. In high-speed digital links, predistortion is often added to compensate for the characteristics in a transmission line. One person’s distortion is another person’s desired signal.

Bob Witte is president of Signal Blue LLC, a technology consulting company.

References 