## TechTalk

What is Nonlinear Distortion?

Ideally, audio gear only changes the size of a wave, which is what amplifiers do, or it changes the media that the wave exists in, which is what a transducer does. When audio gear changes the shape of a wave, the wave is said to be distorted.   Since an audio signal has only two dimensions, which are amplitude (or power) and frequency (or time), it may only be distorted in either or both of these two dimensions.

Nonlinear Distortion is a form of signal processing error that creates signals at frequencies that are not necessarily present in the input. It contrasts with Linear Distortion, which is a form of error that creates no new frequencies beyond those that are present in the input signal.

Linear Distortion changes the relationship of the size and timing of the tones that comprise the input signal. Linear and Nonlinear distortion errors are directly based on frequencies present in the input signal. Therefore the errors that are added by linear or nonlinear distortion are correlated with tones present in the signal.  Noise is a form of error that is not as directly related to the input signal. In other words, noise is uncorrelated with the input signal.

Nonlinear distortion is quantified by measuring the magnitude of each of the new frequencies that are created by the action of the nonlinear distortion mechanisms in the Unit Under Test (UUT). The new frequencies may be "harmonics", so-called because they exist at frequencies that are integer multiples of the tones present in the input signal. For example, the harmonics of a single tone at 1 kHz are at 2 kHz, 3 kHz, 4kHz, and so on. Figure 1 shows the action of a second order (even) amplitude modulation nonlinearity on a pure tone. Figure 2 shows the action of a fifth order (odd) amplitude modulation nonlinearity on a pure tone.

 Figure 1. Frequency domain analysis of a 1 KHz tone distorted by second order (even) Amplitude Modulation Distortion. Please notice the absence of a fundamental tone at 1 KHz, the DC offset at 0 Hz,  and the second harmonic at 2 KHz.
 Figure 2. Frequency domain analysis of a 1 KHz tone distorted by fifth order (odd) amplitude modulation. Notice the fundamental tone at 1 KHz, and harmonics at 3 KHz and 5 KHz.

Even-numbered harmonics as well as a DC offset  are generated by "Even Order Distortion" and are caused by errors that are not symmetrical, as shown in Figure 3. Pure even order distortion also destroys the original signal. Odd numbered harmonics a are generated by "Odd Order Distortion" and are caused by symmetrical errors, as shown in Figure 4. A wave that is symmetrical is said to have "Half-wave symmetry" and has no odd-order distortion. Push-pull or balanced circuits are symmetrical and therefore reduce or practically eliminate Even Order Nonlinear Amplitude Modulation Distortion. A signal that is either all positive, or all negative, is composed of only even order distortion.

 Figure 3. Amplitude domain analysis of a 1 KHz tone with second order (even) Amplitude Modulation Distortion. Please notice the DC offset which displaces the signal vertically. This wave is not the least bit symmetrical. If this was a plot of the position of a loudspeaker cone, the cone would move out.

 Figure 4. Amplitude domain analysis of a 1 KHz tone with third order (odd) Amplitude Modulation Distortion. This wave is symmetrical.

 Figure 5. Amplitude domain analysis of a 1 KHz tone with square root amplitude modulation distortion. This wave is also symmetrical and composed of all odd orders of distortion in amounts that decrease with order. Lower order roots more strongly resemble a square wave.

If more than one tone is being distorted simultaneously, there will be sidebands in addition to harmonics. For example the sidebands of 19 KHz and 20 KHz will lie on 1 KHz multiples above and below 19 and 20 KHz. The 1 KHz multiples are based on the difference in frequency between the original two tones ogf 1 KHz.  The sidebands will be at 18 KHz and 21 KHz, etc. There may or may not be difference tones. The difference tone created by  distorting 19 KHz and 20 KHz would be at 1 KHz. There may be sum tones as well. The sum tone created by  distorting 19 KHz and 20 KHz would be at 39 KHz.

Sum and difference tones may  have sidebands, and sum tones, and difference tones of their own, depending on which distortion mechanisms are involved,  how many different distortion mechanisms exist in the UUT, and how they interact. Figure 6 shows the action of a fourth order amplitude modulation nonlinearity on a pure tone.

 Figure 6. Frequency domain analysis of  19 and 20 KHz tones distorted by fourth order Amplitude Modulation distortion. Notice the absence of fundamental tones at 19 and 20 KHz; DC offset at 0 Hz, difference tones at 1 and 2 KHz;  second harmonics and sidebands in the 37-41 KHz range;  and the fourth harmonics and sidebands in the 76-80 KHz range.

Sum, difference, and sidebands from various signals may appear at the same frequency. When this happens they  produce a single tone at that frequency. Its amplitude is the vector sum of the tones from the signal and various sources of distortion, which appear at that frequency.

Since an audio signal has only two dimensions, which are amplitude (or power) and frequency (or time), it may only be distorted in either or both of these two dimensions  Thus, there are only two types of nonlinear distortion - Amplitude Modulation (AM) Nonlinear Distortion and Frequency Modulation (FM) Nonlinear distortion. AM distortion creates difference tones, sidebands and harmonics.  FM distortion creates only sidebands.

Sidebands that are created by Frequency Modulation Nonlinear Distortion differ from those created by Amplitude Modulation Nonlinear Distortion in that the lower sideband(s) generated by Frequency Modulation Nonlinear Distortion have an opposite polarity, as compared to the carrier and upper sideband.

Samples of pure sound and distorted sound, simulated by MIDI, can be heard by clicking here. Samples of pure sound and musical recordings distorted by various amounts of AM distortion can be heard by clicking here. Similar samples that demonstrate the results of FM distortion can be heard by clicking here.

THD+N is a variation of the basic THD measurement that includes Noise as well as harmonics.

THD requires measuring harmonics that are at higher frequencies than the test signal. If the test signal is at a high frequency, then many harmonics may be filtered out, later on in the process of handling the signal. This leads to misleadingly low estimates of nonlinearity.

THD is not the only means for measuring Nonlinear Distortion. IM is an alternative means for doing this. Historically, IM has been more difficult to measure, but modern FFT's and digital signal generators make measuring IM very feasible. IM measures the sum, difference, and sidebands that nonlinear distortion creates from more than one tone.

IM has the advantage of not requiring the measurement of harmonics that are at frequencies that are many multiples of the frequency of the signal.. Therefore, IM is good for measuring nonlinearity at high frequencies. One disadvantage of IM is that it is harder to determine whether the errors it portrays are symmetrical or not. Another is that it is harder to determine whether the errors it portrays are in the frequency domain or amplitude domain.

Audibility of Nonlinear Distortion

To summarize, there are basically three kinds error signals that are the results of nonlinear distortion:
(1) Harmonics
(2) Sidebands
(3) Difference tones

It is dangerous to presume that just because nonlinear distortion products are generated, that they will be heard. There are two general reasons that distortion is not heard:
(1) The distortion is below the threshold of hearing.
(2) The distortion is masked by other sounds.

Generally, the major reason that harmonics aren't heard is that they are masked by the natural harmonics of sounds created by many musical instruments that appear at the same frequency. When this happens there are slight changes in the amplitude of the harmonics which cause timbre changes. For a timbre change to be perceived, the portion of the harmonic at that frequency that is generated by distortion may need to be relatively large. Concurrent masking by the fundamental and other harmonics appearing at nearby frequencies is also possible, of course.

Predictions of audibility based on changing the timbre of a sound by making audible alterations of its harmonic content often strike closer to home than those we get from concurrent masking. Concurrent masking is often not enough to predict the insensitivity of the ear to higher order harmonics with musical instruments that put out lots of high order harmonics (brass instruments for example).

Sidebands are audible one of two ways. If a sideband is in the same critical band as the carrier and more than 20 Hz away, but within less than 150 Hz of the carrier, then it is perceived as "roughness" . If the sidebands are within 20 Hz of the carrier, then the carrier will appear to have a  loudness that varies or pulsates.   If a sideband is outside the critical band, then it is perceived as a separate concurrent tone and is subject to concurrent masking.

People seem to be able to hear the difference between the sidebands due to AM and FM, at least under some circumstances. Note that for small amounts of modulation, AM and FM produce similar, but with FM the phase of the lower sideband is inverted.

Difference tones follow pretty much the same rules as harmonics and sidebands, but they have the extra added attraction of being both aharmonic and also they fall below the musical sound. Because they are often aharmonic, they may not get as strongly masked by the existing harmonic structure of the musical instruments.

Musical sounds with fundamentals and harmonics above the frequencies where our ear's sensitivity peaks (3-5 KHz) create difference tones that can  fall at lower  frequencies, and/or where there may be few if any sounds to mask them. If the fundamental frequency is high enough (above 4 KHz), the ear may be more sensitive to the difference tone(s) than the musical sound  that caused them.