How is the coefficient of non-linear distortion determined. Nonlinear distortion. Maximum continuous power

of the input signal, to the rms sum of the spectral components of the input signal, sometimes a non-standardized synonym is used - clear factor(borrowed from German). SOI is a dimensionless quantity, usually expressed as a percentage. In addition to SOI, the level of non-linear distortion can be expressed using harmonic distortion factor.

Harmonic Distortion- a value expressing the degree of non-linear distortion of the device (amplifier, etc.), equal to the ratio of the RMS voltage of the sum of the higher harmonics of the signal, except for the first, to the voltage of the first harmonic when a sinusoidal signal is applied to the input of the device.

The harmonic coefficient, like the THD, is expressed as a percentage. Harmonic coefficient ( K G) is related to SOI ( K N) ratio:

measurements

• In the low-frequency (LF) range (up to 100-200 kHz), non-linear distortion meters (harmonic coefficient meters) are used to measure SOI.
• At higher frequencies (MF, HF), indirect measurements are used using spectrum analyzers or selective voltmeters.

Typical THD values

• 0% - the waveform is a perfect sine wave.
• 3% - the waveform is not sinusoidal, but the distortion is not noticeable to the eye.
• 5% - the deviation of the waveform from the sinusoidal is noticeable by eye on the oscillogram.
• 10% is the standard level of distortion at which the real power (RMS) of the UMZCH is considered.
• 21% - for example, a trapezoidal or stepped signal.
• 43% - for example, a square wave signal.

see also

Literature

• Handbook of electronic devices: In 2 tons; Ed. D. P. Linde - M .: Energy,
• Gorokhov P.K. Explanatory dictionary of radio electronics. Basic terms- M: Rus. lang.,

Links

• MAIN ELECTRICAL CHARACTERISTICS OF THE SOUND TRANSMISSION CHANNEL

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THD- THD A parameter that allows taking into account the influence of harmonics and combinational components on the signal quality. Numerically defined as the ratio of the power of non-linear distortion to the power of an undistorted signal, usually expressed as a percentage. [L.M. Nevdyaev ...

THD- 3.9 total distortion ratio of the rms value of the spectral components of the output signal of an acoustic calibrator that are not present in the input signal to the rms value ... ...

THD- netiesinių iškreipių faktorius statusas T sritis fizika atitikmenys: engl. non linear distortion factor vok. Klirrfaktor, m rus. coefficient of non-linear distortion, m pranc. taux de distortion harmonique, m … Fizikos terminų žodynas

UPS Input Current THD Indicates deviations of the UPS input current waveform from a sinusoidal waveform. The larger the value of this parameter, the worse it is for equipment connected to the same mains and the mains itself, in this case it worsens ... ... Technical Translator's Handbook

UPS output voltage THD Characterizes the deviation of the output voltage form from a sinusoidal, usually given for linear (motors, some types of lighting devices) and non-linear loads. The higher this value, the worse quality… … Technical Translator's Handbook

amplifier total harmonic distortion- - [L.G. Sumenko. English Russian Dictionary of Information Technologies. M.: GP TsNIIS, 2003.] Topics Information Technology in general EN amplifier distortion factor … Technical Translator's Handbook

Speaker THD- 89. The coefficient of non-linear distortion of the loudspeaker The coefficient of non-linear distortion Ndp. Harmonic distortion The square root, expressed as a percentage, of the ratio of the sum of squares of the effective values ​​of the spectral components emitted by ... ... Dictionary-reference book of terms of normative and technical documentation

Nonlinear distortion factor of the laryngofone- 94. The coefficient of non-linear distortion of the laryngophone The value of the square root, expressed as a percentage, of the ratio of the sum of the squares of the effective values ​​of the harmonics of the electromotive force developed by the laryngophone during the harmonic movement of air, to ... ... Dictionary-reference book of terms of normative and technical documentation

admissible coefficient of non-linear distortions- - [L.G. Sumenko. English Russian Dictionary of Information Technologies. M.: GP TsNIIS, 2003.] Topics information technology in general EN harmonic tolerance ... Technical Translator's Handbook

- (harmonic coefficient meter) a device for measuring the coefficient of non-linear distortion (harmonic coefficient) of signals in radio engineering devices. Contents ... Wikipedia

AT The entire history of sound reproduction has evolved from attempts to bring the illusion closer to the original. And although the path has been traversed, it is still very, very far from fully approaching live sound. Differences in numerous parameters can be measured, but quite a few remain out of the sight of hardware developers. One of the main characteristics that a consumer with any preparation always pays attention to is non-linear distortion factor (THD) .

And what is the value of this coefficient fairly objectively indicates the quality of the device? The impatient can immediately find an attempt at an answer to this question at the end. For the rest, let's continue.
This coefficient, which is also called the coefficient of total harmonic distortion, is the percentage ratio of the effective amplitude of the harmonic components at the output of the device (amplifier, tape recorder, etc.) to the effective amplitude of the fundamental frequency signal when a sinusoidal signal of this frequency is applied to the input of the device. Thus, it allows one to quantify the nonlinearity of the transfer characteristic, which manifests itself in the appearance in the output signal of spectral components (harmonics) that are absent in the input signal. In other words, there is a qualitative change in the spectrum of the musical signal.

In addition to the objective harmonic distortions present in the audible sound signal, there is the problem of distortions that are absent in real sound, but are felt due to subjective harmonics that occur in the cochlea at high sound pressure values. The human hearing aid is a non-linear system. The non-linearity of hearing is manifested in the fact that when a sinusoidal sound with a frequency f is exposed to the tympanic membrane in hearing aid harmonics of this sound with frequencies 2f, 3f, etc. are born. Since these harmonics do not exist in the primary affecting tone, they are called subjective harmonics.

Naturally, this further complicates the idea of ​​the maximum permissible level of harmonics in the audio path. With an increase in the intensity of the primary tone, the magnitude of subjective harmonics increases sharply and may even exceed the intensity of the fundamental tone. This circumstance gives grounds for the assumption that sounds with a frequency of less than 100 Hz are felt not by themselves, but because of the subjective harmonics they create, falling into the frequency range above 100 Hz, i.e. due to non-linear hearing. The physical causes of the resulting hardware distortions in different devices are of a different nature, and the contribution of each to the overall distortion of the entire path is not the same.

Distortions of modern CD-players have very low values ​​and are almost imperceptible against the background of distortions of other units. For acoustic systems, the most significant are low-frequency distortions caused by the bass head, and the standard specifies requirements only for the second and third harmonics in the frequency range up to 250 Hz. And for a very good sounding speaker system they can be within 1% or even a little more. In analog tape recorders, the main problem associated with physical foundations recording on a magnetic tape is the third harmonic, the values ​​​​of which are usually given in the instructions for information. But the maximum value at which, for example, noise level measurements are always made is 3% for a frequency of 333 Hz. The distortions of the electronic part of tape recorders are much lower.
Both in the case of acoustics and for analog tape recorders, due to the fact that distortions are mainly low-frequency, their subjective visibility drops significantly due to the masking effect (which consists in the fact that the higher frequency is better heard from two simultaneously sounding signals).

So the main source of distortion in your path will be the power amplifier, in which, in turn, the main one is the non-linearity of the transfer characteristics of active elements: transistors and vacuum tubes, and in transformer amplifiers, the non-linear distortion of the transformer is also added, associated with the non-linearity of the magnetization curve. Obviously, on the one hand, the distortion depends on the shape of the nonlinearity of the transfer characteristic, but also on the nature of the input signal.

For example, the transfer response of an amplifier with soft clipping at large amplitudes will not cause any distortion for sinusoidal signals below the clipping level, and as the signal increases above this level, distortions appear and will increase. This nature of the limitation is mainly inherent in tube amplifiers, which to some extent may serve as one of the reasons for the preference of such amplifiers by listeners. And this feature was used by NAD in a series of their sensational "soft-limit" amplifiers produced since the early 80s: the ability to turn on the mode with simulated tube clipping created a large army of fans of NAD transistor amplifiers.
In contrast, the center-cut (notch) characteristic of an amplifier, which is common with transistor models, will distort musical and small sine wave signals, and will decrease as the signal level increases. Thus, the distortion depends not only on the shape of the transfer characteristic, but also on the statistical distribution of the input signal levels, which for musical programs is close to the noise signal. Therefore, in addition to measuring SOI using a sinusoidal signal, it is possible to measure the nonlinear distortions of amplifying devices using the sum of three sinusoidal or noise signals, which, in the light of the foregoing, give a more objective picture of distortion.

Nonlinear distortion factor(SOI or K N) - value for quantitative evaluation of non-linear distortions.

Definition [ | ]

The coefficient of non-linear distortion is equal to the ratio of the rms sum of the spectral components of the output signal, which are absent in the spectrum of the input signal, to the rms sum of all spectral components of the input signal

SOI is a dimensionless quantity and is usually expressed as a percentage. In addition to SOI, the level of nonlinear distortion is often expressed in terms of harmonic distortion factor(CHI or K G) - a value that expresses the degree of non-linear distortion of the device (amplifier, etc.) and is equal to the ratio of the root-mean-square voltage of the sum of the higher harmonics of the signal, except for the first, to the voltage of the first harmonic when a sinusoidal signal is applied to the input of the device.

KGI, as well as KNI, is expressed as a percentage and is associated with it by the ratio

Obviously, for small values ​​of THD and SOI coincide in the first approximation. Interestingly, in Western literature, CHD is usually used, while SOI is traditionally preferred in Russian literature.

It is also important to note that SOI and KGI are only quantitative measures of distortion but not of good quality. For example, the THD (THD) value of 3% does not say anything about the nature of the distortion, i.e. about how harmonics are distributed in the signal spectrum, and what, for example, is the contribution of low-frequency or high-frequency components. So, in the spectra of tube UMZCH, lower harmonics usually predominate, which is often perceived by ear as a “warm tube sound”, and in transistor UMZCH distortion is more evenly distributed over the spectrum, and it is flatter, which is often perceived as a “typical transistor sound” (although this dispute largely depends on personal feelings and habits of a person).

Examples of calculation of CHI[ | ]

For many standard signals, THD can be calculated analytically. So, for a symmetrical rectangular signal (meander)

Ideal sawtooth signal has OGI

and symmetrical triangular

An asymmetric rectangular pulse signal with a ratio of pulse duration to period equal to μ has a CHI

which reaches a minimum (≈0.483) at μ =0.5, i.e. when the signal becomes a symmetrical meander. By the way, filtering can achieve a significant reduction in the THD of these signals, and thus obtain signals close in shape to sinusoidal. For example, a symmetrical rectangular signal (meander) with an initial THD of 48.3%, after passing through a second-order Butterworth filter (with a cutoff frequency equal to the frequency of the fundamental harmonic) has a THD of 5.3%, and if the fourth-order filter - then THD = 0.6% . It should be noted that the more complex the signal at the filter input and the more complex the filter itself (more precisely, its transfer function), the more cumbersome and time-consuming the THD calculations will be. So, a standard sawtooth signal that has passed through a first-order Butterworth filter has a THD no longer of 80.3% but of 37.0%, which is exactly given by the following expression

And the THD of the same signal that has passed through the same filter, but of the second order, will already be given by a rather cumbersome formula

If we consider the aforementioned asymmetric rectangular pulse signal that passed through the Butterworth filter p th order, then

where 0<μ <1 и

for calculation details, see Yaroslav Blagushin and Eric Moreau.

measurements [ | ]

• In the low-frequency (LF) range, non-linear distortion meters (harmonic coefficient meters) are used to measure THD.
• At higher frequencies (MF, HF), indirect measurements are used using spectrum analyzers or selective voltmeters.

The main parameter of an electronic amplifier is the gain K. The power gain (voltage, current) is determined by the ratio of the power (voltage, current) of the output signal to the power (voltage, current) of the input signal and characterizes the amplifying properties of the circuit. The output and input signals must be expressed in the same quantitative units, so the gain is a dimensionless quantity.

In the absence of reactive elements in the circuit, as well as under certain modes of its operation, when their influence is excluded, the gain is a real value that does not depend on frequency. In this case, the output signal repeats the shape of the input signal and differs from it by a factor of K only in amplitude. In the following presentation of the material, we will talk about the gain module, unless there are special reservations.

Depending on the requirements for the output parameters of the AC signal amplifier, there are gain factors:

a) by voltage, defined as the ratio of the amplitude of the variable component of the output voltage to the amplitude of the variable component of the input, i.e.

b) by current, which is determined by the ratio of the amplitude of the variable component of the output current to the amplitude of the variable component of the input:

c) by power

Since , then the power gain can be determined as follows:

In the presence of reactive elements in the circuit (capacitors, inductances), the gain should be considered as a complex value

where m and n are real and imaginary components depending on the frequency of the input signal:

We assume that the gain K does not depend on the amplitude of the input signal. In this case, when a sinusoidal signal is applied to the input of the amplifier, the output signal will also have a sinusoidal shape, but differ from the input in amplitude by a factor of K and in phase by an angle .

A periodic signal of a complex shape, according to the Fourier theorem, can be represented by the sum of a finite or infinitely large number of harmonic components having different amplitudes, frequencies and phases. Since K is a complex value, the amplitudes and phases of the harmonic components of the input signal change differently when passing through the amplifier, and the output signal will differ in shape from the input.

The distortion of the signal when passing through the amplifier, due to the dependence of the parameters of the amplifier on the frequency and does not depend on the amplitude of the input signal, is called linear distortion. In turn, linear distortions can be divided into frequency ones (characterizing the change in the gain modulus K in the frequency band due to the influence of reactive elements in the circuit); phase (characterizing the dependence of the phase shift between the output and input signals on the frequency due to the influence of reactive elements).

Frequency distortion of the signal can be estimated using the amplitude-frequency characteristic, which expresses the dependence of the magnitude of the voltage gain on frequency. The amplitude-frequency characteristic of the amplifier in a general form is shown in fig. 1.2. The operating frequency range of the amplifier, within which the gain can be considered constant with a certain degree of accuracy, lies between the lower and higher cutoff frequencies and is called the bandwidth. The cutoff frequencies determine the decrease in the gain by a given amount from its maximum value at the center frequency.

Entering the frequency distortion factor at a given frequency ,

where is the voltage gain at a given frequency, it is possible, using the amplitude-frequency characteristic, to determine the frequency distortion in any range of the operating frequencies of the amplifier.

Since we have the greatest frequency distortions at the boundaries of the operating range, then when calculating the amplifier, as a rule, we set the frequency distortion coefficients at the lowest and highest cutoff frequencies, i.e.

where are the voltage gains at the highest and lowest cut-off frequencies, respectively.

Usually accepted, i.e., at the cutoff frequencies, the voltage gain decreases to the level of 0.707 of the value of the gain at the middle frequency. Under such conditions, the bandwidth of audio amplifiers designed to reproduce speech and music lies in the range of 30-20,000 Hz. For amplifiers used in telephony, a narrower bandwidth of 300-3400 Hz is acceptable. To amplify pulsed signals, it is necessary to use so-called broadband amplifiers, the bandwidth of which is in the frequency range from tens or units of hertz to tens or even hundreds of megahertz.

To assess the quality of the amplifier, the parameter is often used

For broadband amplifiers, therefore

The opposite of broadband amplifiers are selective amplifiers, whose purpose is to amplify signals in a narrow frequency band (Fig. 1.3).

Amplifiers designed to amplify signals with an arbitrarily low frequency are called DC amplifiers. It is clear from the definition that the lower cutoff frequency of the passband of such an amplifier is zero. The amplitude-frequency characteristic of the DC amplifier is given in fig. 1.4.

The phase response shows how the phase angle between the output and input signals changes with frequency and defines phase distortion.

There are no phase distortions with a linear phase response (dashed line in Fig. 1.5), since in this case each harmonic component of the input signal, when passing through the amplifier, is shifted in time by the same interval. The phase angle between the input and output signals is proportional to the frequency

where is the coefficient of proportionality, which determines the angle of inclination of the characteristic to the x-axis.

The phase-frequency characteristic of a real amplifier is shown in fig. 1.5 with a solid line. From fig. 1.5 it can be seen that phase distortions are minimal within the passband of the amplifier, but increase sharply in the region of cutoff frequencies.

If the gain depends on the amplitude of the input signal, then there are nonlinear distortions of the amplified signal due to the presence of elements with nonlinear current-voltage characteristics in the amplifier.

By setting the law of change, it is possible to design nonlinear amplifiers with certain properties. Let the gain be determined by the dependence , where is the coefficient of proportionality.

Then, when a sinusoidal input signal is applied to the input of the amplifier, the output signal of the amplifier

where are the amplitude and frequency of the input signal.

The first harmonic component in expression (1.6) is a useful signal, the rest are the result of non-linear distortions.

Harmonic distortion can be estimated using the so-called harmonic distortion

where are the amplitude values, respectively, of the power, voltage and current of the harmonic components.

The index determines the harmonic number. Usually, only the second and third harmonics are taken into account, since the amplitude values ​​of the powers of higher harmonics are relatively small.

Linear and non-linear distortions characterize the accuracy of reproduction of the input signal shape by the amplifier.

The amplitude characteristic of quadripoles, consisting only of linear elements, at any value is theoretically an inclined straight line. In practice, the maximum value is limited by the electric strength of the elements of the quadripole. The amplitude characteristic of an amplifier made on electronic devices (Fig. 1.6) is, in principle, non-linear, but it may contain OA sections where the curve is approximately linear with a high degree of accuracy. The operating range of the input signal should not go beyond the linear section (OA) of the amplitude characteristic of the amplifier, otherwise the non-linear distortion will exceed the permissible level.

Nonlinear Distortion Factor (THD)​

Irina Aldoshina​

All electro-acoustic transducers (loudspeakers, microphones, telephones, etc.), as well as transmission channels, introduce their own distortions into the transmitted audio signal, that is, the perceived audio signal is always not identical to the original. The ideology of creating sound equipment, which in the 60s was called High-Fidelity, “high fidelity” to live sound, to a large extent did not achieve its goal. In those years, the levels of distortion of the audio signal in the equipment were still very high, and it seemed that it was enough to reduce them - and the sound reproduced through the equipment would be practically indistinguishable from the original.

However, despite the advances in design and technology that have led to a significant reduction in the levels of all types of distortion in audio equipment, it is still not difficult to distinguish natural sound from reproduced. That is why, at present, research institutes, universities and manufacturing firms in various countries are conducting a large amount of work on the study of auditory perception and the subjective assessment of various types of distortions. Based on the results of these studies, many scientific articles and reports are published. Almost all AES congresses present papers on this topic. Some modern results obtained over the past two or three years on the problems of subjective perception and evaluation of non-linear distortions of an audio signal in audio equipment will be presented in this article.

When recording, transmitting and reproducing musical and speech signals through audio equipment, distortions of the temporal structure of the signal occur, which can be divided into linear and non-linear.

Linear distortion change the amplitude and phase relationships between the available spectral components of the input signal and thereby distort its temporal structure. Such distortions are subjectively perceived as distortions of the signal timbre, and therefore a lot of attention was paid to the problems of their reduction and subjective assessments of their level by specialists throughout the entire period of development of sound engineering.

The requirement for the absence of linear signal distortion in audio equipment can be written in the form:

Y(t) = K x(t - T), where x(t) is the input signal, y(t) is the output signal.

This condition allows only a change in the signal in the scale with a coefficient K and its shift in time by the amount T. It defines a linear relationship between the input and output signals and leads to the requirement that the transfer function H(ω), which is understood as the frequency-dependent ratio of complex amplitudes of the signal at the output and at the input of the system under harmonic influences, was constant in absolute value and had a linear dependence of the argument (that is, phase) on frequency | H(ω) | \u003d K, φ (ω) \u003d -T ω. Since the function 20·lg | H(ω) | is called the amplitude-frequency characteristic of the system (AFC), and φ(ω) is the phase-frequency characteristic (PFC), then ensuring a constant level of frequency response in the reproducible frequency range (reducing its unevenness) in microphones, acoustic systems, etc. is the main requirement for improving their quality. Methods for their measurement are introduced in all international standards, for example, IEC268-5. An example of the frequency response of a modern Marantz control unit with a 2 dB unevenness is shown in Figure 1.

Frequency response of the Marantz reference monitor

It should be noted that this reduction in frequency response is a huge advance in audio design (e.g., the reference monitors presented at the Brussels Exhibition in 1956 had 15 dB flatness), which was made possible by the use of new technologies, materials and design methods.

The effect of uneven frequency response (and PFC) on the subjectively perceived distortion of the sound timbre has been studied in sufficient detail. We will try to review the main results obtained in the future.

Nonlinear distortion are characterized by the appearance in the signal spectrum of new components that are absent in the original signal, the number and amplitude of which depend on the change in the input level. The appearance of additional components in the spectrum is due to the non-linear dependence of the output signal on the input, that is, the non-linearity of the transfer function. Examples of such dependence are shown in Figure 2.

Various types of non-linear transfer functions in hardware

Nonlinearity can be caused by design and technological features of electroacoustic transducers.

For example, in electrodynamic loudspeakers (Figure 3), the main causes include:

Construction of an electrodynamic loudspeaker

Nonlinear elastic characteristics of the suspension and the centering washer (an example of the dependence of the flexibility of the suspensions in the loudspeaker on the value of the displacement of the voice coil is shown in Figure 4);

Suspension Flexibility vs. Voice Coil Displacement

Non-linear dependence of the voice coil displacement on the magnitude of the applied voltage due to the interaction of the coil with the magnetic field and due to thermal processes in the loudspeakers;
- non-linear oscillations of the diaphragm with a large value of the acting force;
- vibrations of the body walls;
- Doppler effect in the interaction of different emitters in the acoustic system.
Nonlinear distortions occur in almost all elements of the audio path: microphones, amplifiers, crossovers, effects processors, etc.
The relationship shown in Figure 2 between input and output signals (for example, between applied voltage and sound pressure for a loudspeaker) can be approximated as a polynomial:
y(t) = h1 x(t) + h2 x2(t) + h3 x3(t) + h4 x4(t) + … (1).
If a harmonic signal is applied to such a nonlinear system, i.e. x(t) = A sin ωt, then the output signal will contain components with frequencies ω, 2ω, 3ω, ..., nω, etc. For example, if we limit ourselves to only a quadratic term, then second harmonics will appear, since
y(t) = h1 A sin ωt + h2 (A sin ωt)² = h1 A sin ωt + 0.5 h2 A² sin 2ωt + const.
In real converters, when a harmonic signal is applied, harmonics of the second, third and higher orders, as well as subharmonics (1/n) ω (Figure 5), may appear.

To measure this type of distortion, the most widely used methods are measuring the level of additional harmonics in the output signal (usually only the second and third).
In accordance with international and domestic standards, the frequency response of the second and third harmonics is recorded in anechoic chambers and the harmonic distortion coefficient of the n-order is measured:
KГn = pfn / pср 100%
where pfn is the RMS value of the sound pressure corresponding to the n-harmonic component. It calculates the total harmonic distortion:
Kg \u003d (KG2² + KG3² +KG4² +KG5² + ...) 1/2
For example, in accordance with the requirements of IEC 581-7, for Hi-Fi loudspeakers, the total harmonic distortion factor should not exceed 2% in the frequency range of 250 ... 1000 Hz and 1% in the range above 2000 Hz. An example of harmonic distortion for a 300 mm (12") woofer versus frequency for various input voltages ranging from 10 to 32 V is shown in Figure 6.

THD versus frequency for different input voltages

It should be noted that the auditory system is extremely sensitive to the presence of non-linear distortions in acoustic transducers. The “noticeability” of harmonic components depends on their order, in particular, hearing is most sensitive to odd components. With repeated listening, the perception of non-linear distortions is aggravated, especially when listening to individual musical instruments. The frequency region of maximum hearing sensitivity to these types of distortions is within 1...2 kHz, where the sensitivity threshold is 1...2%.
However, such a method for estimating the nonlinearity does not allow taking into account all types of nonlinear products that arise in the process of converting a real audio signal. As a result, there may be a situation where a speaker system with 10% THD can be subjectively judged to be superior in sound quality than a system with 1% THD due to the influence of higher harmonics.
Therefore, the search for other ways to assess non-linear distortions and their correlation with subjective assessments continues all the time. This is especially important at the present time, when the levels of non-linear distortions have significantly decreased, and in order to further reduce them, it is necessary to know the real thresholds of audibility, since the reduction of non-linear distortions in equipment requires significant economic costs.
Along with measurements of harmonic components, in the practice of designing and evaluating electroacoustic equipment, methods for measuring intermodulation distortions are used. The measurement technique is presented by GOST 16122-88 and IEC 268-5 and is based on bringing two sinusoidal signals to the emitter with frequencies f1 and f2, where f1< 1/8·f2 (при соотношении амплитуд 4:1) и измерении амплитуд звукового давления комбинационных тонов: f2 ± (n - 1)·f1, где n = 2, 3.
The total intermodulation distortion factor is defined in this case as:
Kim = (ΣnKimn²)1/2
where kim=/pcp.
The cause of intermodulation distortion is the non-linear relationship between the output and input signals, i.e., the non-linear transfer characteristic. If two harmonic signals are applied to the input of such a system, then the output signal will contain harmonics of higher orders and sum-difference tones of different orders.
The form of the output signal, taking into account nonlinearities of higher orders, is shown in Figure 5.

Harmonic distortion products in loudspeakers

The characteristics of the intermodulation distortion factor versus frequency for a low-frequency loudspeaker with voice coils of various lengths are shown in Figure 7 (a - for a longer coil, b - for a shorter one).

Intermodulation distortion factor (IMD) versus frequency for a loudspeaker with a long (a) and short (b) coil

As mentioned above, in accordance with international standards, only the coefficients of intermodulation distortion of the second and third orders are measured in the equipment. Intermodulation distortion measurements can be more informative than harmonic measurements, since they are a more sensitive criterion for non-linearity. However, as the experiments performed in the works of R. Gedds (report at the 115th AES congress in New York) showed, it was not possible to establish a clear correlation between the subjective assessments of the quality of acoustic transducers and the level of intermodulation distortion - the scatter in the results obtained is too large (as can be seen from figure 8).

The relationship of subjective assessments with the value of the coefficient of intermodulation distortion (IMD)

As a new criterion for assessing nonlinear distortions in electro-acoustic equipment, a multi-tone method was proposed, the history and methods of application of which were studied in detail in the works of A. G. Voishvillo et al. (there are articles in JAES and reports at AES congresses). In this case, the input signal is a set of harmonics from the 2nd to the 20th with an arbitrary distribution of amplitudes and a logarithmic frequency distribution in the range from 1 to 10 kHz. The harmonic phase distribution is optimized to minimize the crest factor of a multi-tone signal. The general view of the input signal and its temporal structure are shown in Figures 9a and 9b.

Spectral (a) and temporal (b) view of a multitone signal

Harmonic and intermodulation distortions of all orders are distinguished in the output signal. An example of such distortion for a loudspeaker is shown in Figure 10.

Common products of non-linear distortion when applying a multi-tone signal

A multitone signal in its structure is much closer to real music and speech signals, it allows you to select much more different products of nonlinear distortions (primarily intermodulation) and better correlates with subjective assessments of the sound quality of acoustic systems. With an increase in the number of harmonic components, this method allows obtaining more and more detailed information, but at the same time, computational costs increase. The application of this method requires further research, in particular, the development of criteria and acceptable standards for the selected products of nonlinear distortions from the standpoint of their subjective assessments.
Other methods, such as Voltaire series, are also used to evaluate nonlinear distortion in acoustic transducers.
However, all of them do not provide a clear connection between the assessment of the sound quality of transducers (microphones, loudspeakers, acoustic systems, etc.) and the level of non-linear distortions in them, measured by any of the known objective methods. Therefore, the new psychoacoustic criterion proposed in the report of R. Gedds at the last AES congress is of considerable interest. He proceeded from the considerations that any parameter can be evaluated in objective units, or it can be subjective criteria, for example, temperature can be measured in degrees, or in sensations: cold, warm, hot. The loudness of the sound can be estimated by the level of sound pressure in dB, or you can - in subjective units: background, sleep. The search for similar criteria for non-linear distortions was the goal of his work.
As is known from psychoacoustics, the hearing aid is a fundamentally non-linear system, and its non-linearity manifests itself both at high and low signal levels. The reasons for the nonlinearity are hydrodynamic processes in the cochlea, as well as nonlinear signal compression due to a special mechanism of elongation of the outer hair cells. This leads to the appearance of subjective harmonics and combination tones when listening to harmonic or total harmonic signals, the level of which can reach 15 ... 20% of the input signal level. Therefore, the analysis of the perception of the products of nonlinear distortions created in electroacoustic transducers and transmission channels in such a complex nonlinear system as a hearing aid is a serious problem.
Another fundamentally important property of the auditory system is the masking effect, which consists in changing the hearing thresholds to one signal in the presence of another (masker). This property of the auditory system is widely used in modern systems for compressing audio information when it is transmitted over various channels (MPEG standards). The advances made in reducing the amount of transmitted information through compression using the properties of auditory masking suggest that these effects are also of great importance for the perception and evaluation of non-linear distortions.
The established laws of auditory masking allow us to state that:
- masking of high-frequency components (located above the frequency of the signal-masker) is much stronger than in the direction of low frequencies;
- masking is more pronounced for the nearest frequencies (local effect, Figure 11);
- with an increase in the level of the masker signal, the zone of its influence expands, it becomes more and more asymmetric, and it shifts towards high frequencies.

From this we can assume that the following rules are observed when analyzing nonlinear distortions in the auditory system:
- non-linear distortion products above the fundamental frequency are less important for perception (they are better masked) than low-frequency components;
- the closer to the main tone the products of non-linear distortions are located, the more likely they will become invisible and will not have a subjective value;
- additional non-linear components arising from non-linearity can be much more important for perception at low signal levels than at high ones. This is shown in Figure 11.

Masking Effects

Indeed, with an increase in the level of the main signal, the zone of its masking expands, and more and more distortion products (harmonics, total and difference distortions, etc.) fall into it. At low levels, this zone is limited, so high-order distortion products will be more audible.
When measuring non-linear products on a pure tone, the transducers mainly produce harmonics with a frequency higher than the main signal n f. However, low harmonics with frequencies (1/n)·f can also occur in loudspeakers. When measuring intermodulation distortion (both using two signals and using multi-tone signals), total-difference distortion products arise - both above and below the main signals m f1 ± n f2.
Taking into account the listed properties of auditory masking, we can draw the following conclusions: the products of non-linear distortions of higher orders can be more audible than the products of lower orders. For example, loudspeaker design practice shows that harmonics higher than the fifth are perceived to be much more unpleasant than the second and third, even if their levels are much lower than those of the first two harmonics. Usually their appearance is perceived as rattling and leads to the rejection of loudspeakers in production. The appearance of subharmonics at half and below frequencies is also immediately noticed by the auditory system as an overtone, even at very low levels.
If the order of non-linearity is low, then with an increase in the input signal level, additional harmonics can be masked in the auditory system and not perceived as distortions, which is confirmed by the practice of designing electroacoustic transducers. Speaker systems with a level of non-linear distortion of 2% can be highly rated by listeners. At the same time, good amplifiers should have a distortion level of 0.01% or less, which, apparently, is due to the fact that loudspeakers create low-order distortion products, while amplifiers produce much higher ones.
The harmonic distortion products that occur at low signal levels can be much more audible than at high levels. This seemingly paradoxical statement can also be of practical importance, since nonlinear distortions in electroacoustic transducers and paths can also occur at low signal levels.
Based on the above considerations, R. Gedds proposed a new psychoacoustic criterion for assessing non-linear distortions, which had to meet the following requirements: be more sensitive to higher-order distortions and be more important for low signal levels.
The problem was to show that this criterion is more consistent with the subjective perception of non-linear distortion than the currently accepted methods of assessment: the harmonic distortion factor and the intermodulation distortion factor on two-tone or multi-tone signals.
To this end, a series of subjective examinations was carried out, organized as follows: thirty-four experts with tested hearing thresholds (mean age 21 years) participated in a large series of experiments to evaluate the sound quality of musical passages (for example, male vocals with symphonic music), in which introduced various types of non-linear distortions. This was done by "convolution" of the test signal with non-linear transfer functions inherent in converters of various types (loudspeakers, microphones, stereo phones, etc.).
Initially, sinusoidal signals were used as stimuli, their “convolution” was performed with various transfer functions, and the harmonic distortion coefficient was determined. Then two sinusoidal signals were used and the intermodulation distortion coefficients were calculated. Finally, the newly proposed coefficient Gm was determined directly from the given transfer functions. The discrepancies turned out to be very significant: for example, for the same transfer function, the THD is 1%, Kim is 2.1%, Gm is 10.4%. This difference is physically explainable, since Kim and Gm take into account much more high-order non-linear distortion products.
Auditory experiments were performed on stereo phones with a range of 20 Hz ... 16 kHz, sensitivity 108 dB, max. SPL 122 dB. The subjective score was given on a seven-point scale for each piece of music, ranging from "much better" than the reference piece (i.e., a piece of music "rolled up" with a linear transfer function) to "much worse." Statistical processing of the results of the auditory assessment made it possible to establish a fairly high correlation coefficient between the average values ​​of subjective assessments and the value of the Gm coefficient, which turned out to be 0.68. At the same time, for SOI it was 0.42, and for Kim - 0.34 (for this series of experiments).
Thus, the relationship of the proposed criterion with subjective assessments of sound quality turned out to be significantly higher than for other coefficients (Figure 12).

Relationship between the Gm coefficient and subjective assessments

The experimental results also showed that an electroacoustic transducer with Gm less than 1% can be considered quite satisfactory in terms of sound quality in the sense that nonlinear distortions in it are practically inaudible.
Of course, these results are still not enough to replace the proposed criterion with the parameters available in the standards, such as the harmonic distortion coefficient and the intermodulation distortion coefficient, however, if the results are confirmed in further experiments, then perhaps this is exactly what will happen.
The search for other new criteria is also actively ongoing, as the discrepancy between existing parameters (especially the harmonic distortion factor, which evaluates only the first two harmonics) and subjective sound quality becomes more and more obvious as the overall quality of audio equipment improves.
Apparently, further ways to solve this problem will go in the direction of creating computer models of the auditory system, taking into account nonlinear processes and masking effects in it. The Institute for Communication Acoustics in Germany works in this area under the direction of D. Blauert, which has already been written about in an article dedicated to the 114th AES Congress. Using these models, it will be possible to evaluate the audibility of various types of non-linear distortions in real music and speech signals. However, while they have not yet been created, assessments of non-linear distortions in the equipment will be made using simplified methods that are as close as possible to real auditory processes.