PROCEEDINGS of the23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Nonlinearities in sound field control systems
Elias D. LUMPERT1; Finn T. AGERKVIST2; Franz HEUCHEL3; Jonas BRUNSKOG4
1,2,3,4Technical University of Denmark, Acoustic Technology Group, 2800 Kgs. Lynby, Denmark

ABSTRACT
If a loudspeaker is driven at high levels, its output is likely to be affected by nonlinearities. The result is adistortion of the loudspeaker output which depends on the input signal. When looking at a soundcancellation or sound field control system the introduction of such nonlinearities lowers the performance.
This paper investigates the extent of such effects and looks at strategies to counteract the problem. The
paper specifically focuses on quantifying the effect of compression on the output of the system at the input
frequency, finding a model to represent the coherent distortion and counteract accordingly. A comparison ofdifferent strategies to counteract the effect of nonlinearities is presented. The results show that acompensation based on an input voltage level adapted linear response of a loudspeaker can improve asound zoning algorithm up to 20dB. Alternative compensation methods are discussed and show the benefitof a displacement level dependent compensation at low frequencies.
Keywords: Sound Field Control, Nonlinearities

1. INTRODUCTION
Sound field control systems rely on accurate estimations of the transfer function between source
and the controlled zone and often assume linear time invariance of the source. However, real
loudspeakers behave nonlinear at high output levels. This leads to compression of the fundamental
frequency and a rise of distortion products. In a previous paper Ma (1) shows that nonlinear
distortion degrades the acoustic contrast, a performance metric in a sound zoning system. Further hestates that nonlinear effects on the fundamental component appear to be the main cause for thiscontrast loss.
This paper investigates first approaches to compensate this effect. In contrast to moresophisticated compensation models (2,3) this work aims on developing a fast and easy-to-implement
solution to the problem. The proposed approach is based on a prediction of the change in linear
transfer function at different voltage and displacement levels. Specifically, the compression anddecompression of the linear transfer function due to changing level will be compensated. The
complexity of the subject required limitations and simplifications at several points. Nevertheless,the outcome of the paper shows promising results of an improvement of up to 20dB on a broad
range of frequencies. The effect of distortion products will be touched but not discussed in detail.
The following sections focus on explaining the proposed compensation method and present
experimental results on a real loudspeaker.

2. METHOD
2.1 Compensation Method
The magnitude and phase response of most loudspeakers is dependent on the input signal due toits nonlinear behavior at higher input levels. The first step of the proposed compensation method isto identify those dependencies. First, we measure the transfer function between voltage and nearfield sound pressure of a given loudspeaker at different input gains to characterize the nonlinearityof a loudspeaker by its apparent linear transfer function at different levels. The level dependentchanges can be used to calculate an approximation of the expected change in transfer function at anyother level. This expected change can be described as a function of input level using a polynomial
fitting function of fourth order. This expected change is used to compute a linear compensation filter

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We introduce the estimation signals 𝑥" 𝑡 with correspondent level 𝐿" = 𝐿 𝑥" 𝑡 . One level ischosen as reference level 𝐿&'( . The level can thereby either be a RMS (root-mean-squared) voltage
or RMS voice coil displacement.
To compute the compensation filters one
1. measures transfer functions 𝐻 𝑓, 𝐿" at levels 𝐿" (Fig.1, left),
2. computes the change in transfer function ∆𝐻 𝑓, 𝐿" = 𝐻 𝑓, 𝐿" 𝐻 𝑓, 𝐿&'( compared to thereference level (Fig.1, right),
3. approximates ∆𝐻 using a polynomial regression (4 th order), to get changes for a continuumof input levels: ∆𝐻 𝑓, 𝐿 . These are the linear compensation filters in frequency domain.
To apply the compensation filters one
1. measures the test signal level 𝐿-'.- = 𝐿 𝑢 𝑡 ,2. applies the compensation filter to the test signal 𝑦 𝑡 = 𝐼𝐹𝐹𝑇{∆𝐻 𝑓, 𝐿-'.- 𝐹𝐹𝑇{𝑢 𝑡 }}, and3. plays back 𝑦 𝑡 instead of 𝑢 𝑡 .
In the following, the signals to estimate the compensation are called estimation signals. The signals
the compensation is applied to will be referred to as test signals.

Figure 1 – Left: sketch of a loudspeaker magnitude response at four different levels. Right: differences inmagnitude response at a single frequency for different levels together with polynomial regression.
It has been elaborated by Klippel (4) that nonlinear behaviors of loudspeakers are dependent on
different variables. These variables include current, velocity and displacement. An effective
compensation strategy should be based on the level of a variable that is directly connected to thechange in response. In this work, a simple approach of using the voltage (which is coupled to thecurrent) and the displacement variable was investigated. In the following we focus on threevariations of the compensation procedure:
1) Sweep-voltage approach: the transfer functions are measured with an exponential sweep
excitation. The compensation is based on the RMS voltage level.
Measuring with sweep signals excites only one frequency at a time. This way mainly distortions atthe fundamental frequency are recorded.
2) Noise-voltage approach: the transfer functions are measured with M-Noise 1 , a noise withmusic like crest-factor. The compensation is based on the RMS voltage level.
In contrast to the sweep-voltage approach, this test signal is closer to music which allows
interaction between frequencies and amplitudes when measuring the transfer function. Theinterpretation of the results of this approach is less obvious due to the random structure of amplitudeand frequency content.
3) Noise-displacement approach: the transfer functions are measured with M-Noise. The
compensation is based on the RMS displacement level.
Here, the calculation and application of the compensation is performed using voice coildisplacement instead of the voltage level. This compensation assumes that changes in transfer
function are related to displacement. In contrast to the two preceding approaches this approach
applies more compensation for signals with high displacement (signals with strong low frequencies).

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https://m-noise.org

2.2 Performance Metrics
Sound cancellation applications are based on a primary wave cancelled by a secondary wave. A
perturbation of amplitude or phase of the secondary wave, e.g. due to nonlinear distortion, directlytranslates to an achievable reduction of sound (𝛥𝐿7 . Fig. 2, right illustrates this 𝛥𝐿7 as a function ofamplitude and phase deviation (5). For no error in phase or amplitude, 𝛥𝐿7 approaches negative
infinity, which corresponds to total cancellation. Using the formula of 𝛥𝐿7 the maximal achievablereduction of a sound cancellation application can be calculated given a certain error in transfer
function. From Fig.4, already slight deviations in level and phase compromise a given soundcancellation application. Assuming a constant phase error of ±2.5° and a magnitude error of +1dB agiven sound cancellation application could only achieve an attenuation of primary sound of around-17dB. A reduction of this errors to ±0.5°and +0.2dB improves the performance to -32dB.

Figure 2 – Reduction in amplitude of a primary sound wave as a function of the difference of amplitude andphase between primary and secondary sound wave

3. Results
To test the discussed compensation strategies measurements were made using a 10-inch closed
box subwoofer. This section first presents measurements of the variation of transfer function forchanging level of this loudspeaker. Next, the compensation method is applied.
3.1 Loudspeaker frequency response at different input levels
The measured transfer functions for measurement signals with different RMS voltage levels areshown in Fig. 3.

Figure 3 – 10-inch subwoofer frequency response measured in near-filed at 5 different levels with 15
second exponential sweeps.
The considered frequency range has been limited to 20Hz to 2000Hz, which is the effective range

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of this speaker. The amplitude response shows a rather flat curve from 70 to 700Hz. The resonanceof the loudspeaker lies between 60-70 Hz which explains the drop in transfer function below thatfrequency (6). At higher frequencies (above 400Hz) the response shows irregularities due to theconstruction of the loudspeaker box. Another irregularity can be observed around 160 Hz. To
identify a change in response we chose 0.43 V as a reference measurement. The change in responseis the difference to the reference response. Fig. 4 shows the differences in transfer function to thereference level at 0.43V.

Figure 4 – Differences of magnitude and phase response at different levels relative to responses at 0.43V.
Above 60 Hz the transfer function decreases in amplitude and has a negative phase shift forincreasing level. At high frequencies, this effect might be mainly a product of voltage dependent
nonlinearities, as the voice coil displacement is small. At lower frequencies, the displacement
increases which allows displacement and velocity dependent nonlinearities to have an effect. Below
60Hz the transfer function amplitude shows a rise up to 4.25V and drops at higher levels. Thisbehavior is related to a downwards shift in the impedance resonance frequency. Above 5V
dampening effects due to high displacements start to dominate (e.g. voice coil leaving the magnet or
reluctant force of the suspension (4)). The shape of the various peaks in Fig. 4 indicate a shift infrequency of (anti-)resonances.
3.2 Sweep-Voltage Compensation
Fig.5 shows the measured transfer function with applied compensation of the sweep-voltage
approach. The differences in transfer function for varying level are smaller than the changes
observed in Fig. 4. Around the area of certain resonances in the speaker (e.g. around 160Hz) theperformance of the compensation is less effective.

Figure 5 – Differences in loudspeaker response (Fig.1) with applied compensation filters at different levels
relative to measurement at 0.43V

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In order to relate the influence of such a compensation to a sound cancellation problem, theachievable reduction with and without compensation is shown in Fig.6 (as explained in Sec.2.2).

Figure 6 – Achievable reduction calculated from an error in transfer function. Simulated and measured for asweep measurement at 4.25V compensated for a reference at 0.43V
The Figure shows that, for a sweep test signal, the compensation using the sweep-voltage
approach improves a sound cancellation problem by 10-30 dB on a broad range of frequencies. It
can be seen that the compensation tends to perform worse in the areas with resonances. Note that thetest signal matches the estimation signal which leads to a good performance of the compensation.
Fig.7. shows the performance of the sweep-volt compensation filter applied to other test signals.

Figure 7 – Achievable reduction calculated from the error in transfer function for different signals.
Simulated and measured for signals around 4 - 5 V compensated for a reference at 0.43V
For M-Noise (a) the compensation reaches an improvement of 10-20dB on a broad range offrequencies. Applied to white noise (b) the compensation shows improvement at higher frequencies(above 200Hz) but not between 50-190Hz, where the displacement is high. As two different signals
can have the same voltage level but different displacement levels, the choice of level is critical. The
compensation might be ineffective, if the voltage and displacement levels of the estimation and test
signal are different. We assume that at low frequencies the compression due to nonlinearities ismostly dependent on displacement and at higher frequencies there is more dependency on voltage.
This might be the reason for the poor performance from 50-190Hz, since displacement is high but4323

the voltage level is used to calculate the compensation. The presence of high displacement below
50Hz suggests an inefficient compensation below 50Hz. but Fig.7 shows an improvement inachievable reduction. This is no contradiction since an over- or underestimation of a change intransfer function can still lead to a compensation filter that improves the error in transfer function
compared to the uncompensated state. The compensation filter improves cancellation for both beat 2(c) and music 3 (d) signals on a broad range of frequencies of 5-20dB. Below 30-40 Hz the beat andmusic signals do not contain enough energy to make meaningful statements.
In general, the results indicate that the proposed compensation shows an improvement on a broad
range of frequencies. The extent of the improvement however is dependent on the frequency content
and time variance of the applied signal.
3.3
Noise-voltage and noise-displacement approaches
As has been stated in Sec. 2.1, compensations based on different models were investigated. Apartfrom sweep-voltage approach, a noise-voltage approach and a noise-displacement approach weretested. The comparison of those three approaches is shown in Fig. 8. White noise was chosen as atest signal due to the difference in spectra compared to the estimation signals used to calculate thecompensation. White noise has a flat spectrum in contrast to exponential sweeps and M-Noise,
which have decaying spectra with increasing frequency.

Figure 8 – Achievable reduction calculated from error in transfer function for different compensation
approaches. Simulated and measured for a signal at 4.64V compensated for a reference at 0.43V.
Fig. 8 shows that for a white noise test signal and the sweep-voltage approach (a) or noisevoltage approach (b) there is no improvement visible at frequencies from 50-200Hz. For higher
frequencies, 200-2000Hz, a consistent improvement of 10-20dB is present. The compensation basedon displacement levels (c) shows an improvement of 5-10dB at low levels (below 80 Hz), while
higher frequencies show less improvement (1-4dB). This behavior indicates that at lower
frequencies a change in response is mainly dependent on displacement while at higher frequencies it
is rather connected to voltage levels.
2A simple generated beat signal using kick drum, snare drum and hihat samples in a 4/4 signature standard
beat accompanied by a simple increasing bass line.
3
The music test signal was chosen such that it contained strong low frequency content, dynamic changes
and a broad frequency spectrum

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4. Discussion
The presented approach compensates a nonlinear loudspeaker system using level dependentchanges in the linear transfer function. Results show that such a compensation can improve a soundcancellation system by several dB on a broad range of frequencies. Experiments with otherloudspeakers show similar results. The compensation was performed for 10 different voltage anddisplacement levels spanning a range of around 24dB. While the results show the compensation
around one RMS voltage level (4-5 V), the measurements at other levels show similar results.
The approach is rather simple and therefore limited. Especially if estimation and test signals havelarge spectral differences, the improvement from the compensation approach is small. This is
assumed to be due to the dependency of nonlinear effects on multiple variables (e.g. current,displacement, etc.). A more sophisticated approach should be based on a compensation based on
multiple variables.
Distortion products have not been considered in the presented compensation. It should be notedthat such artifacts compromise a sound field control strategy as has been discussed by Ma (7). Even
though might not be the primary issue, they become more relevant the more the compression of thefundamental is compensated.
The frequency resolution to calculate the compensation was 1Hz. The usage of a high resolution
uncovers details and allows to compensate narrowband changes like resonance shifts. The effect ofcompression of the fundamental due to nonlinearities is expected to be similar for similarfrequencies. Only looking at the compression of the fundamental, a coarse resolution might not only
perform better but also limit the computational cost.
Two of the discussed compensation methods are dependent on the RMS voltage of the signal. In
section 2.1 it is argued that instead of current level the voltage level is used to calculate and apply
the compensation due to their interconnection. The relation between current and voltage isdependent on the impedance. But the impedance changes with level (e.g. DC resistance or resonancefrequency shift). The interconnection between voltage and current is therefore level dependent.
Using the RMS current instead of RMS voltage might improve the compensation strategy.
Addressing such limitations could further improve the presented compensation approach ofnonlinearities in sound cancellation systems.

5. Conclusion
We showed that nonlinear effects in loudspeakers can be partly compensated by applying a level
dependent linear compensation filter. A compensation approach based on a sweep test signal
measured at different voltage levels was able to counterbalance the effect of (de-)compression on
the excited frequency. The compensation was applied to different signals. For test signals withsimilar frequency content as the estimation signal, results show an improvement in soundcancellation of 10-20dB on average from 20-2000Hz. If test and estimation signal have different
spectra the compensation approach shows only small improvements.

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