# Reduction of nonlinear distortion in condenser microphones using a simple post-processing technique

Petr Honzík, Antonin Novak

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Petr Honzík, Antonin Novak. Reduction of nonlinear distortion in condenser microphones using a simple post-processing technique. Journal of the Acoustical Society of America, 2025, 157, pp.699 - 705. 10.1121/10.0035579. hal-05393636

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# Reduction of nonlinear distortion in condenser microphones using a simple post-processing technique

Petr Honzik<fnref n="1" />*, Antonin Novak<fnref n="2" />

## Abstract

In this paper, an approach for effectively reducing nonlinear distortion in single-backplate condenser microphones is introduced, i.e., most microelectromechanical systems (MEMS) microphones, studio recording condenser microphones, and laboratory measurement microphones. This simple post-processing technique can be easily integrated on external hardware such as an analog circuit, microcontroller, audio codec, digital signal processing unit, or within the Application Specific Integrated Circuit chip in a case of MEMS microphones. It effectively reduces microphone distortion across its frequency and dynamic range, and relies on a single parameter, which can be derived from either the microphone's physical parameters or a straightforward measurement presented in this paper. An optimal estimate of this parameter achieves the best distortion reduction, whereas overestimating it never increases distortion beyond the original level. The technique was tested on a MEMS microphone. The findings indicate that for harmonic excitation, the proposed technique reduces the second harmonic by approximately 40 dB, leading to an effective reduction in the total harmonic distortion. The efficiency of the distortion reduction technique for more complex signals is demonstrated through two-tone and multitone experiments, where second-order intermodulation products are reduced by at least 20 dB.

The archived file is not the final published version of the article P. Honzik & A. Novak (2025), "Reduction of nonlinear distortion in condenser microphones using a simple post-processing technique", The Journal of the Acoustical Society of America. Vol. 157(2), pp. 699-705.

The definitive publisher-authenticated version is available online at https://doi.org/10.1121/10.0035579,
Readers must contact the publisher for reprint or permission to use the material in any form.

<footnote n="1">¹ Department of Radioelectronics, Faculty of Electrical Engineering, Czech Technical University in Prague, Technická 2,
166 27 Praha, Czech Republic</footnote>

<footnote n="2">²Laboratoire d'Acoustique de l'Université du Mans (LAUM), UMR 6613, Institut d'Acoustique - Graduate School
(IA-GS), CNRS, Le Mans Université, France</footnote>

<footnote>*petr.honzik@fel.cvut.cz</footnote>
# 1. Introduction

Condenser microphones, including MEMS (Micro-Electro-Mechanical Systems) microphones, are widely used in various applications due to their high sensitivity, wide frequency response, and compact size [1]. However, nonlinear distortion [2], which affects audio quality and measurement accuracy, presents two challenges: it is difficult to measure accurately [3, 4] and even more difficult to reduce effectively [5, 6, 7, 8].

Nonlinear distortion in microphones manifests itself as unwanted harmonic [9] and intermodulation [10] products, which affect the acquisition accuracy of the acoustic pressure signal. This distortion arises from a number of sources [11]. These include mechanical properties of the diaphragm [12], variations in air gap damping between the membrane and backplate due to thickness changes, acoustic nonlinearities caused by high acoustic pressure in the air gap, cavity stiffness, preamplifier characteristics, and the nonlinear capacitance change of the microphone capsule [9]. Among these sources, the nonlinear capacitance change is considered the primary contributor to distortion in condenser microphones. According to Djuric [13], the second harmonic, which is associated with quadratic nonlinearity of the microphone, contributes close to 90% of the total harmonic distortion.

MEMS microphones, which have gained popularity in recent years for their integration into mobile devices and acoustic measurement systems, are inherently susceptible to nonlinear distortion due to their small size and unique construction. Previous studies have shown that the nonlinear behavior of MEMS microphones can significantly affect the accuracy of acoustic measurements, particularly at high sound pressure levels [14, 15].

Despite the advancements in microphone technology, addressing nonlinear distortion remains an important area of research. One approach to achieving a reduction of the second harmonic in single-backplate microphones is to insert a negative capacitance in the preamplifier circuit [5]. Fletcher [6] suggests reducing nonlinear distortion by using a shallow parabolic backplate, although this method introduces manufacturing complications. Another technique for reducing distortion caused by the nonlinear behavior of condenser microphone capsules is the implementation of "dual backplate" technology [7]. This method uses two backplates to reduce the nonlinear effects associated with changes in capsule capacitance, but is generally more expensive due to increased complexity and manufacturing issues. In addition, a method applicable exclusively to CMOS MEMS microphones exploits the capacitance-voltage nonlinear characteristics of a pMOS capacitor to address MEMS nonlinearity at the ASIC level [8].

In this context, the present study introduces a novel post-processing technique that has the potential to significantly reduce distortion levels of all single-backplate condenser type microphones. This easy-to-implement technique can be integrated into various hardware configurations, including analog circuits, microcontrollers, audio codecs, DSP units, and ASIC chips for MEMS microphones.

The paper is structured as follows: In section 2, we provide an overview of microphones distortion and its measurement technique used in this work. Section 3.1 elaborates on the model of the distortion in condenser microphones employed in our study. In Section 3.2, we present the proposed technique in detail, highlighting its simplicity and feasibility. Finally, Section 4 presents the measurement results, demonstrating how effectively the proposed method reduces microphone distortion.

# 2. Microphone distortion measurement

Measuring the distortion of microphones poses significant challenges, particularly in the absence of linear sources. In our research, we use a recently published method [16] enabling to measure and analyze the distortion of microphones at high-pressure levels, as detailed further in this section.

The measurement setup employed in this work involves two identical 6.5" loudspeakers placed face to face, separated by a 2 cm high plastic cylindrical piece with a central aperture, with a volume of approximately $10^{-5}$ m³ (see Fig. 1). These loudspeakers are connected in parallel to create a push-push configuration, which generates pressure excitation inside the aperture with a low distortion. The MEMS microphone under test and the reference
<figure><img src="image_4.png" /><figcaption>**Figure 1.** Schematic view of the measurement setup.</figcaption></figure>

1/8" Pressure Microphone GRAS 40DP are positioned approximately 1 mm apart within the center of the aperture. Note that the MEMS microphone used in this study is an arbitrarily chosen single back plate microphone available on the market (CUI devices, CMM-2718AB-38308-TR). The supply voltage used here is 3.3 V. Similar results and conclusions were obtained for other single-backplate MEMS microphones.

The signal generation and acquisition are managed using an RME Fireface 400 sound card. A sine signal is sent from the sound card to the loudspeakers, which generates the sound pressure inside the aperture. The output of the
<figure><img src="image_5.png" /><figcaption>**Figure 2.** Example of spectra of the measured sound pressure (a) with the reference microphone, (b) with the MEMS microphone under test. Note that thanks to the harmonic correction, there are no higher harmonics measured by the reference microphone.</figcaption></figure>

microphone under test is connected directly to the sound card input, while the reference microphone is connected through a GRAS 26AC preamplifier and Brüel&Kjær Type 2609 conditioning amplifier. To ensure a pure harmonic excitation inside the aperture, we apply the nonlinear harmonic correction technique [16]. As a result, the pressure signal inside the aperture, measured by the reference microphone — whose distortion is negligible at the measured sound pressure levels — contains no unwanted higher harmonics or intermodulation products, as these components are effectively suppressed to the noise level. This measurement procedure is repeated for excitation levels ranging from 85 to 128 dB SPL.

Fig. 2 shows an example of the spectra of the measured sound pressure for a 1 kHz excitation at a level of 116 dB SPL. The upper figure (fig. 2(a)) presents the efficiency of the applied harmonic correction. The higher harmonics measured by the reference microphone are suppressed to the noise level resulting in a pure harmonic excitation inside the aperture (this remains valid at all the measured levels from 85 to 128 dB SPL). The lower figure (fig. 2(b)) shows the spectra of the sound pressure simultaneously measured by the microphone under test.
Note that the thermal noise of the reference microphone is approximately 52 dB(A) SPL according to the datasheet, whereas the thermal noise of MEMS microphones is typically below 30 dB(A) SPL.

<figure><img src="image_6.png" /><figcaption>**Figure 3.** Sound pressure level of first three harmonics for the microphone under test as a function of reference sound pressure level. Measurement results are denoted by dots, model results are depicted by dashed lines.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Figure 4.** Time-domain signal of the voltage at the output of the microphone under test for different levels of the incident pressure. Note that the electrical signals for the four highest sound pressure levels (122, 124, 126, and 128 dB SPL) are saturated.</figcaption></figure>

The results of such a measurement are shown in fig. 3. The measured levels of the first harmonic (blue points), second harmonic (orange points), and third harmonic (green points) at the output of the MEMS microphone under test are shown for a 1 kHz excitation. These levels are recalculated to an equivalent input sound pressure level through the measured sensitivity of the MEMS microphone. The results are depicted as a function of sound pressure level measured by the reference microphone. The theoretical levels of the harmonic components, given by the model described below, are represented by dashed lines of corresponding colors. Up to approximately 120 dB SPL the second harmonic fits well the predicted value, whereas the third harmonic is several dBs higher. This discrepancy is likely caused by acoustic nonlinearities inside the microphone, such as variations in air gap damping due to thickness changes and acoustic nonlinearities caused by high acoustic pressure in the air gap which are not accounted for in the model. This observation is consistent with previously published research [14, 17]. Above 120 dB SPL the
microphone output signal presents asymmetric clipping (see fig. 4), originating probably from the electronic part of the microphone, and the measured data cannot be predicted by the model at these levels.

# 3. Technique for reduction of distortion

## 3.1 Model of condenser microphone distortion

In this section, we provide a concise summary of the nonlinear model of a single-backplate condenser microphone [14].

The relationship between the output voltage $u(t)$ and the change in capacitance $dC(t)$ can be expressed as

$$u(t) = -U_0 \frac{dC(t)}{C}, \qquad (1)$$

where $U_0$ represents the polarization voltage, and where the distortion is caused by nonlinear behavior of the capacitance change $dC(t)$. This equation can be further expanded to the following expression:

$$u(t) = U_0 \frac{C_0}{C_P + C_0} \left[ \frac{\bar{\xi}(t)}{h_g} - \left( \frac{\bar{\xi}(t)}{h_g} \right)^2 + \left( \frac{\bar{\xi}(t)}{h_g} \right)^3 - \dots \right], \qquad (2)$$

where $C_P$ represents the parasitic capacitance, $C_0$ is the static capacitance of the microphone, $h_g$ denotes the air gap thickness between the membrane and backplate, and $\bar{\xi}$ represents the mean displacement of the membrane over the backplate area. Eq. (2) can be rewritten as

$$u(t) = K_0 [y(t) - y^2(t) + y^3(t) - \dots], \qquad (3)$$

where, $K_0$ represents a constant that depends on the capacitance and polarization voltage

$$K_0 = U_0 \frac{C_0}{C_P + C_0}, \qquad (4)$$

and $y$ represents the relative mean membrane displacement with respect to the air gap thickness. Eq. (3) provides a simplified form to model the distortion characteristics in the electrical output of the condenser microphone.

## 3.2 Reduction of distortion

As previously mentioned, the primary distortion component in single-backplate microphones is the second harmonic. Furthermore, recent work [14] has confirmed this finding for single-backplate MEMS microphones, showing that the above mentioned model accurately predicts this distortion component regardless of sound pressure level and frequency. Consequently, the critical objective in distortion reduction is to effectively suppress mainly the quadratic component. Therefore, we can preserve the powers of $y(t)$ up to the second order in eq. (3), leading to

$$u(t) = K_0 [y(t) - y^2(t)]. \qquad (5)$$

The procedure for distortion reduction through inversion of eq. (5) is described in this section.

First, we denote $u_{lin}(t) = K_0 y(t)$ the distortion free output signal. Then, the output voltage given by eq. (5) becomes

$$u(t) = u_{lin}(t) - \frac{1}{K_0} u_{lin}^2(t), \qquad (6)$$

and one can find an inverse function to eq. (6) providing an estimate of the linear output signal as

$$u_{lin}(t) = \frac{K_0}{2} \left[ 1 - \sqrt{1 - \frac{4u(t)}{K_0}} \right]. \qquad (7)$$

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>
<figure><img src="image_10.png" /><figcaption>Figure 5. Block schema of the proposed distortion reduction technique.</figcaption></figure>

Using Taylor series expansion around zero, the estimate of the linear output signal given by eq. (7) can be further simplified to

$$u_{\text{lin}}(t) \approx u(t) + \frac{1}{K_0} u^2(t). \qquad (8)$$

A block schema for such a distortion reduction technique is depicted in fig. 5.

The constant $K_0$ (the only parameter necessary for the reduction of distortion using eqs. (7) or (8)) can be calculated from the polarization voltage $U_0$ and the static ($C_0$) and parasitic ($C_P$) capacitances of the microphone (see eq. (4)). Since these parameters are not always known, a simple method for estimation of $K_0$ is proposed hereafter.

Considering a pure harmonic excitation of the microphone membrane at the frequency $f_0$, the mean displacement to air gap thickness ratio can be described by $y(t) = y_m \sin(2\pi f_0 t)$, where $y_m$ denotes the amplitude. The absolute values of the first and second component of the Fourier series of the nonlinear output $u(t)$ (5) can be expressed respectively as $V_1 = K_0 y_m$, corresponding to frequency $f_0$, and $V_2 = K_0 y_m^2/2$, corresponding to frequency $2f_0$. The constant $K_0$ can be thus estimated from these measured spectral components

$$K_0 = \frac{V_1^2}{2V_2}. \qquad (9)$$

Such an estimate is independent of input excitation level and frequency. Fig. 6 shows the estimate of $K_0$ according to eq. (9) for different input levels at 1 kHz (blue points) using the measured data from fig. 3, resulting in an estimated coefficient value of $K_0 = 8.85$ V (gray dashed line). Note that if a gain $G$ is applied to the output of the microphone before the correction procedure, then $GK_0$ must be used instead of $K_0$.

# 4. Results

In this section the performance of the proposed distortion reduction technique is studied for different types of input signals. The impact of incorrect estimation of the parameter $K_0$ is also discussed.

## 4.1 Harmonic distortion

First, the microphone was excited by a pure sine tone, which leaded to the presence of higher harmonics at its output due to the distortion, as shown in fig. 7 (blue line). Then, the distortion reduction technique has been applied on the microphone output signal, resulting to a significant reduction of the second harmonic component of the output signal spectrum - see orange line in fig. 7 (shifted to left for clarity). The upper graph of fig. 7 shows the result when applying the complete inverse function given by eq. (7), the lower graph presents the result of the simplified equation (8). In both cases the second harmonic component is reduced by approximately 40 dB, while the

<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /></figure>
<figure><img src="image_13.png" /><figcaption>**Figure 6.** Estimated value of the coefficient $K_0$ as a function of reference sound pressure level. The measurement results are denoted by dots, and the selected optimal value $K_0$ is represented by the gray dashed line.</figcaption></figure>

other harmonics stay almost intact (the slight increase of the third harmonic in the lower graph being negligible). Since eq. (8) provides very similar results to eq. (7) with lower computational costs, all the following results are obtained using this simple technique.

Fig. 8 presents the Total Harmonic Distortion (THD) of the microphone at 1 kHz without any postprocessing (blue points) and when applying the distortion reduction technique (orange crosses) as a function of the input sound pressure level. Except for very low levels, where the second harmonic is hidden in background noise, and very high levels, where signal clipping appears (the Acoustic Overload Point, defined as the 10% THD level, appears near 123 dB SPL, contrary to the 130 dB SPL stated in the datasheet), THD is reduced 10 times near 120 dB SPL, or even 50 times near 110 dB SPL. Note that the THD obtained using the distortion reduction technique never exceeds the original THD (without any processing).

## 4.2 Impact of $K_0$ uncertainty

The reduction of distortion described above was achieved using the value of $K_0$ estimated with eq. (9). However, if we use eq. (4), which includes parameters $U_0$, $C_0$, and $C_P$ that may not all be known precisely, $K_0$ could be estimated inaccurately. To study the effect of uncertainty in $K_0$ on THD, we repeated the previous test for different values of $K_0$. Fig. 9 shows the dependence of THD on the varying value of $K_0$ using both eq. (7) (orange stars) and eq. (8) (green circles) compared to THD when no distortion reduction is applied (blue line), for input sound pressure level of 110 dB SPL at 1 kHz. First, it is clear again that the difference between results of eq. (7) and eq. (8) is negligible. Next, the estimated value of $K_0 = 8.85$ V from eq. (9) (vertical dashed gray line in fig. 9) fits well with the minimum THD value. Furthermore, fig. 9 shows that although the best distortion reduction is achieved with the optimal estimated value of $K_0$, THD is reduced to some extent in very large range of values of $K_0$. However, if the value of $K_0$ is largely underestimated (below 50% of the optimal value), the THD can exceed its original value. On the other hand, overestimating the value of $K_0$ by 50% still leads to a reduction of THD by a factor of two. Indeed, overestimating the value of $K_0$ never causes the increase of THD above the level obtained without any distortion reduction. This is obvious from eq. (8), where the quadratic term decreases with increasing $K_0$ and the result $u_{lin}(t)$ converges to the original distorted signal $u(t)$.

## 4.3 Intermodulation distortion

To evaluate the robustness of the proposed method under intermodulation distortion, two complex signals are used to excite the microphone: (i) a two-tone signal consisting of 1440 Hz and 1680 Hz components, with a total RMS
<figure><img src="image_14.png" /><figcaption>**Figure 7.** Spectral analysis of microphone signal for 1 kHz excitation. The blue spectrum corresponds to the original (unprocessed) signal, the orange spectrum (shifted to the left) corresponds to the signal processed by the proposed technique (Eq. (7) upper graph, Eq. (8) lower graph).</figcaption></figure>

(Root Mean Square) value of 8 Pa, corresponding to 109 dB SPL for each component; and (ii) a multitone signal consisting of 100 Hz, 300 Hz, 500 Hz, 1 kHz, 1.8 kHz, 3.4 kHz, 6.1 kHz, 11.1 kHz, and 20 kHz components, with a total RMS value of 5 Pa, corresponding to approximately 98.5 dB SPL for each component. All spectral components in both signals have random phases to improve the crest factor [18]. The harmonic correction technique[16], previously used for single sine excitation, is also used here to ensure pure two-tone and multitone excitation. This technique is applicable to any periodic signal and effectively ensures that only the desired spectral components are present.

Fig. 10 depicts the spectrum of resulting signal when applying the distortion reduction technique (orange line) and without any distortion reduction (blue line). The upper graph shows the result for the two-tone signal, where the second order harmonics and intermodulation products at 240 Hz, 2880 Hz, 3120 Hz and 3360 Hz are reduced by more than 25 dB. For the multitone signal the intermodulation products are reduced by approximately 20 dB on average, as shown in the lower graph.
<figure><img src="image_15.png" /><figcaption>**Figure 8.** Total harmonic distortion as a function of reference sound pressure level at 1 kHz. The unprocessed values are represented by blue dots, the values after applying distortion reduction are denoted by orange crosses.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>**Figure 9.** Total harmonic distortion of the MEMS microphone at 1 kHz and 110 dB SPL plotted as a function of the coefficient $K_0$. The unprocessed value is represented by the blue line, the technique applying the distortion reduction from Eq. (7) by the orange stars, and the one from Eq. (8) by the green circles.</figcaption></figure>

## 5. Conclusion

In conclusion, this paper presents a new technique aimed at reducing nonlinear distortion in single-backplate condenser microphones, e.g., MEMS microphones. The proposed technique offers a simple and efficient solution, requiring minimal post-processing that can be easily implemented on an ASIC chip or in external hardware (analog circuit, microcontroller, audio codec, DSP unit, etc.) and that reduces the microphone distortion within its frequency and dynamic range.

One of the key advantages of this technique is its reliance on a single parameter, $K_0$, which can be deduced either from the microphone physical parameters (polarisation voltage, static and parasitic capacitance) or from a straightforward measurement. With a correct estimation of the parameter $K_0$, the total harmonic distortion can be reduced by a factor of 10 or even 50 depending on excitation level. Remarkably, even an incorrect estimation of $K_0$ by 50% still leads to substantial reductions in total harmonic distortion, highlighting the robustness of this approach.

The method has very low computational cost, requiring only two multiplications and one addition per sampling

<figure><img src="image_17.png" /></figure>

<figure><img src="image_18.png" /></figure>
<figure><img src="image_19.png" /><figcaption>**Figure 10.** Spectral analysis of microphone signal for a two-tone (upper graph) and multi-tone (lower graph) excitation. The blue spectrum corresponds to the original (unprocessed) signal, the orange spectrum (shifted to the left) corresponds to the signal processed by the proposed technique.</figcaption></figure>

period, making its contribution to signal postprocessing latency negligible. Since the algorithm does not involve feedback, stability issues are not expected, even in the presence of rounding errors. However, care must be taken when implementing the square of large signals in eq. (8), as this may lead to arithmetic overflow on certain hardware platforms (in this case, the multiplication by $1/\sqrt{K_0}$ can be performed prior to the square calculation).

The efficiency of the distortion reduction technique of more complex signals is demonstrated through two-tone and multitone experiments. Both experiments show an important reduction of intermodulation components created by the microphone by around 20 dB. This finding further strengthens the reliability and applicability of the proposed technique.

Considering the ease of implementation, low-cost nature, and robustness of this method, it holds great promise for improving the performance of single-backplate microphones, including MEMS microphones.
Acknowledgments

This work was supported by the project 23-07621J of the Czech Science Foundation (GAČR) "Otoacoustic emissions in normal cochlea and cochlea with endolymphatic hydrops: modeling and experiments", by the Grant Agency of the Czech Technical University in Prague, Grant No. SGS23/185/OHK3/3T/13 and by the Institut d'Acoustique - Graduate School (IA-GS), Le Mans, France. A patent application for this invention has been filed with the French National Institute of Industrial Property (INPI) under the application number FR2404994 on May 16, 2024.

Author Declarations

The authors have no conflicts to disclose.

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