# Perceptual evaluation of an auralization model for pulse width modulation noise

Salomé Wanty, Etienne Parizet, Nicolas Totaro, Martin Glesser

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Salomé Wanty, Etienne Parizet, Nicolas Totaro, Martin Glesser. Perceptual evaluation of an auralization model for pulse width modulation noise. Applied Acoustics, 2026, 246, 10.1016/j.apacoust.2026.111235. hal-05497807

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# Perceptual evaluation of an auralization model for pulse width modulation noise

Salomé Wanty<sup>a, b,*</sup> <sup>□</sup>, Etienne Parizet<sup>a □</sup>, Nicolas Totaro<sup>a □</sup>, Martin Glesser<sup>b □</sup>

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## HIGHLIGHTS

* An auralization model for e-motor noise is validated through a perceptual experiment.

* INDSCAL analysis provides similar 2D perceptual spaces for real and virtual stimuli.

* Low-frequency noise and rattle noise affect the perception of electric motor noise.

* The similarity of pleasantness ratings demonstrates the relevance of the auralization model.

## GRAPHICAL ABSTRACT

### Perceptual evaluation of an auralization model for PWM noise.

An auralization model for the electromagnetic noise of electric motors supplied by PWM was implemented by the authors from existing methods. Are the simulated signals reliable enough to perform sound quality analysis ?

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### INDSCAL analysis

The perceptual space of PWM electromagnetic noise can be correctly described from a set of computed sounds.

Test bench measurement on an induction machine

VS

Noise synthesis

with the auralization model

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### RM-ANOVA

The model enables to correctly rank different strategies.

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Wanty et al. 2026

## ARTICLE INFO

### Keywords:

- Electric motor noise

- Noise synthesis

- Pulse width modulation

- Acoustic perception

- Individual difference scaling

- Pairwise comparison

## ABSTRACT

Electric motors have become a part of our daily lives, making the question of their noise essential for our acoustic comfort. The sound they emit is often described as tonal or whistling due to the presence of many harmonics, one source of which is a power supply technique called pulse width modulation (PWM). During the design process, auralization models can be used to evaluate the effect of PWM harmonics on sound quality. Engineering-grade models, which are based on simplifying assumptions, are used in the early stages of design when little input data is available. With these models, a direct perceptual comparison between measured and simulated sounds would inevitably reveal significant differences. However, if the model can reliably predict the perceptual sound space, it can be a valuable tool for assessing sound quality. This paper presents a methodology to evaluate the capability of a simplified engineering model to simulate the main attributes of PWM noise. To this end, the authors implemented an auralization model to synthesize the noise emitted by an electric motor from its supply signals. Some sound stimuli were collected from measurements and simulations to conduct a perceptual experiment. The measured and simulated sounds were evaluated separately within two sets of stimuli. The experiment included similarity and pleasantness evaluations. Comparing the results obtained by Individual Difference Scaling (INDSCAL) showed great coherence between the two sound sets, suggesting that the simulated stimuli were evaluated similarly to the measured stimuli. Pleasantness ratings yielded the same result. Therefore, the auralization model appears to reliably reproduce the main sound dimensions underlying the perception of PWM noise.

<footnote><sup>a</sup> INSA Lyon, LVA, UR677, 69621 Villeurbanne, France</footnote>

<footnote><sup>b</sup> SIMULIA Manatee, Dassault Systèmes, 59650 Villeneuve d'Ascq, France</footnote>

<footnote>* Corresponding author at: INSA Lyon, LVA, UR677, 69621 Villeurbanne, France.
Email address: salome.wanty@3ds.com (S. Wanty).</footnote>
# 1. Introduction

The increasing prevalence of electric motors in the automotive industry raises the question of their sound signature. Their operating principle relies on electromagnetic phenomena, which generate specific noises that are often described as tonal, whistling or whining [1,2]. Among these mechanisms, pulse width modulation (PWM) is necessary to create an alternating current from the constant voltage delivered by a battery. PWM generates sideband harmonics that contribute significantly to the noise emitted by the motor [3,4]. These harmonics therefore exert a considerable influence on sound quality, and their characterization is imperative for the design of motors whose sound signature is controlled.

The following technical details are provided for completeness and to introduce the context of PWM-related noise. A significant number of studies concerning PWM noise are grounded in recorded sounds [1,3]. In other studies, simulation is employed to generate vibration and noise harmonics of an electric motor, generally with semi-analytical approaches [5]. Current in the stator's winding can be used to calculate the magnetic flux density in the air gap (between the stator and the rotor) using finite element methods (FEM). This provides the force density according to the Maxwell stress tensor [4,6-9]. Vibrations and noise can then be computed by FEM [4,7-10], or analytically [10]. In most cases, the current provided as input to the simulation model is measured on a test bench [8,9]. When this current is simulated, PWM harmonics can be neglected [11] or considered analytically as a Fourier series [10], which makes it difficult to study new strategies. The accurate numerical representation of the current harmonics from PWM requires the use of an equivalent electrical circuit including the inverter. This approach can present complexities when implemented at variable speeds or with random PWM strategies.

This study aims to assess whether an auralization model can be used to evaluate the sound quality of a motor under various PWM strategies. Indeed, it is unrealistic to expect a simplified model to exactly reproduce a measured signal (in the sense that a listener cannot hear any difference between the two signals). Nonetheless, auralization can be useful if (i) it enables the correct ranking of different strategies in terms of sound quality and (ii) if the perceptual space of electromagnetic noise can be correctly described using a set of computed sounds. To verify this, a perceptive experiment can be conducted on two sets of sounds. One of them gathers recorded sounds obtained from several control strategies, while the other is composed of sounds computed from the model using the same strategies. By comparing the results obtained with the two sound sets, it is possible to assess whether the model can be used for the two objectives mentioned above. This approach has been used for various applications, including wind turbine noise [12] and aerodynamic noise in cars [13].

In this context, the authors implemented an auralization model to study the noise emitted by an electric motor under steady or transient operating conditions. Various PWM strategies are considered thanks to a parametric formulation of intersective PWM. A parametric formulation of intersective PWM allows for the consideration of multiple PWM strategies. The calculation of the current flowing in the winding from the command signals is based on the temporal approach described by Li [14]. The winding function and rotor magnetomotive force resulting from magnetic FEA analysis are combined with the current to obtain the electromagnetic excitations in the air gap, as explained by Le Besnerais [15]. The stator's external surface displacement is computed from the radial component of those excitations using Tooth Wave Frequency Response Functions, according to the experimental method developed by Pile [16]. The noise is then synthesized directly from the stator's envelope vibrations, under the assumption of Equivalent Radiated Power (ERP) [17], which assumes that the motor's outer surface radiates like a plate with a unit efficiency. The model neglects certain phenomena, such as the mechanical noise due to the rotation of the rotor, and potential defects like eccentricity and unbalanced magnetic pull. The validation method, which involves the separate evaluation of measured signals and

simulated signals, is applied here to assess the model's ability to predict noise signals that are sufficiently reliable for sound perception studies.

Section 2 presents the two sets of sound stimuli, which were obtained from test bench measurements and simulations, respectively. The perceptual experiment conducted to assess the sound reliability of the model is presented in Section 3 in terms of evaluation methods and experimental conditions. The results of the pairwise comparisons and pleasantness assessments are presented and discussed in Section 4.

# 2. Noise stimuli

## 2.1. Motor description

To assess the quality of the sound representation provided by the model for this motor, two types of signals were collected with several PWM strategies. The measured set is composed of recorded sounds, and the virtual set gathers simulated signals. The stimuli are sounds emitted by the motor of a washing machine (Candy/Hoover 41046224) supplied by different PWM strategies. It is a Squirrel Cage Induction Machine (SCIM) whose main specifications are given in Table 1.

## 2.2. Selection of PWM strategies

PWM is a technique used to chop the steady voltage delivered by the battery through a system of switches (typically insulated-gate bipolar transistors) to obtain a succession of pulses creating a fundamental voltage at the desired frequency. As illustrated in Fig. 1(a) for one phase, the computation of these voltage pulses is performed with an intersective

<table><caption>**Table 1**
Motor specifications.</caption><tr><td>Pole Pair number</td><td>1</td></tr><tr><td>Number of stator slots</td><td>24</td></tr><tr><td>Number of rotor slots</td><td>28</td></tr><tr><td>Nominal Power</td><td>500 W</td></tr><tr><td>Nominal Speed</td><td>18,000 rpm</td></tr><tr><td>Nominal Current</td><td>5 A</td></tr></table>

<figure><img src="image_16.png" /><figcaption>(a) Intersection of the command waveform (SVPWM) with the triangular carrier</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>(b) Resulting square voltage signal</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>(c) Frequency signature of PWM sideband harmonics (with arbitrary amplitudes)</figcaption></figure>

<caption>**Fig. 1.** Illustration of the PWM technique, and its harmonics.</caption>
<figure><img src="image_19.png" /><figcaption>Fig. 2. Waveforms of the eight PWM strategies studied. Grey lines represent the fundamental electrical frequency.</figcaption></figure>

technique. The process involves comparing the command signal of electrical frequency $f_e$ with a triangular carrier wave of switching frequency $f_{swi}$. The generated PWM square signal in Fig. 1(b) represents the commutation state of the phase: 1 if the switch is open, -1 if it is closed. The inductance of the stator winding tends to smooth the pulses, resulting in a quasi-sinusoidal phase current with the desired frequency and amplitude. However, the process generates sideband harmonics, located around multiples of the switching frequency, as shown in Fig. 1(c). In the induced noise and vibrations, they have frequencies $f = m f_{swi} + n f_e$ where $m \in \mathbb{N}^*$ and $n \in \mathbb{N}$ [10]. Their amplitudes vary depending on the strategy and the operating point (speed and torque) [18]. Each transition from one commutation state to another generates losses. Therefore, it is not possible to set $f_{swi}$ outside the audible range to limit the impact of PWM harmonics on sound quality, as this would significantly degrade the system's performance [1].

Several PWM strategies have been developed to improve the electrical performance, employing novel command waveforms [19]. For the present study, twenty strategies were tested, and eight of them were selected for the perceptual experiment. The selection was based on their timbre singularity to retain strategies that are well distinct from each other. Their characteristic waveforms are shown in Fig. 2. The first three PWM strategies, space vector PWM (SVPWM), discontinuous PWM min (DPWMMIN) and DPWM0, are commonly used in the automotive industry [19]. Their design allows for an optimized use of the direct voltage delivered by the battery, reducing commutation losses. The five other strategies involved in the study (SPPWM and DPWM6 to 9) are new. The authors proposed them for their unique voltage spectra, which result in new noise signatures. However, they may not be more effective than existing strategies in terms of fundamental voltage or commutation loss.

## 2.3. Measured stimuli

The experimental tests and recordings were carried out on the bench presented in Fig. 3. The motor, presented in Section 2.1, can be controlled by various PWM strategies. A Python script generates the PWM command signals which are converted to analog signals by a multichannel sound card (PreSonus Audiobox 1818VSL). After amplification from 5 V to 15 V, the command signals are delivered to the 3-phase inverter (Semikron SEMITEACH IGBT), which generates the PWM voltage signals from the 44 V DC signals provided by the direct source (EA 3050B). The motor runs without load to reduce the level of extra noise from the load machine or the brake.

Electromagnetic noise, including PWM harmonics, is predominant at low rotation frequencies [20]. The speed was therefore set to 300 rpm, which corresponds to an electrical frequency of 5 Hz on this motor. A switching frequency of 1 kHz was used to locate PWM sideband harmonics in the high-sensitivity area of the human ear [21].

A microphone (1/2" PCB 378B02) was placed 25 cm above the motor. The recording environment being non-anechoic, acoustic panels were placed around the test bench to attenuate reverberation and limit

<figure><img src="image_20.png" /><figcaption>Fig. 3. Test bench used for recordings and measurements.</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>Fig. 4. Diagram of the auralization model process.</figcaption></figure>

outside noise. An acquisition frequency of 51.2 kHz was used for all measurements with a DEWESOFT Sirius analyzer.

## 2.4. Virtual stimuli

The auralization model to be validated can be used to synthesize the electromagnetic noise emitted by electric motors with the process presented in Fig. 4, formulated entirely in the time domain. The model is not intended to faithfully reproduce the noise emitted by the motor but to provide a sufficient level of prediction to help make choices during the design process, therefore with a limited number of parameters. In the current industrial context, the objective is not to provide completely realistic noise, as this would require excessive computing resources. The goal is to reproduce the main acoustic phenomena that will influence the perception of the noise emitted by the motor in order to make design choices based on the acoustic and psychoacoustic performance of the motor.

The model was implemented in Python (v3.12) by combining existing methods that were adapted for temporal calculation, including PWM. The current calculation model proposed by Li [14] has been
<figure><img src="image_22.png" /><figcaption>Fig. 5. Experimental setup for TWFRF measurements. Five points on the external surface of the stator were impacted with the hammer to collect the response of the teeth.</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>Fig. 6. SPL spectrum from experiment (a) and simulation (b), with strategy SVPWM.</figcaption></figure>

updated to include the PWM process, with different strategies. The system of differential equations linking voltage and current is solved using the Runge-Kutta solver from the SciPy library [22]. The approach described by Le Besnerais [15] is used to compute the electromagnetic radial excitations from the current. Tooth Wave Frequency Response Functions (TWFRF) are used to compute the stator's external surface displacement from the radial excitations in the air gap. Each TWFRF characterizes the vibration response of the stator's outer envelope to a unitary electromagnetic wave excitation [16] to describe the mechanical behavior of the stator. For this study, the TWFRF were measured on a twin motor with an impact hammer, according to the experimental setup shown in Fig. 5. The motor was mounted with the same mechanical boundary conditions as during the recordings on the full bench. Because the air gap is very thin, the rotor was removed to place the accelerometers on the teeth tips. The measurement was carried out twice, with 12 accelerometers on the even or odd teeth, respectively, to collect all teeth contributions [16]. PWM force harmonics have a spatial order of either 0 (breathing) or 2p (ovalization) [15]. Only those two TWFRF were therefore considered. Since the model is implemented in the time domain, the TWFRFs have been converted to impulse responses using Inverse Fast Fourier Transform (IFFT) and then convolved with the force signals. The resulting noise is finally synthesized under the assumption of Equivalent Radiated Power, considering the stator's outer surface [17].

The noise was simulated with operating conditions identical to those of the test bench recordings. The sampling frequency was set at 96 kHz to ensure an accurate representation of the current calculated by the solver. The computation time was around 100 s to compute signals lasting 3 s for each strategy.

## 2.5. Measured to virtual stimuli comparison

The SPL spectra obtained from measurement and simulation are shown in Fig. 6 for the SVPWM strategy. The overall level is

overestimated by the simulation, with an average error of less than 3 dBA for all strategies.

Below 200 Hz, the model does not perfectly capture spatial harmonics. This can be explained by the approximations made on the motor geometry in the magnetic finite element analysis and by the absence of mechanical vibration sources in the model. The resonance around 800 Hz is correctly captured by the TWFRF method. The frequency of all PWM harmonics is correctly captured by the model, and the amplitudes of the main ones are given in Table 2. Around 1 kHz, simulated harmonics have amplitudes higher than the measured ones, with a maximum difference of 4 dB, but the structure of the group is respected. Around 2 kHz and 3 kHz, the measured noise harmonics present an amplitude asymmetry. For example, the harmonic at $3f_{swi} + 5f_e$ is 9 dB higher than the harmonic at $3f_{swi} - 5f_e$, whereas they are supposed to be identical. This could be explained by a defect in the winding, creating slightly unbalanced currents. Since the model considers an ideal motor (with perfectly balanced currents, without eccentricity or unbalanced magnetic pull), it does not represent this effect, and simulated signals seem purer than measured signals. This could exaggerate certain acoustic phenomena that are smoothed out by the motor's defects in the measurements or, on the opposite, miss some additional noise created by the defects. This explains why it is not possible to compare them directly. Instead, a set of pairwise comparisons can be carried out on the two sets separately, and an Individual Difference Scaling (INDSCAL) analysis can help identify the sound attributes present in both types of signals to compare their perceptual spaces.

The measured and simulated noise spectra have a similar global structure and comparable levels, but clear differences are visible. If the main PWM harmonics are correctly represented, one may wonder whether they are sufficient to ensure the quality of the sound representation, despite the noticeable errors.
<table><caption>**Table 2**
Comparison of the main PWM harmonics amplitude between measurement and simulation.</caption><thead><tr><th></th><th>Measured SPL<br>[dB ref 2 × 10<sup>-5</sup>Pa]</th><th>Simulated SPL<br>[dB ref 2 × 10<sup>-5</sup>Pa]</th></tr></thead><tbody><tr><td>f<sub>swi</sub> ± f<sub>e</sub></td><td>73</td><td>77</td></tr><tr><td>f<sub>swi</sub> ± 3f<sub>e</sub></td><td>82</td><td>83</td></tr><tr><td>f<sub>swi</sub> ± 5f<sub>e</sub></td><td>70</td><td>74</td></tr><tr><td>2f<sub>swi</sub></td><td>68</td><td>70</td></tr><tr><td>2f<sub>swi</sub> - 2f<sub>e</sub></td><td>67</td><td>73</td></tr><tr><td>2f<sub>swi</sub> + 2f<sub>e</sub></td><td>72</td><td>73</td></tr><tr><td>3f<sub>swi</sub> - f<sub>e</sub></td><td>79</td><td>79</td></tr><tr><td>3f<sub>swi</sub> + f<sub>e</sub></td><td>74</td><td>79</td></tr><tr><td>3f<sub>swi</sub> - 5f<sub>e</sub></td><td>78</td><td>78</td></tr><tr><td>3f<sub>swi</sub> + 5f<sub>e</sub></td><td>69</td><td>78</td></tr></tbody></table>

## 2.6. Stimuli preparation

The stimuli duration was limited to 2 s, with a linear fade in and out of 0.2 s. As this study is focused on PWM noise, a high-pass filter ($f_e$ = 100 Hz) was applied to all stimuli. It eliminates spatial harmonics and potential mechanical noise. The overall level difference between measured signals was less than 5 dBA, but all stimuli were presented with an equalized loudness level [23]. Otherwise, loudness differences may disrupt the timbre perception during the similarity evaluation [24].

# 3. Perceptual experiment

## 3.1. Participants

Thirty-eight persons (24 men, 14 women, aged between 20 and 52, average = 26) participated in the experiment. They were 19 students from the Mechanical Engineering Department of INSA Lyon and members of the laboratory. None of them reported hearing impairments, and none of them had previously carried out a sound evaluation on the noise of electric motors. Remuneration was provided to all the students who participated.

## 3.2. Sound evaluation method

The auralization model aims at studying the electromagnetic noise emitted by an electric motor supplied by different PWM strategies. All stimuli are placed in the same operational conditions, so they all have the same harmonic spectrum support [15]. Only the relative amplitudes of the peaks vary from one strategy to another, giving rise to variable acoustic phenomena (e.g., masking, modulation). Timbre is then the only variable element between our stimuli, and a suitable assessment approach should be adopted. It is a multidimensional characteristic, and several types of perceptive experiments can give access to these dimensions. For example, semantic differentials involve providing listeners with a series of scales corresponding to specific characteristics (e.g., low frequency/high frequency, regular/irregular...) [25,26]. Participants'

responses provide profiles of the different stimuli, which can then be linked to an overall assessment of the sounds. However, this method requires that the terms defining each scale be interpreted the same way by all listeners, which is not always guaranteed for complex timbre characteristics. Moreover, these characteristics must have been defined in advance when designing the experiment. In the case of new sets of sounds (as in this study, because new PWM strategies have been used), this may require preliminary experiments. It can lead the experimenter to use many scales, which makes the task longer and more difficult for the listeners. To overcome these limitations, another procedure consists of asking listeners to evaluate the similarities between stimuli presented in pairs. A multidimensional analysis uses this set of similarities to place the stimuli in a space of as few dimensions as possible so that the distances between the stimuli represent the measured dissimilarities. The dimensions of this space can then be related to timbre characteristics used by listeners to compare the stimuli. This makes it possible to find new sound attributes whose relevance had not been anticipated by the test designers [24]. To link these characteristics to the pleasantness of the sounds, the listeners rate this pleasantness in a second experiment, the results of which are explained with the dimensions highlighted in the first similarity evaluation experiment. The experiment performed for this study included pairwise comparisons and pleasantness evaluations for each set successively, as presented in Fig. 7 and described in the following.

### 3.2.1. Similarity evaluation

Similarities perceived between the different strategies were assessed by pairwise comparisons. All possible pairs were presented for each set of stimuli, that being $n(n-1)/2 = 28$ pairs, with $n = 8$ being the number of stimuli per set. Random variables were used to define the order of presentation of the two sets. The stimuli were numbered randomly for each participant, and the successive pairs were created using Ross series [27], which are constructed from double-entry tables. These series avoid successive appearances by separating pairs involving the same stimulus by a number of pairs between $(n-3)/2$ and $(n-1)/2$. They also balance the presentation order within a pair so that a stimulus appears as many times in the first position as in the second position.

Fig. 8 shows the interface developed for the experiment. By clicking on the button, participants heard both sounds of a pair, separated by 0.2 s of silence. They assessed their similarity by adjusting a slider on a non-numerical continuous response scale ranging from “very different” to “very similar” as shown in Fig. 8. The corresponding numerical scale ranged from 0 for “very different” to 100 for “very similar”. Scales defined with a pair of words are more effective than graduated numerical scales because they increase the number of possible values and reduce the bias induced by the numbers [25]. After validation of the answer, the slider was repositioned to the center of the scale to indicate the transition to the next pair, and a progress bar at the bottom of the page was updated. In order to measure the listeners’ consistency during the experiment, the first pair of each sound set was presented again in the last position, which increased the number of evaluated pairs to 29

<figure><img src="image_24.png" /><figcaption>Fig. 7. Schematic representation of the data collected during the test.</figcaption></figure>
<figure><img src="image_25.png" /><figcaption>Fig. 8. Pairwise comparison interface in the first part of each sound set evaluation.</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>Fig. 9. Pleasantness evaluation interface in the second part of each sound set evaluation.</figcaption></figure>

per set. A large variation between the first and second evaluation of the repeated pair could be explained by some training effect, a variation in concentration, or a focus on other attributes of the stimuli.

### 3.2.2. Pleasantness evaluation

Pleasantness evaluations of all stimuli were collected during the second part of the experiment. Listeners used scales from “unpleasant” to “pleasant”, as shown in Fig. 9. The corresponding numerical scale ranges from 0 for “unpleasant” to 100 for “pleasant”. The evaluations of the two sound sets were performed at two different times, after pairwise comparisons of each set. This ensured a sufficient delay (an average of 10 minutes) between the evaluations of the two sets to limit bias in the evaluation process. Participants were instructed to evaluate the stimuli individually. However, the eight stimuli of each set were presented on the same page, with multiple listening allowed. A form of comparative listening may impact the evaluations, but it makes it easier to adjust the evaluation from one stimulus to another and provides more precise answers.

### 3.3. Experimental conditions

The experiment took place in a soundproof booth, and sounds were presented with HD 650 Sennheiser headphones. The output level was calibrated to 80 dBA using a binaural head (Cortex MK2), employing both versions of s1. Since all strategies create harmonics around the same frequencies and with comparable amplitudes, the overall output level is similar for all strategies.

At the beginning of the experiment, each listener received written instructions on the test procedure and the tasks to perform on the interface. Any questions were answered orally by the experimenter. Among the frequently asked questions were the possibility of taking a break, the number of times a stimulus could be listened to, the time limit for the experiment, and the need to memorize the signals. Eight stimuli from both sound sets were played for the participants to familiarize themselves with the type of noise to be evaluated.

## 4. Results

### 4.1. Individual difference scaling

The variation between repeated assessments of the same pair (in the first and last positions of each set) and a hierarchical classification of the correlation between individual similarity answers (using the function “linkage” from the SciPy library [22]) confirmed the agreement between the participants, which does not present any significant disparity.

Individual similarity matrices were analyzed by Individual Difference SCALing (INDSCAL). This method is a type of metric multidimensional scaling, which considers similarity data as relative distances between stimuli. It transforms these distances into positions in a multidimensional space, with R dimensions [28,29]. Stimuli are then located in the so-called perceptual space so that distances between the points correctly represent the measured similarities. INDSCAL analysis has the advantage of taking into account the inter-individual variations, since it considers all individual similarity matrices and not only the average. In order to model the differences in appreciation, each listener is associated with a set of R weighting factors to be applied to the axes of the perceptual space. The model considers distances in the perceptual space as weighted Euclidean distances. For subject i, the distance $d_{jk}^i$ between stimuli j and k is defined as

$$d_{jk}^i = \sqrt{\sum_{r=1}^{R} w_r^i (x_{jr} - x_{kr})^2}, \quad (1)$$

where $x_{jr}$ is the coordinate of the jth stimulus on the rth axis, and $w_r^i$ is the individual weighting factor of the ith subject for the rth axis. The INDSCAL algorithm described by Carroll and Chang was implemented in Python (v3.12) for this study [30]. The stimuli coordinates and individual weights are computed by minimization of least-square differences between measured similarities $\delta_{jk}^i$ and distance estimates $d_{jk}^i$. From the similarity matrices, the algorithm provides a global perceptual space where all stimuli are positioned and a set of individual weighting factors assigned to the participants for each dimension, as illustrated in Fig. 10. The weighting factors adjust the global space to optimize individual responses.

The INDSCAL algorithm [30] minimizes mean squared errors between measured similarities and estimated distances. However, the global error resulting from the analysis is measured by stress, defined as

$$\text{stress} = \frac{1}{I} \sum_{i=1}^{I} \frac{\sum_{j,k} (\delta_{jk}^{i} - d_{jk}^{i})^{2}}{\sum_{j,k} (\delta_{jk}^{i})^{2}}, \quad (2)$$

with I being the number of listeners, so that a perfect representation of the similarities would lead to a null stress value. The scree plots in Fig. 11 show the evolution of stress as a function of the number of dimensions. Adding new dimensions minimizes the gap between measured and estimated similarities, which are represented by distances. The maximum error of 0.03 obtained with one dimension corresponds to an error of around 15% between the similarities and the corresponding distances estimated by the algorithm. The representation is better for the measured sound set. This difference between sets may arise from the simplifying assumptions of the auralization model, especially the lack of potential motor defects. Indeed, the test bench certainly has defects (e.g., eccentricity, unbalanced magnetic pull, winding defect), which can create additional harmonics and therefore modify the acoustic properties of the sound. The auralization is, however, based on the assumption of an ideal model, with fewer harmonics. Some sound phenomena, which were masked or smoothed by the test motor’s defects, can emerge in the simulated sounds. The model therefore tends to exaggerate the sound characteristics of the motor. This explains why it would be necessary to add a dimension to correctly represent the virtual perceptual space. However, to allow comparison, the same number of dimensions is used
<figure><img src="image_27.png" /><figcaption>Fig. 10. Transformation of the similarity matrices into a global perceptual space and some individual weighting factors with INDSCAL algorithm.</figcaption></figure>

<figure><img src="image_28.png" /><figcaption>Fig. 11. Scree plots of the stress resulting from the INDSCAL algorithm.</figcaption></figure>

in both sound sets. Given the limited number of stimuli per set, the study is limited to two dimensions in the following.

The stress function can be distorted by the presence of outliers or if the regression trend is not linear. To avoid these effects, Shepard diagrams are frequently used to assess the quality of the representation provided by the INDSCAL algorithm [44]. Fig. 12 shows evaluated similarities on the x-axis against the corresponding estimations on the y-axis for each sound set. The relationship between measured and estimated similarities seems to be linear, monotonic, and increasing. There are no real outliers, although some points contribute more than others to stress. The points lie close to the regression line, which confirms the INDSCAL representation's validity. However, the regression line is not aligned with the diagonal, which represents the ideal case of equality between measured and estimated similarities. That shows the existence of a bias, which is common to obtain with that kind of method [28]. This

is coherent with the stress scree plot, showing that a third dimension would be necessary to completely describe the perceptual space if more stimuli were available.

The positions resulting from INDSCAL analysis depend on the responses given by the participants. To assess the stability of the estimated positions according to the sample of listeners, 99 % confidence ellipses are calculated using a jackknife method [31]. The corresponding 2D perceptual spaces are represented in Fig. 13. Recorded and simulated stimuli are close together for all tested strategies. The similarities evaluated in both sound sets lead to comparable relative distances, which means that the eight stimuli were appreciated in the same way in both cases. Moreover, the stimuli positions confirm the variety of PWM strategies, which are evenly distributed in the perceptual space.

To ease the comparison between the two sound sets, Fig. 14 shows the stimuli positions obtained with the measured set compared to those obtained with the virtual set for both dimensions. The Pearson correlation coefficient between the two sets is 96 % for dimension 1 and 95 % for dimension 2, which confirms that the perceptual criteria on which the listeners based their judgments are present in both sound sets. The model seems to correctly describe the sound dimensions allowing the differentiation of the different PWM strategies. INDSCAL analysis therefore allows us to conclude in favor of the model's reliability.

## 4.2. Sound dimensions explanation

Participants measured a level of similarity between pairs of stimuli, without indication about which sound characteristics to focus on, or any method to differentiate them. This provided spontaneous responses. For each stimulus, INDSCAL analysis then provided coordinates on two dimensions, which correspond to perceptual attributes used by the listeners to differentiate the stimuli. These axes quantify so-called sound dimensions, which can be identified a posteriori to understand what influences the perception of electromagnetic noise.

<figure><img src="image_29.png" /><figcaption>Fig. 12. Shepard diagrams of the individual similarities, for the measured noise (a) and virtual noise (b).</figcaption></figure>
<figure><img src="image_30.png" /><figcaption>Fig. 13. Superimposition of the perceptual spaces obtained by INDSCAL, with 99 % jackknife confidence ellipses. Markers are full for the measured set, and empty for the virtual set.</figcaption></figure>

Listening to the different stimuli coupled with their position in the perceptual space allows identifying the sound attributes underlying the perception of the stimuli. When listening to the stimuli, it seems that the presence of noise in low frequencies can explain the first dimension. Similarly, the second dimension can be interpreted as the presence of impulsive regular noises, which could be explained by the presence of modulation between the different harmonics. This noise is called rattle noise in the following. Different psychoacoustic metrics were calculated on the measured noise stimuli to try to describe the phenomena identified by listening. The objective metrics were compared with the positions of the stimuli in the perceptual space to identify which of them were the most relevant to explain the relative positioning of the signals.

Low-frequency loudness is defined as the sum of specific loudness below the switching frequency (1 kHz), as

$$LF-Loudness = \sum_{z=0}^{8} N'(z) \qquad (3)$$

where $N'(z)$ is the specific loudness of the $z^{th}$ critical band (z in Bark) according to ISO 532–1:2017. It was computed with the Python library MOSQITO [32]. According to Fig. 15(a), low-frequency loudness presents a Pearson correlation coefficient of 89 % with the first dimension, which confirms the link between the first dimension and the presence of low-frequency noise, as anticipated.

Regarding the second dimension, which was identified as the presence of rattle noise, the analysis of the signals with the classic sound quality metrics was not conclusive. Fluctuation index, roughness, and

tonality [33,34] did not show any significant correlation with the positions of the signals on this axis. This shows the need for new metrics to qualify the noise of electric motors, which presents new specificities compared to the historical noise of internal combustion engines (ICE). A new perceptual experiment was initiated by the authors to characterize the perception of rattle noise and explain its source in the case of electric motor noise [35]. It allowed the development of a new index, called harmonic richness, whose calculation is based on the density of prominent peaks around the switching frequency, as

$$Harmonic richness = 1 - \frac{400}{\sum_{i} E_{i} + 220} \qquad (4)$$

with $E_i$ being the prominence of the $i^{th}$ peak, computed according to the Prominence Ratio procedure [33] within a 300 Hz wideband around the switching frequency. An optimization algorithm was applied to the coefficients of the expression to match the results of the perceptual experiment [35]. The Pearson correlation coefficient between the positions on the second dimension provided by INDSCAL and the harmonic richness values is 81 %, as shown in Fig. 15(b).

### 4.3. Pleasantness evaluation

The 16 stimuli (8 strategies and 2 sets) were assessed by the 38 participants on a pleasantness-unpleasantness scale, but their evaluation is referred to as pleasantness level hereafter for simplicity. The question is whether the type of signal (measured or virtual) induces a change in the pleasantness perception of the different strategies. In other words, is there a significant interaction between the type of signal and the different PWM strategies on the appreciation of PWM noise?

The values averaged over the 38 participants are shown in Fig. 16. Despite the simultaneous presentation of all stimuli, the answers did not show a proper ranking shape (they were not regularly distributed on the scale). All stimuli were perceived as rather unpleasant, and results are similar between both sound sets. The pleasantness levels can be interpreted in light of the two main sound dimensions identified in the previous section to explain why some strategies were preferred over others. The preferred strategies, s2 and s5, are gathered in the lower right corner of the perceptual space, which corresponds to a small low-frequency loudness and a low presence of rattle noise.

To refine the analysis, the 16 responses of each participant were analyzed with the “linkage” function from the Scipy library [22], using Ward’s method on Euclidean distances. According to the dendrogram in Fig. 17, two groups stand out with identical headcounts of 19 listeners. Participants that are joined together below the dashed line were placed in the same group. This suggests that two different types of behavior were adopted. Although INDSCAL analysis showed that all participants heard the same sound characteristics, the classification

<figure><img src="image_31.png" /><figcaption>Fig. 14. Comparison of the INDSCAL positions with both sound sets, for dimension 1 (a) and 2 (b).</figcaption></figure>
<figure><img src="image_32.png" /><figcaption>Fig. 15. Comparison of the INDSCAL positions with the relevant sound quality metrics, for dimension 1 (a) and 2 (b).</figcaption></figure>

revealed a separation in how they evaluated the stimuli's pleasantness level. This could mean that the two groups of listeners made a different judgment on the same sound characteristics. To account for the difference in judgment revealed by the classification, the responses collected in the two groups are separated. The values averaged over the 19 participants of each group are shown in Fig. 18. The range of responses is lower in the second group, with responses extending over 30 % of the scale, compared to 45 % in the first group. The listeners of group 2 globally rated all sounds as unpleasant and used only the lowest part of the scale. For this reason, the correlation between the two sets of sounds seems stronger within group 1.

A Shapiro-Wilk test showed that the sample of evaluations is not from a normally distributed population for either group (p-value < 0.01 in both cases). However, the literature indicates that ANOVA is considered a robust test against the assumption of normality [36,37]. A repeated measures ANOVA (RM-ANOVA) was therefore applied to the participants' responses, taken separately according to their group. It considered

two within-subjects factors: the PWM strategy and the sound set. The results from JASP [38] are presented in Table 3. Within the two groups, the PWM strategy is a significant factor. Since all strategies have singular sound signatures, it seems coherent that their pleasantness levels vary. This significance confirms the timbre variety of the chosen strategies. The analysis also revealed a significant interaction between the sound set and strategy, in group 2 only. For these listeners, the pleasantness differences between strategies depend on the sound set. In order to analyze this effect, a post-hoc test was carried out with Tukey's HSD method [38,39]. The evaluations of the eight strategies were compared across sound sets. The results showed that only s7 (strategy DPWM8) was judged significantly differently depending on the sound set (T=3.5, p<0.001). This suggests that the difference in perception between the two groups is limited to this stimulus.

According to the INDSCAL results, s7 is the highest stimulus on the second dimension, therefore the stimulus with the strongest rattle. Fig. 19 shows that the simulated signal overestimates harmonics around

<figure><img src="image_33.png" /><figcaption>Fig. 16. Mean values obtained for the eight stimuli, depending on the sound set.</figcaption></figure>

<figure><img src="image_34.png" /><figcaption>Fig. 17. Dendrogram based on similarity correlations between listeners.</figcaption></figure>

<figure><img src="image_35.png" /></figure>

<figure><img src="image_36.png" /><figcaption>Fig. 18. Mean values obtained for the eight stimuli, depending on the sound set, within (a) group 1 and (b) group 2.</figcaption></figure>
<table><caption>**Table 3**
RM-ANOVA on the pleasantness responses of the two groups.</caption><thead><tr><th rowspan="2"></th><th colspan="3">Group 1</th><th colspan="3">Group 2</th></tr><tr><th>Set</th><th>Strategy</th><th>Interaction</th><th>Set</th><th>Strategy</th><th>Interaction</th></tr></thead><tbody><tr><td>Total variance</td><td>412</td><td>32,636</td><td>3276</td><td>2647</td><td>5645</td><td>5225</td></tr><tr><td>Degree of freedom</td><td>1</td><td>7</td><td>7</td><td>1</td><td>7</td><td>7</td></tr><tr><td>Adjusted variance</td><td>412</td><td>4662</td><td>468</td><td>32,647</td><td>806</td><td>746</td></tr><tr><td>η<sup>2</sup><sub>g</sub></td><td>0.0040</td><td>0.2429</td><td>0.0312</td><td>0.0341</td><td>0.0701</td><td>0.0652</td></tr><tr><td>F-value</td><td>0.74</td><td>19.91</td><td>2.02</td><td>5.02</td><td>3.95</td><td>4.31</td></tr><tr><td>p-value</td><td>0.4023</td><td>&lt;0.001</td><td>0.0565</td><td>0.0378</td><td>&lt;0.001</td><td>&lt;0.001</td></tr></tbody></table>

<figure><img src="image_37.png" /><figcaption>**Fig. 19.** SPL spectrum from experiment (a) and simulation (b), for the stimulus s7.</figcaption></figure>

the first two multiples of the switching frequency. This can lead to an exaggeration of rattle noise in the simulation, since this specific noise appears in the presence of several closely prominent peaks. Listening to the measured and simulated versions of s7 confirmed this observation: the motor defects, which create asymmetry in PWM harmonics, reduce the presence of rattle. Since the model does not include these defects, it overestimates the presence of rattle noise, resulting in lower pleasantness assessments.

Overall, we can conclude that the differences between measured and simulated stimuli resulted in minimal differences in perception in seven out of eight cases. The difference in perception on s7 for half of the participants may stem from an exaggeration of the rattle noise by the model.

## 5. Conclusion

A perceptual experiment was performed to confirm the reliability of PWM noise representation provided by an auralization model. INDSCAL analysis of pairwise comparisons provided two perceptual spaces, one for measured noise and one for virtual noise. Their similarity accounted for the relevance of the model. The two dimensions of the perceptual space corresponded to the presence of low-frequency noise and rattle noise. The pleasantness evaluation of all stimuli also presented similar results between measured and simulated sounds for seven out of eight strategies. This confirmed the reliability of the auralization from the point of view of sound perception. The model, which already includes new PWM strategies, can therefore be used to assess the sound quality of new electric motor configurations without increasing the production costs. Making the model more complex (e.g., by considering defects) would allow for a better representation of the perceptual space, particularly for signals with a lot of rattle noise, which is currently exaggerated by the model. To ensure the model's complete validity, it would be necessary to study other operating conditions. Indeed, the noise spectrum can vary significantly with the rotation speed and load.

## Funding sources

This work is part of a thesis co-funded by the French National Association for Research and Technology (ANRT), and the software

company Dassault Systèmes. It is performed within the framework of the LABEX CeLyA (ANR-10-LABX-0060) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-16-IDEX-0005) operated by the French National Research Agency.

## CRediT authorship contribution statement

**Salomé Wanty:** Writing – review & editing, Writing – original draft, Visualization, Software, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **Etienne Parizet:** Writing – review & editing, Supervision, Resources, Methodology, Formal analysis, Conceptualization. **Nicolas Totaro:** Writing – review & editing, Supervision, Resources, Methodology, Conceptualization. **Martin Glesser:** Writing – review & editing, Supervision, Resources, Methodology, Conceptualization.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Data availability

Data will be made available on request.

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Author biography

<figure><img src="image_38.png" /></figure>

Salomé **Wanty** was born in France. She received the M. degree in mechanical engineering from University of Technology de Compiègne, France, in 2022. She is currently working towards a Ph.D. degree in acoustics at the Laboratory of Vibration and Acoustics of the National Institute of Applied Sciences in Lyon, France, in partnership with SIMULIA Manatee in Dassault Systèmes.

<figure><img src="image_39.png" /></figure>

Etienne **Parizet** was born in France. He holds a doctorate in Acoustics from Le Mans University (France), received in 1992. He worked for Renault's research department between 1987 and 2000, before joining INSA in Lyon, as a professor. His research focuses on the perception of sound and vibration, as well as noise annoyance.

<figure><img src="image_40.png" /></figure>

Nicolas **Totaro** was born in France. He received the Ph.D. degree in acoustics from INSA Lyon, France in 2004. After graduating, he was a lecturer at the Laboratoire Vibrations Acoustique at INSA Lyon until 2020, when he was appointed full professor. His main research interests are numerical modelling in vibroacoustics and acoustic source identification.

<figure><img src="image_41.png" /></figure>

Martin **Glesser** was born in France. He received the Ph.D. degree in acoustic from University of La Rochelle, France in 2006. After graduation, he worked as a contract researcher at the Laboratoire de Mécanique et d'Acoustique de Marseille, France. From 2008 to 2020 he was an acoustic expert at Alstom Transport. In 2020, he joined Eomys Engineering, that became part of Dassault Systèmes in 2024, as an R&D engineer in the field of noise and vibration of electric motors.