Sound quality modeling and active control for electric buses based on complete ensemble empirical mode decomposition with adaptive noise

Abstract

Sound quality research is an important development direction for electric bus interior noise control, and this paper proposes a coherent method of sound quality evaluation, modeling, prediction and active control. Firstly, an objective evaluation parameter is constructed based on complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) and energy characterization; The structural parameters of the extreme gradient boosting (XGBoost) algorithm is adaptively optimized by particle swarm optimization (PSO) to establish the sound quality prediction model. Secondly, an active sound quality control method based on CEEMDAN and filtered-x least mean square (FxLMS) is presented. Finally, through the training and testing of sound quality data of 64 electric bus noise samples under different working conditions, the acoustic comfort evaluation model based on PSO-XGBoost algorithm is validated to have high prediction and fitting accuracy. In addition, the active controls based on CEEMDAN-FxLMS algorithm are implemented for four signal samples in a semi-anechoic room, and the results indicate that their acoustic comforts are effectively improved. The feasibility of the proposed method for sound quality modeling and active control is thus verified.

Keywords

electric bus sound quality active control CEEMDAN FxLMS

Introduction

In the latest released national standard of limits and measurement methods for bus interior noise in 2024, sound quality was included as a key assessment content, which is the first time in the domestic and international standardization of interior noise limits for buses.¹ Practice has indicated that vehicle interior sound quality is the most direct factor affecting people’s subjective feelings, and an excellent acoustic environment facilitates the physical and mental health of driver and passengers, and significantly improves users' satisfaction with vehicles.² Consequently, the current electric bus noise control from noise reduction to sound quality has important practical significance for building vehicle core competitiveness and seizing the international high-end market.

As the drive motor replaces the traditional engine, the noise inside electric bus is more prominent without the engine noise masking effect. Many noise types that are not easily detected in fuel vehicles are more prominent, such as air conditioning noise, electromagnetic noise, tire road noise, and institutional transmission noise³; the noise signals collected inside the bus are observed after mixing of these noise sources, which makes the in-vehicle noise control problem more complicated. In particular, the characteristic order of motor electromagnetic noise is high, and the subjective feeling is a harsh whistling sound, which has a significant negative impact on the whole vehicle acoustic comfort.⁴ It is foreseeable that the improvement of vehicle interior sound quality is urgent and imperative for the development of electric buses.

Subjective and objective evaluations are the primary problem for sound quality research. Subjective evaluation intuitively reflects the human auditory experience, but there are shortcomings such as cumbersome, time-consuming and many interfering factors, and its results are easily influenced by the evaluator’s physiology and psychology.⁵ Therefore, on the basis of obtaining the noise database, objective parameters and subjective evaluation results are treated as independent and dependent variables respectively, and their functional relationship (i.e., sound quality modeling) is sought through data fitting, which has become a hot topic in vehicle noise research. The current sound quality modeling methods fall into two general categories: the first one is based on mathematical statistics, mainly including multiple linear regression,⁶ Kriging model,⁷ and grey theory⁸; the second one is based on machine learning algorithms to simulate the human neural network’s ability of extracting and processing the information features, which includes back-propagation neural network,⁹ deep learning,¹⁰ extreme gradient boosting (XGBoost),¹¹ and so on. Among them, XGBoost was applied to the vehicle sound quality nonlinear modeling in the recently 2 years.^11,12 For example, Zhang and others utilized the XGBoost algorithm to establish a vehicle interior sound quality model and verified its effectiveness.¹³ It should be noted that XGBoost, firstly proposed by Prof. Chen in 2016, adopts the second-order Taylor expansion to optimize the objective function and introduces a regular term to improve the model’s generalization ability for effectively controlling the overfitting problem.¹⁴ Therefore, fusing intelligent algorithms to fully exploit the high-precision mapping ability of XGBoost algorithm is a promising direction.

In addition, for the objective evaluation of vehicle sound quality, since the vehicle interior noise in the actual driving has significant time-domain variability at different speeds, introducing the time-domain signal processing method to establish new objective parameter is an effective evaluation method, mainly including continuous wavelet transform,¹⁵ Wigner-Ville distribution,¹⁶ empirical mode decomposition (EMD),¹⁷ variable mode decomposition (VMD), and so on. For example, Zuo et al. proposed a feature parameter based on complete ensemble EMD combined with Hilbert transform for hybrid vehicle sound quality modeling.¹⁸ It is worth pointing out that using EMD, the original signal is decomposed into several intrinsic mode functions (IMFs) with the advantage of being completely adaptive to time-varying signals, but it has the drawbacks of mode aliasing and pseudo-modality.¹⁹ To address this problem, Wu and Huang proposed an improved EMD method based on noise-assisted analysis, that is, ensemble empirical mode decomposition (EEMD).²⁰ Subsequently, Torres and others presented a method of complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) to improve the completeness of EEMD and reduce the number of spurious modes.²¹ Relevant reports have performed theoretical analysis, practical signal decomposition and quantitative comparison of these empirical signal processing methods. For example, Zaiem et al. exhibited that CEEMDAN enhances the signal extraction and reduces the residual noise through the non-stationary fault signal processing, which is superior to other empirical methods such as EMD, EEMD, CEEMD, Hilbert vibration decomposition, and VMD²²; Zhang proposed a fault diagnosis method by the enhanced complementary CEEMDAN, and proved that its accuracy is higher than that of VMD through experiment cases.²³ Obviously, CEEMDAN is an advanced empirical signal decomposition method for dealing with time-domain characterized noise signals generated by different working conditions.

The current research on vehicle sound quality mainly focuses on passenger cars,^{2,5–8,11–14} while there are fewer studies on electric buses. With the continuous improvement of people’s demand for in-vehicle auditory comfort, the study of electric bus interior sound quality has gradually become an emerging field. In this paper, we propose a complete coherent method of the sound quality evaluation, modeling, prediction, and control based on CEEMDAN, as seen in Figure 1, with an expectation to providing a key technical support for improving electric bus interior sound quality.

Figure 1.

The strategy for vehicle interior sound quality modeling and active control.

Proposed method of sound quality evaluation and modeling

Calculating objective parameter and establishing subject-object mapping model are two important steps for sound quality prediction. Since the subjective perception of the human ear is affected by the differentiated time-domain characteristics of vehicle interior noise at different speeds. The CEEMDAN handles time-domain nonlinear signals well and reduces the reconstruction error of EEMD effectively and is characterized by high decomposition efficiency.²⁴ Therefore, in this section, CEEMDAN is utilized to extract the features of electric bus interior noise signals under different driving conditions, and energy eigenvalue is defined as an objective parameter for sound quality evaluation. In addition, aiming at improving the sound quality prediction accuracy, particle swarm optimization (PSO) algorithm with fast search speed is combined with XGBoost. Its realization process is drawn in Figure 2, and the specific steps are described below.

Step 1: Import the sound pressure time-domain data of original signals and set the parameters including standard deviation, the numbers of decomposition layers, and noise additions, then perform CEEMDAN to obtain the IMFs and residual components. Its decomposition principle is given as:

Figure 2.

Block diagram of objective evaluation and sound quality modeling.

First, assume that the added noise is Gaussian white noise $w_{i}$ (n) with a standard deviation $ε_{0}$ is added to the original signal $x (n)$ , where $i$ = 1, 2, …, $I$ , and $I$ is the number of noise additions. What’s more, the repeated addition of noise followed by decomposition and averaging effectively mitigates noise interference while preserving the intrinsic signal characteristics. Accordingly, the constructed signal is expressed as

x_{i} (n) = x (n) + ε_{0} w_{i} (n)

(1)

Second, any time-domain signal is assumed to be decomposed into a combination of IMFs. Therefore, performing an I-times decomposition of the signal $x_{i} (n)$ yields the first order component, that is,

{\bar{IMF}}_{1} (n) = \frac{1}{I} \sum_{i = 1}^{I} {IMF}_{i} (n)

(2)

Then, the first order residual component obtained in the first decomposition stage (k = 1) is expressed as

r_{1} (n) = x (n) - {\bar{IMF}}_{1} (n)

(3)

The second modal function is derived for the new residual signal $r_{1} (n)$ + $r_{1} N_{1} w_{i} (n)$ by performing EMD as:

{\bar{IMF}}_{2} (n) = \frac{1}{I} \sum_{i = 1}^{I} N_{1} (r_{1} (n) + r_{1} N_{1} w_{i} (n))

(4)

And so on, when k = 2, 3, …, K, the kth residual signal is

r_{k} (n) = r_{k - 1} (n) - {\bar{IMF}}_{k} (n)

(5)

By executing EMD on $r_{k} (n)$ , the (k+1)th modal function is obtained as

{\bar{IMF}}_{k + 1} (n) = \frac{1}{I} \sum_{i = 1}^{I} N_{k} (r_{k} (n) + r_{k} N_{k} w_{i} (n))

(6)

Until the residual signal is unable to be decomposed further, the final residual is

R (n) = y (n) - \sum_{k = 1}^{K} {\bar{IMF}}_{k} (n)

(7)

Thus, the original time-series signal is represented as:

y (n) = \sum_{k = 1}^{K} {\bar{IMF}}_{k} (n) + R (n)

(8)

The aforementioned assumptions and principles render CEEMDAN effective for decomposing time-domain signals; however, its decomposition performance may be compromised when handling highly complex and rapidly varying signals under certain extreme conditions.

Step 2: Calculate and normalize the energy eigenvalues. Since the energy feature is closely related to human auditory perception, the human ear has different sensitivity to the energy distribution at different frequencies. Accordingly, the above method of CEEMDAN is used to decompose the noise signal into multiple IMFs and residual components, and the energy values of each IMF and residual component are calculated using the following formula.

E_{k} = \sum_{t = 1}^{T} {| f_{k} (t) |}^{2}

(9)

where

f_{k} (t)

denotes the value of the kth IMF in the sequence and T is the signal length. At the same time, in order to facilitate subsequent calculation and statistics, the obtained energy values are normalized, that is.

p_{k} = \frac{E_{k}}{\sum_{k = 1}^{K} E_{k}}

(10)

Step 3: Obtain the objective parameters of all acoustic signals through the mentioned CEEMDAN-energy. Combining with the subjective evaluation results, the sound quality database is established and divided into training and test sets.

Step 4: Set the basic parameters of PSO, namely the maximum iteration number, population number, inertia weight, individual and population learning factors, and the main optimization parameter range of XGBoost.

Step 5: Update particle’s position and velocity by the following formulas⁹:

V_{i d}^{k + 1} = ω V_{i d}^{k} + c_{1} r_{1} (P_{i d}^{k} - X_{i d}^{k}) + c_{2} r_{2} (P_{g d}^{k} - X_{i d}^{k})

(11)

X_{i d}^{k + 1} = X_{i d}^{k} + V_{i d}^{k + 1}

(12)

where

ω

is the inertia factor; i = 1,2,…,n, and n is the number of particles; d = 1,2,… ,D, and D is the dimension of the particle swarm; k represents the number of current iterations; V_id and X_id are the velocity and positions of the particle id, respectively, and P_id and P_gd stands for the optimal positions (i.e., the best fitness) experienced by the particle id and population gd; c₁ and c₂ are the learning factors; and r₁ and r₂ are the random numbers distributed in the interval of [0, 1].

Step 6: Construct the initial XGBoost model based on each particle position, and take the mean square error (MSE) between the model’s predicted and test values as the particle fitness function

M S E = \frac{1}{N} \sum_{i = 1}^{N} {(y_{i} - {\bar{y}}_{i})}^{2}

(13)

where N is the number of samples;

y_{i}

and

{\bar{y}}_{i}

represent the

i

th predicted and test values, respectively.

Step 7: Calculate the fitness of each particle. Each particle represents a potential solution that is utilized to build the corresponding XGBoost model, which is trained and tested using the data set, whereby the fitness of each particle is recorded accordingly; each particle’s current fitness is compared with its historical best fitness, and if it is better than the individual historical best, it is replaced by the current particle and its parameters are recorded; subsequently, the population’s historical best is compared, and if it is better than the population’s historical best, the current particle is replaced by the population historical best, and its parameters are recorded.

Step 8: Determine whether the maximum iteration number or accuracy requirement are reached, if the conditions are met, then stop iteration; otherwise, perform Step 9.

Step 9: Update the position and velocity of each particle according to equations (11) and (12) to generate new population; return to Step 7 for the next round of iterations.

Step 10: Take the optimal position of particle swarm as the structural parameters of XGBoost, establish the sound quality evaluation model based on PSO-XGBoost, and evaluate its prediction accuracy and fitting effect by test data.

Accordingly, it is worth noting that the final predicted value, specifically referring to acoustic comfort in this study, can be mathematically expressed as:

y_{i} = \sum_{k = 1}^{K} f_{k} (x_{i})

(14)

where

f_{k} (x_{i})

is the prediction score of the kth tree classifier to

x_{i}

for XGBoost model.

Proposed method of active sound quality control

In order to construct the technical connection between sound quality evaluation model and its active control, two thoughts are raised: firstly, the CEEMDAN-based signal feature decomposition method is presented for ASQC; secondly, calculate the objective energy eigenvalues of the signals after being controlled, and bring them into the established model to predict and evaluate sound quality.

At present, FxLMS has become the mainstream algorithm of active control due to its advantages of simple implementation and small computation amount and is widely used in the field of vehicles.^25–27 Aiming at the time-domain feature signals, this section proposes an ASQC method combining CEEMDAN with FxLMS algorithm, that is, the signals are decomposed by CEEMDAN, and then the obtained IMFs are controlled by the FxLMS algorithm, the principle of which is depicted in Figure 3, and its specific process is described as follows.

Figure 3.

ASQC based on CEEMDAN-FxLMS.

Firstly, the signal sample x(n) is subjected to feature decomposition by CEEMDAN to obtain N sub-signals labeled as [x₁, x₂, …, x_N], namely,

x (n) = \sum_{i = 1}^{N} x_{i}

(15)

where

x_{i}

is the ith sub-signal, namely, the ith IMF.

Secondly, after filter control for each sub-signal, the corresponding output signal y(n) is obtained as

{\begin{cases} y_{1} = x_{1}^{T} W_{1} (n) \\ y_{2} {= x^{T}}_{2} W_{2} (n) \\ \dots \\ y_{N} {= x}_{n}^{T} W_{n} (n) \end{cases}

(16)

y (n) = \sum_{i = 1}^{N} y_{i} s (n)

(17)

y(n) is superimposed with the desired signal d(n) to form the error signal e(n), which is the input to the FIR filter, denoted as

e (n) = d (n) + y (n)

(18)

Finally, the following iterative formula of weight coefficient vector is adopted:

W (n + 1) = W (n) - 2 μ e (n) r (n)

(19)

where

μ

is the iteration step and r(n) stands for the filtered-x signal. In the ANC system, the real-time noise control is realized by updating the filter weight.

The above theoretical analysis reveals that the computational complexity of the standard FxLMS algorithm, serving as the benchmark, is 2M+2L+1, where M and L denote the filter orders of the secondary channel and the ANC system, respectively. In contrast, the proposed method involves decomposing the input signal into N IMFs via CEEMDAN and processing them in parallel with the FxLMS algorithm, resulting in a computational complexity of 2NM+2NL+1. This demonstrates that the number of decomposition layers is a critical factor influencing computational complexity, and thus its selection requires a balance between decomposition accuracy and computational efficiency.

The next work is to verify the feasibility of the above proposed CEEMDAN-based methods of sound quality modeling and active control through real vehicle noise acquisition tests, signal feature decomposition and computation, and data analysis.

Modeling and active control of electric bus interior sound quality

Noise signal acquisition tests

To secure the representatives of the interior noise environment at quasi-steady speeds, a total of eight different kinds of electric buses, numbered from A to H, were selected for the test.²⁸ According to the domestic standard of limits and measurement methods for bus interior noise (GB/T 25982-2024), the seats from driver and rear were selected as the measurement points in Figure 4 with the air conditioning on and off as two different working conditions, which is to focus on the noise at the long-time driver’s ear, and the motor and drive axle noise in the rear, respectively; Since 30 km/h and 50 km/h are representative and commonly used driving speeds on urban roads, whose noise characteristics broadly affect the subjective perception of passengers, the prototypes run at these two quasi-steady speeds with the usual deviation of ±1 km/h on the same test track. To better replay the noise received by human ears in the subsequent subjective evaluation, a portable Squadriga II binaural acquisition system from HEAD acoustics and head-mounted BHS II high-fidelity headphones were applied to collect real-time interior noise signals. The test scene and instruments are shown in Figure 5. Because the more time-consuming the evaluation process, the lower the evaluation accuracy, 64 electric bus noise samples with a duration of 5 s were finally obtained after screening and editing. Note that the acquisition instruments simultaneously obtained two time-domain signals from the left and right ears for each sample, and the waveforms and spectra of sample 1, for example, are displayed in Figure 6, from which it can be seen that the noise signals from two ears have the same time-domain waveforms and characteristic spectra, with energy primarily concentrated in the frequency band below 1000 Hz.

Figure 4.

Distribution of measurement points inside an electric bus.

Figure 5.

The test scene and instruments.

Figure 6.

Signal characteristics of sample 1: (a) Time domain waveform; (b) A-weighted spectrum.

Subjective evaluation tests

To avoid the influence of the information leakage such as measurement points, speed, and working conditions on the subjective evaluation of sound quality, all the noise samples were coded and randomly sorted. The results are listed in Table 1.

Table 1.

The codes and numbers of all noise samples.

AFD3	AFD5	AFM3	AFM5	AND3	AND5	ANM3	ANM5
49	18	9	37	1	32	13	21
BFD3	BFD5	BFM3	BFM5	BND3	BND5	BNM3	BNM5
44	27	36	20	46	8	30	41
CFD3	CFD5	CFM3	CFM5	CND3	CND5	CNM3	CNM5
24	19	10	26	54	48	51	2
DFD3	DFD5	DFM3	DFM5	DND3	DND5	DNM3	DNM5
31	55	40	4	62	12	29	57
EFD3	EFD5	EFM3	EFM5	END3	END5	ENM3	ENM5
3	34	15	22	58	11	35	63
FFD3	FFD5	FFM3	FFM5	FND3	FND5	FNM3	FNM5
60	43	7	52	5	16	64	25
GFD3	GFD5	GFM3	GFM5	GND3	GND5	GNM3	GNM5
23	33	28	61	17	59	39	53
HFD3	HFD5	HFM3	HFM5	HND3	HND5	HNM3	HNM5
14	47	6	45	38	50	42	56

Note: The code meaning takes serial number 1 as an example: AND3 represents the noise samples obtained from A electric bus under the conditions of turning on the air conditioner and driver’s position with the speed of 30 km/h.

In this case, acoustic comfort is taken as the evaluation index, and a subjective evaluation system based on rank score comparison (RSC) is developed, in which the index is divided into 5 levels: poor, accepted, satisfied, good, and excellent comfortable, and two comfort values are assigned to each level, thus constituting a range of acoustic comfort values with the interval of [1,10]; The jury of this subjective evaluation tests were composed of NVH engineers, drivers and acoustic experts with rich experience in contact with bus noise, totaling 22 people with a male to female ratio of 9:2 and the age distribution of 24–52 years. The specific evaluation process is illustrated in Figure 7, and the final acoustic comfort results of all evaluators for the 64 noise samples were acquired.

Figure 7.

Subjective evaluation process based on RSC.

In order to ensure the validity of the evaluation data, Spearman was utilized to calculate the correlation between the subjective data of the 22 evaluators, and the calculation results are plotted in Figure 8. It is observed that the average correlation coefficients of all the evaluators are not less than 0.7, indicating that the subjective evaluation based on the RSC has a favorable consistency. After averaging, the acoustic comfort values of 64 samples are obtained and listed in Table 2.

Figure 8.

Correlation analysis of subjective evaluation results.

Table 2.

Acoustic comfort values of 64 noise samples.

1	2	3	4	5	6	7	8
5.17	3.33	6.42	5.33	4.67	3.25	2.58	3.42
9	10	11	12	13	14	15	16
5.75	5.25	4.58	4.83	4.42	4.67	2.33	5.42
17	18	19	20	21	22	23	24
3.17	7.25	2.92	7.08	4.92	2.08	5.33	3.84
25	26	27	28	29	30	31	32
2.33	3.08	3.92	2.5	4.25	5.08	3.92	2.5
33	34	35	36	37	38	39	40
2.42	4.25	1.75	7.5	5.08	1.42	2.58	6.75
41	42	43	44	45	46	47	48
6.92	2.5	4	5.17	2.33	3.84	4.83	3.08
49	50	51	52	53	54	55	56
7.67	4.12	3.67	2.58	2.67	4.25	5.58	1.92
57	58	59	60	61	62	63	64
5.33	3.58	2.58	5.32	4.33	4.84	1.58	3.13

Calculation of objective parameters

Based on the time-domain analysis results in Figure 6, this section calculates the objective eigenvalues of all noise signals based on the CEEMDAN-energy method using the noise signal from left ear as the representative. To improve the computing speed, the sample is divided into five sub-signals with a duration of one second, and each sub-signal is decomposed to obtain seven IMFs and one residual component. Accordingly, a noise signal is decomposed and 40 energy eigenvalues are generated as [x₁, x₂, …, x₄₀]. Taking sample 1 as an example, its noise signal decomposition and energy calculation process are presented in Figure 9.

Figure 9.

Objective parameter calculation based on CEEMDAN-energy.

The objective energy eigenvalues of all noise samples are gained by Figure 9 and their results are listed in Table 3. It is stated that after several signal decompositions on the noise samples, IMF energy calculations, and their distributional consistency analyses, the CEEMDAN parameters for this case include the numbers of decomposition layers, added noise, and the standard deviation of 7, 100, and 0.02, respectively. The selection of the decomposition layers was determined by balancing decomposition accuracy and computational efficiency. Figure 10(a) illustrates the energy characteristics of IMFs for sample 1 under different decomposition layers. As observed, the energy of IMFs decreases with increasing decomposition layers, and when it reaches 7, the energy of subsequent IMFs declines significantly, providing no meaningful feature information. Consequently, the number of decomposition layers is chosen as 7. In order to assess the time-varying nature of the energy eigenvalues, sample 1 is also taken as an example, and the decomposed IMFs under the 5-s time are plotted in Figure 10(b), which shows that the significant energy distributions change with time, reflecting the time-varying characteristics of noise signals in the decomposition process.

Table 3.

Objective parameters for all noise samples.

Order	x ₁	x ₂	x ₃	x ₄	x ₅	…	x ₃₉	x ₄₀
1	1.01436E-05	6.29E-06	8.55E-05	0.000534	0.00441436	…	0.599764	0.386397
2	5.52113E-05	0.000147	0.000992	0.011933	0.00914969	…	0.385227	0.548833
3	4.57878E-05	0.000443	0.001398	0.001429	0.00446980	…	0.372927	0.596634
4	8.28632E-06	8.76E-06	3.35E-05	0.000493	0.00089351	…	0.721475	0.243843
5	1.62613E-06	3.71E-06	1.57E-05	0.000128	0.00029696	…	0.676096	0.320096
6	6.00145E-05	0.000299	0.002223	0.004541	0.046332986	…	0.244779	0.02343
7	5.10278E-06	1.29E-05	0.000165	0.000649	0.00148219	…	0.593707	0.38077
8	3.46142E-05	2.53E-05	0.000158	0.004092	0.002058878	…	0.706383	0.288157
9	2.45993E-05	8.04E-06	3.96E-05	0.000904	0.014392159	…	0.61995	0.374995
10	4.81421E-05	1.68E-05	9.57E-05	0.003687	0.003524272	…	0.279748	0.677628
11	2.77264E-05	0.000328	0.001807	0.002297	0.003817777	…	0.180048	0.793712
12	2.22841E-05	8.24E-06	4.78E-05	0.00051	0.000615591	…	0.579358	0.405745
13	8.68755E-05	1.86E-05	6.91E-05	0.000738	0.004894307	…	0.69515	0.290533
14	5.44224E-05	7.14E-05	0.000327	0.000346	0.122642587	…	0.056317	0.01505
15	0.000234875	0.000216	0.001241	0.001524	0.003278821	…	0.097346	0.86662
16	1.63172E-05	8.96E-06	2.72E-05	0.000161	0.000532788	…	0.830067	0.158902
17	3.23161E-05	1.43E-05	0.000103	0.000584	0.001864988	…	0.674145	0.318015
18	3.41265E-06	2.95E-06	3.94E-05	0.000237	0.001712107	…	0.657552	0.339642
19	2.63307E-05	4.72E-05	0.000251	0.001803	0.00214206	…	0.623725	0.360316
20	9.78313E-05	1.2E-05	2.67E-05	0.000308	0.000961825	…	0.514419	0.475305
21	2.25862E-06	8.03E-06	0.000117	0.001054	0.006508455	…	0.450954	0.530511
…	…	…	…	…	…	…	…	…
50	6.03527E-06	1.11E-05	3.18E-05	0.000528	0.000554387	…	0.479217	0.507933
51	2.74391E-05	3.35E-05	0.000233	0.008823	0.005451378	…	0.273606	0.680448
52	3.64053E-05	2.34E-05	6.9E-05	0.000952	0.001446354	…	0.380853	0.523646
53	4.30279E-06	1.18E-05	0.000113	0.002012	0.003063324	…	0.598164	0.386097
54	0.000125584	4.2E-05	0.000182	0.00102	0.00168085	…	0.443344	0.548386
55	5.94127E-06	3.81E-06	3.89E-05	0.000829	0.00046076	…	0.693023	0.297449
56	5.69016E-05	0.000865	0.010246	0.008097	0.009349688	…	0.137733	0.050833
57	6.79102E-05	1.71E-05	7.57E-05	0.000956	0.001725893	…	0.315366	0.667771
58	1.53245E-05	0.000301	0.001234	0.001295	0.002671101	…	0.3457	0.622312
59	1.93491E-05	9.99E-06	8.54E-05	0.000593	0.03514278	…	0.810067	0.137038
60	4.98231E-06	4.44E-06	1.01E-05	6.53E-05	0.00015052	…	0.577756	0.312843
61	1.44938E-06	6.06E-06	4.62E-05	0.000551	0.00370729	…	0.700658	0.291527
62	5.39211E-06	9.45E-06	7.27E-05	0.000377	0.00100828	…	0.699833	0.294395
63	0.000198389	0.000402	0.002557	0.003725	0.00543617	…	0.211458	0.728114
64	2.19966E-05	1.38E-05	9.05E-05	0.000521	0.00178147	…	0.22424	0.531392

Figure 10.

The relationship between energy value and decomposition layers (a) and time (b).

Correlation analysis of subjective and objective evaluation

To quantitatively assess the relationship between the energy characteristics of noise samples in Table 3 and their corresponding acoustic comfort levels in Table 2, the Spearman coefficient was employed to plot the correlation distribution between them, as shown in Figure 11. The visualization results indicate that the correlation of different energy eigenvalues with subjective evaluation varies, ranging between −0.6 and 0.4. Notably, 70% of the energy eigenvalues exhibit moderate correlations with subjective acoustic comfort, effectively encompassing and capturing the most relevant features for sound quality evaluation.

Figure 11.

Results of correlation analysis.

Acoustic comfort modeling

The subjective and objective evaluation results in Tables 2 and 3 constitute the sound quality database of 64 noise samples in this case, and the energy eigenvalues and acoustic comfort are taken as the independent and dependent variables of the model, respectively. Aiming at improving the robustness and generalization of the electric bus interior sound quality evaluation model, the sample data amount of 7 sample buses, that is, the data of the first 56 noise samples, was used as the training set, accounting for 87.5%, and the data amount of one sample bus, namely, the data of the last eight samples, was taken as the test set. Otherwise, it is found that according to Table 1, the test samples are from different speeds, positions and working conditions, which contributes to assessing the generalization ability of the sound quality evaluation model. Notably the prediction accuracy and performance of the XGBoost model are jointly influenced by the combination of structural parameters, the optimization ranges of structural parameters were scheduled near the default values of the XGBoost, which is detailed in Table 4, and the algorithms were optimized within this structural parameter range.

Table 4.

Parameter meaning and optimization range of the XGBoost.

Parameter	Physical meaning	Optimization range
a	Number of decision trees	(10, 300)
b	Iterative step size of the decision tree	(0.01, 0.3)
c	Maximum depth of the specified tree	(1, 6)
d	Proportion of random samples per tree	(0.5, 1)
e	Proportion of feature random sampling	(0.5, 1)
f	Regularization term of L₁	(0, 10)
g	Regularization term of L₂	(0, 10)
h	Minimum loss value for leaf node branch	(0, 5)
i	Maximum increment of tree weight estimation	(0, 5)
z	Minimum sample weight on a leaf node	(0, 10)

The PSO parameters are provided as follows: the numbers of particle swarms and iterations are 40 and 200, respectively, and the initial values of weights and learning factors are 0.8. According to the sound quality modeling process in Figure 2, the joint algorithm based on PSO and XGBoost is compiled and run in MATLAB. The model satisfies the iteration conditions after training the sound quality data of 56 noise samples, and the optimal model parameters based on PSO-XGBoost algorithm are gained and listed in Table 5. It should be noted that after Ackely function test and fitness calculation, the search process is close to the global optimal solution and convergence, as illustrated in Figure 12.

Table 5.

Optimal structural parameters of the PSO-XGBoost model.

Parameter	a	b	c	d	e	f	g	h	i	z
Value	175	0.251	5	0.771	1	2.216	0.052	0.981	3.14	5.168

Figure 12.

Results of global search: (a) Ackely function; (b) convergence.

In order to assess the regression model’s prediction accuracy, the average relative error (ARE), root mean square error (RMSE) and correlation coefficient (R²) are adopted as the evaluation indexes, in which the smaller values of ARE and RMSE, the higher the prediction accuracy of the regression model. In addition, the higher R² indicates the better fitting effect. The formulas for these indexes are expressed as follows, where the meanings of $y_{i}$ and ${\bar{y}}_{i}$ are the same as that of the equation (13).

A R E = \frac{1}{N} \sum_{i = 1}^{N} | \frac{{\bar{y}}_{i} - y_{i}}{{\bar{y}}_{i}} | \times 100 %

(20)

R M S E = \sqrt{\frac{\sum_{i = 1}^{N} {({\bar{y}}_{i} - y_{i})}^{2}}{N}}

(21)

R^{2} = 1 - \frac{\sum_{i = 1}^{N} {({\bar{y}}_{i} - y_{i})}^{2}}{\sum_{i = 1}^{N} {(\vec{y} - y_{i})}^{2}}

(22)

where N is the number of samples, and N = 8 in this case;

\vec{y}

is averaged for the dependent variable.

Based on the test set, the evaluation indexes of this regression model are calculated by equations (20) and (22), and their prediction values, accuracy, and fit correlation are described in Table 6 and Figure 13.

Table 6.

Test and predicted values of the PSO-XGBoost model.

Order	57	58	59	60	61	62	63	64
Test	5.33	3.58	2.58	5.32	4.33	4.84	1.58	3.13
Prediction	5.15	3.57	2.62	5.27	4.49	4.79	1.70	3.12

Figure 13.

Prediction results of the PSO-XGBoost model: (a) Percentage error; (b) Fitting effect.

In terms of prediction accuracy, Figure 13(a) illustrates that the maximum percentage error and ARE of the PSO-XGBoost model are 7.59% and 2.35% with the RMSE of 0.0995, while its R² is 0.9979 as calculated by equation (22) and seen in Figure 13(b).

To validate the superiority of the constructed PSO-XGBoost model in terms of prediction and fitting accuracies, three commonly utilized sound quality modeling algorithms, namely, BPNN,⁹ long short-term memory (LSTM), and standard XGBoost,¹¹ are introduced. Similarly, the first 56 noise samples in Tables 2 and 3 serve as training data, as well as the last eight samples as test data. After writing the code of each algorithm in the MATLAB platform, importing the sound quality data, and executing multiple training and running, the structural parameters set for each model are as follows: for BPNN, the numbers of neurons in the input, hidden and output layers are 40, 48, and 1, respectively, and their corresponding activation functions are ReLU and Linear, with the learning rate and the iteration number of 0.001 and 500; for LSTM, the number of hidden units, Dense layer neurons are 128 and 16, respectively, and the learning rate, number of iterations, and regularization factor are 0.001, 500, and 0.0001; for standard XGBoost, the values of structural parameters a to z in Table 4 are 200, 0.05, 5, 0.8, 0.8, 5, 5, 1, 1, and 1 in turn. Ultimately, the prediction results for the three models are listed in Tables 7 and 8.

Table 7.

Relative errors of three sound quality models.

Model	57	58	59	60	61	62	63	64
BPNN	2.63%	7.26%	10.47%	4.89%	6.24%	8.06%	3.16%	8.63%
LSTM	2.25%	9.78%	0.39%	5.26%	2.77%	7.44%	10.76%	0.64%
XGBoost	1.69%	3.07%	1.94%	3.76%	5.54%	3.72%	10.76%	2.56%

Table 8.

Prediction results of three sound quality models.

Model	ARE	RMSE	R²
BPNN	6.42%	0.2569	0.9591
LSTM	4.91%	0.2204	0.9699
XGBoost	4.13%	0.1533	0.9854

Comparison of Table 8 and Figure 13 reveals that the sound quality models with low to high for ARE and RMSE, and high to low for R² are PSO-XGBoost, XGBoost, LSTM, and BPNN, respectively, fully justifying that on the basis of the subjective and objective evaluation data of this case, the established PSO-XGBoost model with the structural parameters in Table 5 has the best acoustic comfort prediction accuracy and fitting effect, which serves as an evaluation model of electric bus interior sound quality in the uniform driving conditions.

Active control of acoustic comfort

In order to improve vehicle interior sound quality, in this section, signal samples from different buses with low acoustic comfort values, labeled as 35, 38, 45, and 63 in Table 1, are selected for active control using CEEMDAN-FxLMS constructed in Figure 3. The controlled error signals are treated as new samples, and their objective parameters are computed by CEEMDAN-energy of Figure 2 and brought into the above established PSO-XGBoost model, whereby the acoustic comfort values of these samples are predicted without the need to organize further subjective evaluation tests.

Similarly, for each noise sample with a duration of 5 s after performing CEEMDAN, a total of 40 sub-signals were obtained by decomposition, and the code based on CEEMDAN-FxLMS was programmed in the Simulink of MATLAB. To validate the effectiveness and real-time performance of the proposed algorithm, an ANC platform was constructed and illustrated in Figure 14, which consists of several sensors, loudspeakers and an Arduino-DUE-based controller with the capable of real-time noise signal acquisition, processing, and control. Further added is that the constructed ANC test platform operates in a precision-grade semi-anechoic room conforming to ISO 3745, with the background noise value of ≤15 dB(A) and the cut-off frequency of ≤80 Hz, which effectively eliminates the external ambient noise interference to ensure the consistency of the ANC test conditions and the reproducibility of the test process. As a result, by conducting ANC tests with the sampling frequency of 50 kHz in the semi-anechoic room, the noise signals (Figure 15), spectra (Figure 16), and their corresponding ASPLs (Figure 17) of the four samples before and after controls were obtained. These ANC physical test data provide an important basis for evaluating the noise reduction effect and real-time response performance of the proposed algorithm, proving its potential and superiority in different noise environments.

Figure 14.

ANC platform and tests.

Figure 15.

Signal comparison results before and after controls for the samples of 35, 38, 45, and 63 corresponds to (a) to (d).

Figure 16.

Spectrum comparison results before and after controls for the samples of 35, 38, 45, and 63 corresponds to (a) to (d).

Figure 17.

Comparative results of ASPL before and after controls.

The ANC physical test results demonstrate that the four different noise signals after control have significant noise reduction effects compared with the original signals. Specifically, Figure 15 displays that the error signals converge to narrow-band waveforms with substantially reduced amplitudes after control implementation. This is further corroborated by Figure 16, where the controlled spectra exhibit general attenuation in mid-low frequency components, and quantitatively confirmed by the decreased ASPL values presented in Figure 17. Consequently, these consistent ANC test findings collectively validate that the proposed CEEMDAN-FxLMS-based algorithm maintains both effectiveness and robustness in noise suppression across varying acoustic environments.

To quantitatively contrast sound quality, the objective parameters of the error signals after controls were calculated, and their specific energy eigenvalues are given in Table 9. When these parameters are directly brought into the PSO-XGBoost model established above, with the combination of the parametric structural parameters listed in Table 5, the acoustic comfort values are obtained, and their comparison results are illustrated in Figure 18.

Table 9.

Objective parameter values for the four error signals.

Order	x ₁	x ₂	x ₃	x ₄	x ₅	…	x ₃₉	x ₄₀
35	1.77547E-06	1.12E-05	9.88E-05	0.000250	0.00039406	…	0.026824	0.969078
38	4.71122E-05	0.000197	0.000630	0.000732	0.60234637	…	0.038383	0.013073
45	2.34592E-05	0.000524	0.002677	0.006955	0.14334242	…	0.080925	0.026739
63	2.56865E-06	2.35E-05	0.000260	0.000317	0.00041483	…	0.007168	0.989955

Figure 18.

Comparison results of acoustic comfort before and after controls.

In summary, the signal samples controlled by CEEMDAN-FxLMS have greater reduction in spectral amplitude, which is consistent with the predicted improvement in acoustic comfort. It is evident that for the objective evaluation data of sound quality decomposed by CEEMDAN, the high-precision acoustic comfort prediction model based on PSO-XGBoost is an alternative to the repetitive subjective evaluation tests to reduce the cost of human and material inputs. The consistent results of qualitative spectral and quantitative acoustic comfort prediction demonstrate the feasibility and effectiveness of the proposed method of ASQC, as well as the robustness of the sound quality prediction model constructed by the optimal parameter combination.

Discussions

In this paper, the effectiveness of the proposed CEEMDAN-based coherent strategy of sound quality modeling and active control for acoustic comfort prediction and noise reduction is verified through data analysis and ANC physical tests using electric bus noise samples under quasi-steady speed conditions as an application case, and the following is a further discussion of the performance analysis of the proposed method in terms of robustness as well as convergence.

(1) Comparative analysis with state-of-the-art approaches in sound quality modeling demonstrates that the CEEMDAN-PSO-XGBoost-based model achieves superior prediction accuracy (ARE = 2.35%) and fitting performance (R² = 0.99) compared to recent reports,^29–31 but the algorithmic modeling process involves data processing, parameter setting, etc., which makes it difficult to compare the superiority from the theoretical aspect; it is evident that the method in this paper provides a tool for sound quality prediction with the advantages of innovation and accuracy, yet due to the pre-processing of CEEMDAN as an objective evaluation, there is space for improvement in computational efficiency. In addition, to prevent the sound quality modeling process from falling into overfitting, PSO was utilized to optimize the regularization terms L₁ and L₂ inherent to the XGBoost algorithm, which are mainly for controlling the model complexity and reducing overfitting. Figure 19 reveals that with the PSO iterations, these two regularization parameters are constantly changing, and finally converge to the fixed values listed in Tables 5, that is, L₁ = 2.216 and L₂ = 0.052, which indicates that the established PSO-XGBoost model achieves the regional optimal solution.

(2) The ANC algorithm needs to track the time-varying noise input signals to adaptively adjust the controller. To evaluate the real-time tracking performance of the proposed CEEMDAN-FxLMS-based ANC algorithm for four distinct noise signals, the mean square error (MSE) is introduced as an evaluation parameter,³² defined as $φ_{M S E} = 10 \log_{10} {{e (n)}^{2}}$ , with smaller values indicating better tracking performance. As clearly demonstrated in Figure 20, the established ANC model initially exhibits relatively large MSE values during the early iteration stages, while all MSE curves progressively stabilize with increasing iteration counts. Overall, the proposed ANC method performs consistent convergence stability across diverse noise environments, confirming its adaptive performance.

Figure 19.

Optimization process for two regularization terms.

Figure 20.

MSE results for the samples of 35, 38, 45, and 63 correspond to (a) to (d).

Conclusions and future work

This paper investigates the technical linkages between sound quality modeling and active control, proposes the CEEMDAN-based method involving sound quality evaluation, modeling, prediction and active control by feature decomposition for time-domain noise signals, and presents its feasibility through the electric bus noise samples under quasi-steady speed conditions. The following key findings are summarized.

(1) With the effective feature extraction of 64 electric bus interior noise signals, the objective parameters of sound quality based on CEEMDAN-energy are calculated, and the subjective evaluation test with acoustic comfort as the index is accomplished. Consequently, a high-precision sound quality prediction model is established based on PSO-XGBoost. An active control method based on CEEMDAN-FxLMS is proposed, and for the four noise samples with poor sound quality, spectrum analysis and acoustic comfort prediction for the error signals are completed via ANC platform in the semi-anechoic room. The comparison results indicate that the sound quality of the samples are improved.

(2) The case proves that the combinations of CEEMDAN with PSO-XGBoost and FxLMS are effective to form a coherent strategy for acoustic comfort evaluation, modeling, prediction, and active control, which provides a technological solution for electric bus sound quality improvement.

(3) The established models of sound quality prediction and active control in this paper are suitable for electric buses driving at constant speeds, and the next step is the extension of more unsteady conditions including acceleration and deceleration to explore the new sound quality evaluation and active control methods, and verify their robustness under complex alternating conditions. Meanwhile, it is also a promising future work for the active sound quality control of specific noise sources, such as electric drive axle noise.

Footnotes

ORCID iD

Enlai Zhang

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study is supported by Natural Science Foundation of Fujian Province (2023J011438), National Natural Science Foundation of China (12004136), Major Educational Research Project of Fujian Province (FBJY20230154), Fuxiaquan National Independent Innovation Demonstration Zone Collaborative Innovation Platform Project (3502ZCQXT2024008) and Open Fund of Fujian Provincial Key Laboratory of Advanced Design and Manufacture for Bus Coach (Xiamen University of Technology).

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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