Development and DSP Implementation of an Optimized Multi-Channel Active Control System for Vehicle Interior Engine Noise Using Local Secondary Path Equalization

Abstract

Engine noise is a predominant source of noise in the cabin of internal combustion engine vehicles and new energy hybrid vehicles. The conventional multi-channel active noise control (ANC) system, based on the adaptive notch filtered-X least mean square algorithm, is commonly employed to mitigate such multi-tonal noise. However, the computational efficiency and convergence performance of this system may be significantly hindered by the large estimated secondary path length and the frequency-dependent convergence behavior. To overcome these limitations, this paper proposes a computationally efficient and fast-converging multi-channel ANC system by incorporating a local secondary path (LSP) equalization method. The proposed method enhances the convergence speed by equalizing the magnitude responses of estimated secondary paths and reduces the computational complexity through an improved LSP modeling approach. Accordingly, a set of low-order equalized LSP models with normalized amplitude-frequency responses is generated and employed for reference filtering. A computational complexity analysis comparing the conventional system, a recent cost-effective system, and the proposed system is presented. Numerical simulations are conducted to evaluate the convergence speed and noise attenuation performance of these three systems. Additionally, real vehicle experiments are performed using a digital signal processing controller. The results demonstrate that the proposed multi-channel ANC system achieves a superior noise reduction effect. Under accelerated conditions, the average attenuation of the second-order noise component at the four error microphones is measured at 4.4 dB(A), 6.2 dB(A), 13.4 dB(A), and 10.0 dB(A). These findings confirm the practical effectiveness of the proposed multi-channel ANC system.

1. Introduction

The noise performance of a vehicle is a critical determinant of its overall quality and brand perception, particularly in the medium- and high-end automobile market. It serves as a key factor in enhancing brand value and plays a significant role in shaping consumer purchasing decisions. A quiet in-cabin acoustic environment not only alleviates passenger fatigue but also substantially improves ride comfort. To maintain a competitive edge, automobile manufacturers are placing increasing emphasis on vehicle interior noise control. In recent years, active noise control (ANC) technology has been widely employed for sound field control within the passenger compartments of various complex electromechanical systems, including automobiles [1,2,3], ships [4,5], and aircraft [6,7]. Based on the principle of acoustic interference cancellation, this technology offers distinct advantages over conventional passive noise control methods, particularly in low-frequency noise attenuation and parameter tuning flexibility. The most widely used algorithm in ANC systems is the filtered-X least mean square (FXLMS) algorithm [8,9].

Engine noise, characterized by prominent low-frequency multi-harmonic components, constitutes the main noise source within the cabin of internal combustion engine vehicles. With the rapid development of hybrid and electric vehicles, the application of ANC technology has become increasingly important for improving vehicle interior acoustic comfort. In hybrid electric vehicles and range-extended electric vehicles, the engine may operate intermittently under different driving conditions, producing prominent low-frequency harmonic and order-related noise components. In contrast, pure electric vehicles eliminate the conventional engine noise but introduce new dominant noise sources associated with electric motors, inverters, and gear systems, which are often characterized by prominent tonal and harmonic components. These changes in powertrain noise characteristics pose new challenges for vehicle interior noise control and further highlight the potential of ANC technology for suppressing low-frequency and tonal noise components. To mitigate this type of noise, a specialized ANC system adopting the adaptive notch FXLMS algorithm with two-weight notch filters has demonstrated a certain degree of effectiveness [10,11,12]. However, the large estimated secondary path length and frequency-dependent convergence behavior impose significant constraints on computational complexity and convergence speed, particularly in multi-channel, multi-frequency, or time-varying noise scenarios [13,14]. In mass-production applications of ANC systems, the substantial computational burden necessitates high-rate data processing capabilities in digital signal processing (DSP) controllers, consequently resulting in higher costs [15]. Moreover, the slow convergence speed degrades the system’s real-time performance, making it challenging to effectively track and suppress nonstationary noises [16].

In the past few years, various techniques have been proposed to mitigate the computational complexity of the conventional ANC system mentioned above. These include the delay compensation method [17], the bandpass filter method [18], the filtered-error structure method [19,20], the local secondary path (LSP) estimation method [13,21], and the complex signal representation method [22]. However, our previous research has revealed that some of these approaches involve trade-offs, such as inaccuracies in secondary-path estimation or delayed convergence [23]. More recently, frequency-point-selective updating and partial-update strategies have also been investigated to reduce the computational burden of ANC systems [24,25]. Despite these advances, achieving high computational efficiency while maintaining fast convergence remains challenging. To further improve computational efficiency, Zhang et al. [26] proposed a computationally efficient ANC system based on a novel delayed-filtered reference method. This method effectively simplifies the filtering process of the reference signal, leading to a substantial enhancement in computational efficiency. Nevertheless, the challenge of achieving rapid convergence remains.

Various approaches have also been developed to improve the convergence and tracking performance of ANC systems, including the variable step-size method [27,28,29], time-frequency domain transformation method [30,31], and subband adaptive filtering method [32,33]. Recent studies have further explored intelligent and data-driven strategies, including intelligent optimization of control parameters, deep-learning-assisted secondary path modeling, and adaptive control, to improve the adaptability and performance of ANC systems under dynamic operating conditions [34,35,36]. Despite these advances, the additional adaptation, optimization, or modeling procedures introduced by these methods may increase algorithmic complexity. More importantly, the convergence characteristics of narrowband ANC systems remain strongly frequency-dependent. In the FXLMS algorithm, the optimal step size varies with frequency, making it difficult to achieve consistently fast convergence across multiple frequency components or under time-varying frequency conditions. Generally, when the amplitude-frequency response (AFR) of the secondary path exhibits high dynamic variation across frequencies, a larger step size is required at frequencies with lower magnitude, whereas a smaller step size is preferred at frequencies with higher magnitude. This behavior is primarily related to the inverse relationship between the optimal step size and the maximum eigenvalue of the autocorrelation matrix of the filtered reference signal. To address this issue, Thomas et al. [37] proposed an eigenvalue equalization method that modifies the magnitude coefficients of the estimated secondary path while preserving its phase response. By reducing the eigenvalue variation of the autocorrelation matrix of the filtered reference signal, this approach can significantly improve the convergence performance.

Based on the above research results, the conventional LSP estimation method [13,21] can effectively reduce the computational complexity of ANC systems by avoiding the estimation of the complete long secondary path. However, its performance is limited by the accuracy of the estimated local secondary paths, and the mismatch among different secondary paths may lead to slow convergence and degraded noise reduction performance. On the other hand, the eigenvalue equalization method [37] can improve the convergence speed of adaptive ANC algorithms by reducing the eigenvalue spread of the input signal correlation matrix. However, this method focuses primarily on convergence improvement and does not address the high computational complexity of multi-channel ANC systems. To overcome the above limitations, this study combines the advantages of both approaches and proposes a novel LSP equalization method. The proposed method not only enhances the convergence speed by equalizing the magnitude responses of estimated secondary paths but also achieves low computational complexity by employing an improved LSP estimation method. Based on the proposed LSP equalization method, an efficient multi-channel ANC system is developed for vehicle interior engine noise control. Compared with conventional approaches, the proposed system achieves a better trade-off among computational efficiency, convergence speed, and noise attenuation performance, thereby providing an effective solution for practical vehicle-cabin noise-suppression applications. The key contributions of this study are as follows:

• A computationally efficient and fast-converging multi-channel narrowband ANC system is developed for vehicle interior engine noise by incorporating an LSP equalization method, with the objective of achieving an effective balance between computational efficiency, convergence performance, and noise attenuation.
• A comprehensive computational complexity analysis is conducted to quantitatively demonstrate the computational advantages of the proposed system over the conventional system and a representative cost-effective system.
• The proposed system is systematically validated through numerical simulations and real-vehicle DSP experiments, demonstrating improved convergence speed and noise attenuation performance and confirming its potential for practical real-time applications.

The remainder of this paper is structured as follows. Section 2 provides a detailed description of the proposed multi-channel ANC system and presents a computational complexity comparison with both conventional and recently developed ANC systems. Section 3 and Section 4 present numerical simulations and real-time experiments, respectively, to validate the effectiveness of the proposed approach. Finally, conclusions are drawn in Section 5.

2. The Proposed Multi-Channel ANC System

2.1. Multi-Channel Adaptive Notch Filtering

Figure 1 shows the block diagram of the multi-channel ANC system based on the adaptive notch FXLMS algorithm. This system employs K two-weight adaptive notch filters in parallel to control K frequency components. Moreover, the multi-channel ANC system includes M secondary sources and L error microphones. $P_l(z)$ denotes the true primary path between the noise source and the lth error microphone. $S_{ml}(z)$ denotes the true secondary path between the mth secondary source and the lth error microphone. ${\widehat{S}}_{ml}(z)$ denotes the estimate of $S_{ml}(z)$. $d_l(n)$ and $e_l(n)$ denote the primary noise signal and the residual noise signal at the lth error microphone, respectively, both expressed in Pa. The definitions of the remaining symbols are provided in the following derivation.

Assuming that the kth frequency channel is employed to control the jth harmonic order of the vehicle interior engine noise, the corresponding reference frequency can be determined as

$$f_k=j{\displaystyle \frac{RPM\cdot z}{60\tau }} \left(\mathrm{Hz}\right)$$

(1)

where z denotes the number of engine cylinders, τ represents the number of engine power strokes, and RPM refers to the engine speed, expressed in revolutions per minute (rpm). The synchronous engine speed signal can be acquired through the CAN protocol output signal or the ignition pulse signal.

The reference cosine and sine signals corresponding to the frequency ${\omega }_k$ in the kth frequency channel are then expressed by

$$x_{k1}(n)=\cos ({\omega }_{k}n)$$

(2)

$$x_{k2}(n)=\sin ({\omega }_{k}n)$$

(3)

where n is the time index (or sample point), ${\omega }_k=2\pi f_k/f_s$ [38,39], and $f_s$ is the sampling frequency in Hz.

In the case of multiple harmonics, the output of the mth controller with K reference signals is calculated by

$$y_m(n)={\displaystyle \sum _{k=1}^K{y}_{mk}(n)}$$

(4)

$$y_{mk}(n)=w_{mk1}(n)x_{k1}(n)+w_{mk2}(n)x_{k2}(n)=w_{mk}^{\mathrm{T}}(n)x_k(n)$$

(5)

where $w_{mk}(n)={\left[w_{mk1}(n),w_{mk2}(n)\right]}^{\mathrm{T}}$ is the weight vector of the mth controller with the kth reference signal, and $x_k(n)={\left[x_{k1}(n),x_{k2}(n)\right]}^{\mathrm{T}}$ is the kth reference signal vector.

The residual noise signal $e_l(n)$ at the lth error microphone may be written as

$$e_l(n)=d_l(n)+{\displaystyle \sum _{m=1}^M{\displaystyle \sum _{i=0}^{I-1}{s}_{ml,i}}(n)y_m}(n-i)$$

(6)

where ${\left\{s_{ml,i}(n)\right\}}_{i=0}^{I-1}$ are the impulse response coefficients of the true secondary path $S_{ml}(z)$ with length I.

The FXLMS algorithm is employed to minimize the instantaneous squared error signal. Consequently, the weight vector of the mth controller corresponding to the kth reference signal is updated as

$$w_{mk}(n+1)=w_{mk}(n)-\mu {\displaystyle \sum _{l=1}^L{e}_l(n)x_{mlk}^{\prime }(n)}$$

(7)

where $x_{mlk}^{\prime }(n)={\left[x_{mlk1}^{\prime }(n),x_{mlk2}^{\prime }(n)\right]}^{\mathrm{T}}$ is the filtered reference vector, given by

$$x_{mlk1}^{\prime }(n)={\displaystyle \sum _{i=0}^{\widehat{I}-1}{\widehat{s}}_{ml,i}(n)}{x}_{k1}(n-i)$$

(8)

$$x_{mlk2}^{\prime }(n)={\displaystyle \sum _{i=0}^{\widehat{I}-1}{\widehat{s}}_{ml,i}(n)}{x}_{k2}(n-i)$$

(9)

where ${\left\{{\widehat{s}}_{ml,i}(n)\right\}}_{i=0}^{\widehat{I}-1}$ are the impulse response coefficients of the estimated secondary path ${\widehat{S}}_{ml}(z)$ with length $\widehat{I}$.

As observed from Equations (8) and (9), $2\widehat{I}$ multiplications per iteration are required to compute the filtered reference signals for each frequency component in every channel. However, a large length $\widehat{I}$ for the estimated secondary path is typically necessary to ensure high estimation accuracy, which imposes a considerable computational burden on multi-channel, multi-frequency ANC systems. Additionally, the frequency-dependent problem becomes evident in the case of multi-frequency or time-varying frequency noise, as selecting an appropriate step size for each frequency component to achieve optimal system performance is challenging. To address these two issues, we propose a novel local secondary path (LSP) equalization method in the following section.

2.2. The LSP Equalization Method

The proposed LSP equalization method consists of three parts—an offline modeling test, amplitude-frequency response (AFR) equalization and LSP modeling—as shown in Figure 2. Generally, the AFR of the secondary path exhibits significant fluctuations, particularly in complex enclosed acoustic environments. After convolving the reference signal with the secondary path model, the spectral characteristics of the filtered reference signal undergo changes. This leads to an excessively large dynamic range in the eigenvalues of the filtered reference signal’s autocorrelation matrix, resulting in substantial frequency dependence of the system’s control performance. To mitigate this effect, this study first employs the commonly used random white-noise method based on the least mean square (LMS) algorithm to perform offline modeling of the global secondary path, yielding the global secondary path model ${\widehat{S}}_{ml}(z)$, as shown in Figure 2a. Subsequently, the global secondary path model undergoes frequency-domain amplitude equalization to transform its fluctuating amplitude-frequency response into a flat amplitude-frequency response while preserving its phase characteristics [14], resulting in the equalized global secondary path model ${\widehat{S}}_{ml}^{\prime }(z)$, as depicted in Figure 2c.

Assuming the impulse response coefficients of the global secondary path model ${\widehat{S}}_{ml}(z)$ are ${\widehat{s}}_{ml,i}(n), i=0,\dots \widehat{I}-1$, performing an Nfft-point fast Fourier transform (FFT) on these time-domain coefficients yields the frequency-domain secondary path:

$$\widehat{S}=\mathrm{FFT}\left(\left\{{\widehat{s}}_{ml,0}(n),{\widehat{s}}_{ml,1}(n),\dots {\widehat{s}}_{ml,\widehat{I}-1}(n)\right\},Nfft\right)$$

(10)

where Nfft is set equal to the sampling frequency f_s to ensure sufficient frequency resolution.

Then, we divide $\widehat{S}$ at each frequency bin by its amplitude to obtain the normalized frequency-domain secondary path ${\widehat{S}}^{\prime }$:

$${\widehat{S}}^{\prime }={\displaystyle \frac{\widehat{S}}{abs(\widehat{S})}}$$

(11)

Finally, the normalized frequency-domain secondary path ${\widehat{S}}^{\prime }$ undergoes an Nfft-point inverse FFT (IFFT) to revert to the equalized time-domain secondary path model:

$$\left\{{\widehat{s}}_{ml,0}^{\prime }(n),{\widehat{s}}_{ml,1}^{\prime }(n),\dots {\widehat{s}}_{ml,\widehat{I}-1}^{\prime }(n)\right\}=\mathrm{first} \widehat{I} \mathrm{coefficients} \mathrm{of} \mathrm{IFFT}\left({\widehat{S}}^{\prime },Nfft\right)$$

(12)

where ${\widehat{s}}_{ml,i}^{\prime }(n), i=0,\dots \widehat{I}-1$ are the impulse response coefficients of the equalized global secondary path model ${\widehat{S}}_{ml}^{\prime }(z)$. Since the global secondary path model should maintain the same length before and after AFR equalization, the first $\widehat{I}$ coefficients are truncated after performing the IFFT on ${\widehat{S}}^{\prime }$. This results in the ARF of the equalized global secondary path model in Figure 2d not being perfectly flat.

A long finite impulse response (FIR) filter (e.g., $\widehat{I}=256$) is usually required to model the global secondary path, so as to achieve effective noise reduction over the entire frequency range. However, this will significantly increase the computational complexity of the algorithm. For multi-tonal noise, due to its narrowband frequency characteristics, our previous work [23,40,41] adopted a pure-tone excitation method to advance the local secondary path modeling approach initially proposed by Delega [13]. This method is also applied here to reduce the computational complexity of the multi-channel ANC system. The equalized global secondary path model ${\widehat{S}}_{ml}^{\prime }(z)$ is locally modeled via pure-tone excitation, as follows:

A cyclic LMS module developed in MATLAB (2021a) performs cyclic local modeling on ${\widehat{S}}_{ml}^{\prime }(z)$ with a 1 Hz frequency gradient, thereby yielding the equalized LSP model ${\widehat{S}}_{mlk}^{\prime }(z)$ at each individual frequency point within the target frequency range, as shown in Figure 2e. The 1 Hz frequency gradient is set to ensure a high degree of consistency between the frequency characteristics of the LSP model corresponding to each frequency point and those of the global secondary path model at the same frequency point. For example, if the frequency range to be controlled is 0 to 600 Hz, pure-tone excitation modeling can be performed sequentially for each frequency point at a 1 Hz frequency gradient based on the LMS algorithm, which ultimately yields LSP models at 600 distinct frequencies. The frequency band covered by each local model is reduced to a single frequency point. Consequently, setting the local model length ${\widehat{I}}^{\prime }$ to 2 enables precise approximation of the global secondary path model at each frequency point, greatly reducing the computational complexity of the algorithm. It can be seen from Figure 2f that, taking the local models at 150 Hz, 300 Hz, and 450 Hz as examples, the amplitude-frequency responses of the equalized local models perfectly match those of the equalized global model at the corresponding frequencies, demonstrating the accuracy of the local modeling of the equalized global secondary path. For clarity and completeness, Algorithm 1 provides a concise pseudocode representation of the proposed LSP equalization method, complementing the workflow illustrated in Figure 2.

Algorithm 1. Implementation procedure of the proposed LSP equalization method

Input: white-noise excitation v(n), target frequencies {${\omega }_k$}, global model length $\widehat{I}$, local model length ${\widehat{I}}^{\prime }$, and LMS step sizes ${\mu }_g$ and ${\mu }_l$. Output: Equalized LSP models ${\left\{{\widehat{s}}_{mlk,i}^{\prime }(n)\right\}}_{i=0}^{{\widehat{I}}^{\prime }-1}$. a. Offline modeling test: 1: Generate white-noise excitation v(n). 2: Apply v(n) to the secondary path and acquire the corresponding response dv(n) using the error microphone. 3: Initialize the global model ${\widehat{s}}_{ml}(n)={\left\{{\widehat{s}}_{ml,i}(n)\right\}}_{i=0}^{\widehat{I}-1}$. 4: for n = 1, …, N do 5:      Calculate the global model output $v^{\prime }(n)={\displaystyle \sum _{i=0}^{\widehat{I}-1}{\widehat{s}}_{ml,i}(n)}v(n-i)$. 6:      Calculate the modeling error $e_v(n)=d_v(n)-v^{\prime }(n)$. 7:      Update the global model using the LMS algorithm: ${\widehat{s}}_{ml}(n+1)={\widehat{s}}_{ml}(n)+{\mu }_{g}v(n)e_v(n)$. 8: end for b. AFR equalization: 9: Calculate the frequency response of ${\left\{{\widehat{s}}_{ml,i}(n)\right\}}_{i=0}^{\widehat{I}-1}$ using FFT. 10: Normalize the amplitude spectrum in the frequency domain. 11: Obtain the equalized global model ${\left\{{\widehat{s}}_{ml,i}^{\prime }(n)\right\}}_{i=0}^{\widehat{I}-1}$ using IFFT. c. LSP modeling: 12: for k = 1, …, 600 do 13:      Generate the pure-tone excitation $x_k(n)=\cos ({\omega }_{k}n)$. 14:      Generate the desired response $d_k(n)={\displaystyle \sum _{i=0}^{\widehat{I}-1}{\widehat{s}}_{ml,i}^{\prime }(n)}{x}_k(n-i)$. 15:      Initialize the local model ${\widehat{s}}_{mlk}^{\prime }(n)={\left\{{\widehat{s}}_{mlk,i}^{\prime }(n)\right\}}_{i=0}^{{\widehat{I}}^{\prime }-1}$. 16:      for n = 1, …, N do 17:          Calculate the local model output $x_{mlk}^{\prime }(n)={\displaystyle \sum _{i=0}^{{\widehat{I}}^{\prime }-1}{\widehat{s}}_{mlk,i}^{\prime }(n)}{x}_k(n-i)$. 18:          Calculate the error signal $e_k(n)=d_k(n)-x_{mlk}^{\prime }(n)$. 19:          Update the local model using the LMS algorithm: ${\widehat{s}}_{mlk}^{\prime }(n+1)={\widehat{s}}_{mlk}^{\prime }(n)+{\mu }_l{x}_k(n)e_k(n)$. 20:      end for 21:      Store the equalized local model ${\widehat{s}}_{mlk}^{\prime }(n)={\left\{{\widehat{s}}_{mlk,i}^{\prime }(n)\right\}}_{i=0}^{{\widehat{I}}^{\prime }-1}$. 22: end for

The obtained equalized LSP models are then employed for filtering operations in the multi-channel ANC system. During the active control process, the system automatically invokes the local models corresponding to the reference frequencies. Therefore, the proposed LSP equalization method is also applicable to multi-frequency or time-varying frequency noise scenarios. By addressing the challenges associated with the frequency dependence and the large secondary path length, the proposed LSP equalization method significantly enhances both the convergence speed and computational efficiency of the system. Based on Equations (8) and (9), the reference signals filtered by the equalized LSP models can be expressed by

$$x_{mlk1}^{\prime }(n)={\displaystyle \sum _{i=0}^{{\widehat{I}}^{\prime }-1}{\widehat{s}}_{mlk,i}^{\prime }(n)}{x}_{k1}(n-i)$$

(13)

$$x_{mlk2}^{\prime }(n)={\displaystyle \sum _{i=0}^{{\widehat{I}}^{\prime }-1}{\widehat{s}}_{mlk,i}^{\prime }(n)}{x}_{k2}(n-i)$$

(14)

where ${\left\{{\widehat{s}}_{mlk,i}^{\prime }(n)\right\}}_{i=0}^{{\widehat{I}}^{\prime }-1}$ are the impulse response coefficients of the equalized LSP model ${\widehat{S}}_{mlk}^{\prime }(z)$ that correspond to the frequency ${\omega }_k$ between the mth secondary source and the lth error microphone.

2.3. Computational Complexity Analysis

In this section, the computational costs of the conventional system, Zhang’s [26] model, and the proposed four-channel ANC systems are evaluated in terms of the number of multiplications.

Table 1. Computational complexity of the conventional, Zhang, and proposed four-channel ANC systems in terms of multiplications, where m = 1, 2, 3, 4 and l = 1, 2, 3, 4.

System	Computational Procedure	Computational Expression	Multiplications
Conventional	Controller output	$y_{mk}(n)=w_{mk1}(n)x_{k1}(n)+w_{mk2}(n)x_{k2}(n)$	$8K$
	Reference filtering	$x_{mlk}^{\prime }(n)={\displaystyle \sum _{i=0}^{\widehat{I}-1}{\widehat{s}}_{ml,i}(n)}{x}_k(n-i)$	$32K\widehat{I}$
	Weight updating	$w_{mk}(n+1)=w_{mk}(n)-\mu {\displaystyle \sum _{l=1}^L{e}_l(n)x_{mlk}^{\prime }(n)}$	$40K$
	Total		$32K\widehat{I}+48K$
Zhang’s	Controller output	$y_{mk}(n)=w_{mk1}(n)x_{k1}(n)+w_{mk2}(n)x_{k2}(n)$	$8K$
	Reference filtering	Offline: $c_k=\left[\begin{array}{cc}\cos {\omega }_k& \sin {\omega }_k\\ -\sin {\omega }_k& \cos {\omega }_k\end{array}\right]$, $z_{mlk}={\displaystyle \sum _{i=0}^{\widehat{I}-1}{\widehat{s}}_{ml,i}(n)}{c_k}^{-i}$ Online: $x_{mlk}^{\prime }(n)=z_{mlk}{x}_k(n)$	$64K$
	Weight updating	$w_{mk}(n+1)=w_{mk}(n)-\mu {\displaystyle \sum _{l=1}^L{e}_l(n)x_{mlk}^{\prime }(n)}$	$40K$
	Total		$112K$
Proposed	Controller output	$y_{mk}(n)=w_{mk1}(n)x_{k1}(n)+w_{mk2}(n)x_{k2}(n)$	$8K$
	Reference filtering	$x_{mlk}^{\prime }(n)={\displaystyle \sum _{i=0}^{{\widehat{I}}^{\prime }-1}{\widehat{s}}_{mlk,i}^{\prime }(n)}{x}_k(n-i)$	$32K{\widehat{I}}^{\prime }$
	Weight updating	$w_{mk}(n+1)=w_{mk}(n)-\mu {\displaystyle \sum _{l=1}^L{e}_l(n)x_{mlk}^{\prime }(n)}$	$40K$
	Total		$32K{\widehat{I}}^{\prime }+48K$

summarizes the computational operations involved in each iteration, where the multiplication requirements of different procedures, including controller output calculation, reference signal filtering, and adaptive weight updating, are separately analyzed. The total computational cost of each system is obtained by summing the multiplication operations of all procedures. The notations used here are consistent with those defined in previous sections, where K represents the number of tonal components, and $\widehat{I}$ and ${\widehat{I}}^{\prime }$ denote the lengths of the global and local secondary path estimates, respectively.

It can be observed from Table 1 that the three systems share a similar computational framework, with the main difference lying in the reference signal filtering process. And when ${\widehat{I}}^{\prime }=2$, the proposed system and Zhang’s system require the same number of multiplications, namely $112K$. This indicates that both approaches can achieve a comparable reduction in computational cost compared with the conventional structure by avoiding direct filtering through the complete secondary path model.

For further evaluation, Figure 3 compares the multiplication operations of the conventional and proposed four-channel ANC systems under different combinations of K and $\widehat{I}$, where ${\widehat{I}}^{\prime }=2$ is adopted. The results demonstrate that the proposed system consistently requires considerably fewer operations than the conventional system for all investigated cases. For instance, under the typical condition of $K=3$ and $\widehat{I}=256$, the computational requirement of the proposed system is only 1.36% of that of the conventional system. As K or $\widehat{I}$ increases, the computational burden of the conventional system grows significantly, whereas the proposed system experiences only a marginal increase, maintaining a low computational load due to the incorporation of the LSP estimation method. Furthermore, the proposed four-channel ANC system has the same computational advantage as Zhang’s four-channel ANC system. Consequently, both methods can effectively reduce processor requirements and improve the feasibility of real-time implementation. In particular, the proposed system provides a more flexible solution for vehicle interior noise control applications where computational resources are limited.

3. Numerical Simulations

To evaluate the convergence speed and noise attenuation performance of the proposed multi-channel ANC system, the algorithms derived in Section 2 are implemented and evaluated through numerical simulations conducted in the MATLAB environment. The correctness and effectiveness of the algorithm implementation are further assessed by comparing the proposed system with the conventional system and Zhang’s multi-channel ANC system under identical simulation conditions. Both synthesized multi-tonal noise signals and real interior engine noise signals are employed as the primary noise signals. Before evaluating the noise control performance, two important parameters of the proposed LSP equalization method are investigated in Section 3.1 and Section 3.2. Section 3.1 analyzes the influence of the local model length on the accuracy of LSP modeling, while Section 3.2 investigates the influence of the frequency resolution, defined as the frequency interval between adjacent reference frequencies used for LSP modeling, on the noise control performance under a time-varying frequency noise condition.

The real engine noise signals are recorded at the four seat headrests of a standard four-seater passenger vehicle operating at a constant speed of 3500 rpm. The secondary path models are obtained through system identification using an NI cRIO-9040 controller in the vehicle. Figure 4 illustrates the schematic diagram of the ANC device configuration in the real vehicle, where four secondary sources, S1, S2, S3, and S4, are installed on the four car doors, respectively, and four error microphones, E1, E2, E3, and E4, are placed near the seat headrests. The 16 secondary paths are modeled as FIR filters with a model length of 256, as shown in Figure 5. This model length is selected based on the measured secondary path characteristics, providing sufficient temporal length to accurately characterize the measured secondary paths over the frequency range of interest, thereby avoiding significant modeling truncation errors and ensuring reliable noise reduction performance of the benchmark conventional ANC system. The sampling frequency is set to 4096 Hz, considering both the frequency characteristics of low-frequency engine noise and the processing capability of commonly used experimental hardware. Since the ANC system mainly focuses on low-frequency harmonic components below 500 Hz, the selected sampling frequency provides sufficient bandwidth for signal acquisition, secondary path modeling, and active noise control while avoiding unnecessary computational overhead.

The adaptive step size is another important parameter that affects the convergence speed and steady-state noise reduction performance of ANC systems. This parameter is carefully selected to achieve an appropriate trade-off between convergence speed and steady-state noise reduction performance, particularly when comparing the control performance of different ANC systems. The same parameter selection principle is consistently maintained throughout the subsequent simulations and experiments.

3.1. Influence of Local Model Length on Modeling Accuracy

The local model length ${\widehat{I}}^{\prime }$ is an important parameter affecting both the modeling accuracy and computational complexity of the proposed LSP equalization method. Since the equalized LSP model is constructed for individual narrowband frequency components, a relatively short model length is sufficient to characterize its frequency response. To investigate the influence of the local model length on the secondary path modeling accuracy, the secondary path from the secondary source S1 to the error microphone E1, as illustrated in Figure 5, is selected as an example based on the measured secondary paths of the real vehicle experimental platform. Different local model lengths of 2, 5, and 8 are considered, and their corresponding frequency responses at multiple frequencies are compared with those of the equalized global secondary path model, as listed in

Table 2. Comparison of the frequency responses of the equalized local secondary paths with different local model lengths.

		100 Hz		200 Hz		300 Hz		400 Hz
		Magnitude (dB)	Phase (Degree)	Magnitude (dB)	Phase (Degree)	Magnitude (dB)	Phase (Degree)	Magnitude (dB)	Phase (Degree)
Global model length	256	0.41	−293.74	−0.86	−582.97	0.95	−835.58	1.31	−1027.96
Local model length	2	0.39	−293.15	−0.84	−582.84	0.96	−835.79	1.30	−1027.72
	5	0.40	−294.30	−0.88	−582.60	0.94	−835.34	1.32	−1027.99
	8	0.42	−293.37	−0.87	−582.75	0.95	−835.21	1.31	−1027.90

.

Taking the results at 100 Hz, 200 Hz, 300 Hz, and 400 Hz as representative individual frequency points for illustration and quantitative comparison, it can be observed that the equalized local secondary paths obtained with the three different model lengths exhibit highly consistent magnitude and phase responses at the target frequencies. In particular, when the local model length is set to 2, its frequency responses at all the investigated frequencies are almost identical to those of the global secondary path model with a model length of 256. The maximum magnitude difference among the local models with different lengths is less than 0.04 dB, while the phase difference is within approximately 2°. These results indicate that even with a model length of 2, the local model can accurately characterize the frequency response of the secondary path at the corresponding target frequencies, and increasing the local model length provides only negligible improvement in modeling accuracy. However, a longer local model inevitably increases the computational burden of the multi-channel ANC system. Therefore, considering the trade-off between modeling accuracy and computational efficiency, a local model length of 2 is selected in this study.

3.2. Influence of Frequency Resolution on Noise Control Performance

In the proposed LSP equalization method, the equalized LSP models are established at discrete frequencies with a prescribed frequency interval and are subsequently employed for filtering in the ANC system. During the active control process, the system automatically selects the local model corresponding to the current reference frequency. Therefore, the frequency resolution determines the density and number of the available equalized LSP models. This is particularly important for multi-frequency and time-varying frequency noise. For multi-frequency noise containing numerous frequency components, a finer frequency resolution requires more local models to cover the frequency range of interest, thereby increasing the data-processing workload during LSP modeling and the storage requirements for the resulting local models. For time-varying frequency noise, the frequency resolution affects the matching accuracy between the instantaneous reference frequency and the available local model. A finer frequency resolution provides a denser set of local models and enables more accurate model selection, whereas a coarser resolution may lead to a larger frequency mismatch and consequently degrade the noise control performance. Meanwhile, an increased number of local models may introduce additional overhead in model indexing and selection during active control.

To investigate the influence of frequency resolution, a single-channel ANC system is considered as an example. A synthesized acceleration sinusoidal noise signal is employed as the source noise signal, thereby reproducing the time-varying frequency characteristics of engine-order noise under an acceleration condition. The instantaneous frequency is determined from a simulated engine speed that increases linearly from 1000 to 6000 rpm over 10 s and is expressed as

$$RPM(n)=1000+{\displaystyle \frac{500}{4096}}n \left(\mathrm{rpm}\right)$$

(15)

The source noise signal is given by

$$x(n)=\left(0.1+0.4{\displaystyle \frac{n}{4096\cdot 10}}\right)\cdot \sin \left(2\pi {\displaystyle \sum _{j=1}^n\frac{2\cdot RPM(j)}{60\cdot 4096}}\right) \left(\mathrm{Pa}\right)$$

(16)

where the amplitude of the source noise signal increases from 0.1 Pa to 0.5 Pa. Then, the primary noise signal is simulated from the source noise signal and an additional zero-mean white Gaussian noise with the variance 0.01.

The measured secondary path S1-E1 is also utilized here. Five different frequency resolutions, i.e., 0.1 Hz, 0.5 Hz, 1 Hz, 3 Hz, and 5 Hz, are considered in the proposed LSP equalization method. For each frequency resolution, the corresponding set of equalized LSP models is established. During the active control process of the single-channel ANC system, the local model associated with the discrete frequency point closest to the instantaneous reference frequency is automatically selected from the pre-established local model set based on the simulated engine speed. The noise control performances obtained with difference frequency resolutions are compared in Figure 6. The results show that the controlled sound pressure level (SPL) curves obtained with frequency resolutions of 0.1, 0.5, and 1 Hz are almost identical over the investigated speed range, indicating that further refining the frequency resolution below 1 Hz provides negligible improvement in noise control performance. In contrast, the SPL obtained with a frequency resolution of 3 Hz is generally higher than that obtained with 0.1, 0.5, and 1 Hz, while the 5 Hz resolution exhibits the poorest noise attenuation performance. This indicates that a coarse frequency resolution may result in insufficient matching between the instantaneous reference frequency and the available local model, thereby degrading the noise control performance. Considering that the 1 Hz resolution achieves essentially the same control performance as the finer resolutions while requiring fewer local models, a frequency resolution of 1 Hz is selected for the following simulations and experiments.

3.3. Active Control of Synthesized Multi-Tonal Noise Signal

In this case, the four primary noise signals utilized in a four-channel ANC system each comprise three tonal components at frequencies of 117 Hz, 233 Hz, and 350 Hz, along with a white Gaussian noise component with a variance of 0.001. However, the amplitudes of three tonal components in the primary noise signals corresponding to different channels are different. The amplitude coefficients of all sinusoids can be expressed by $\left\{0.3,0.2,0.1;0.25,0.2,0.05;0.3,0.05,0.1;0.2,0.25,0.1\right\}$. The conventional multi-channel ANC system and Zhang’s multi-channel ANC system are used as reference benchmarks. The simulation duration is set to 5 s, and all step sizes are carefully tuned to ensure a fair comparison among the systems. To assess the noise attenuation performance of the systems, the averaged noise reduction (ANR) metric [42] is employed, which is defined as follows:

$$ANR(n)=20{\log }_{10}\left({\displaystyle \frac{A_e(n)}{A_d(n)}}\right) \left(\mathrm{dB}\right)$$

(17)

where $A_e(n)=\xi A_e(n-1)+(1-\xi )\left|e(n)\right|$, $A_d(n)=\xi A_d(n-1)+(1-\xi )\left|d(n)\right|$ with initial conditions $A_e(0)=0$, $A_d(0)=0$, and $\xi =0.999$, is the forgetting factor.

Figure 7 presents a comparison of the noise time waveforms and ANR curves at the four error microphones for the conventional, Zhang, and proposed multi-channel ANC systems. Notably, Zhang’s system exhibits almost exactly the same control performance as the conventional system, resulting in the blue line results overlapping completely with the green line results. This may be attributed to the fact that Zhang’s system only reduces the computational complexity with respect to the conventional system, without altering other performance areas of the system, such as convergence speed and noise attenuation performance. It is evident that the proposed system achieves significantly faster convergence and superior noise attenuation performance compared to both the conventional and Zhang systems. For instance, at error microphone E1, the conventional and Zhang systems fail to reach steady-state convergence within the 5 s iteration period, whereas the proposed system achieves convergence in approximately 1.5 s. These results clearly demonstrate that the proposed multi-channel ANC system enjoys enhanced convergence speed and noise attenuation performance while maintaining low computational complexity.

3.4. Active Control of Real Interior Engine Noise Signal

To further evaluate the control performance of the proposed multi-channel ANC system in comparison with the conventional and Zhang multi-channel ANC systems, real interior engine noise signals from a standard four-seater passenger vehicle equipped with a four-cylinder, four-stroke engine are utilized as the primary noise signals. In this case, the objective is to attenuate the 2nd-, 4th-, and 6th-order harmonic components, corresponding to frequencies of 117 Hz, 233 Hz, and 350 Hz, respectively. The simulation duration is set to 5 s, with all step sizes carefully optimized to ensure a fair comparison among the systems. Additionally, the used A-weighted SPL metric is obtained either through spectral processing or by directly applying the A-weighting filter available in MATLAB.

Figure 8 presents the ANR curves and noise frequency spectrums at the four error microphones for the three multi-channel ANC systems. The comparison results clearly demonstrate that the proposed system exhibits obviously superior noise attenuation performance compared to the conventional and Zhang systems, and the results of these two systems are also overlapped in this case. The proposed system effectively suppresses the 2nd-, 4th-, and 6th-order harmonic components at all four error microphones. For a more detailed analysis, the A-weighted SPL reductions of the 2nd-, 4th-, and 6th-order noises at the four error microphones are computed based on the frequency spectrums shown in Figure 8. As summarized in

Table 3. A-weighted SPL reductions (dB(A)) of 2nd-, 4th- and 6th-order noises at four error microphones for three multi-channel ANC systems.

	Conventional System			Zhang’s System			Proposed System
	2nd Order	4th Order	6th Order	2nd Order	4th Order	6th Order	2nd Order	4th Order	6th Order
E1	4.9	0.4	1.1	4.9	0.4	1.1	11.6	7.8	12.9
E2	8.8	4.0	1.6	8.9	4.0	1.6	19.4	8.1	12.4
E3	12.2	1.9	2.8	12.1	1.7	2.8	20.2	6.9	11.1
E4	6.8	4.8	4.0	6.8	4.8	4.0	16.6	13.5	6.8

, the noise attenuation achieved by the proposed system is substantially greater than that of the other two systems. These findings further validate the effectiveness of the proposed multi-channel ANC system in achieving outstanding noise attenuation performance.

On the other hand, a significant difference in noise attenuation levels is also observed between the synthesized multi-tonal noise and real interior engine noise, and the underlying reasons are elaborated as follows: First, the synthesized multi-tonal noise is composed of only a few pure sinusoidal components. Even when Gaussian white noise is added, the noise remains dominated by tonal components. Once these sinusoidal components are attenuated by the proposed ANC system, a considerable noise reduction effect can be achieved, with the corresponding ANR values exceeding 15 dB, as shown in Figure 7. In contrast, real interior engine noise has a much more complex composition. In addition to the prominent 2nd-, 4th-, and 6th-order harmonics, it contains numerous other frequency components that are not attenuated by the proposed ANC system, leading to ANR values of less than 10 dB. Second, the synthesized multi-tonal noise consists of stationary sinusoidal signals and stationary white noise. Although the real interior engine noise was recorded under a steady 3500 rpm operating condition, the measured noise is not ideally stationary due to random variations during the test, which also exerts an adverse impact on the noise attenuation performance.

Furthermore, the noise attenuation varies among the four error microphones, mainly due to the distinct spectral characteristics of the primary noise at each error microphone. In general, a higher primary noise amplitude indicates greater noise reduction potential. In addition, the secondary path characteristics differ among the source–receiver pairs, resulting in different frequency responses and control effectiveness at the four error microphones. These differences in the primary noise and secondary path characteristics jointly contribute to the variation in achievable noise attenuation at different channels. Therefore, the observed differences in attenuation among the four error microphones are reasonable even under the same ANC configuration.

4. DSP Experiments in the Real Vehicle

4.1. DSP Controller Design for the Multi-Channel ANC System

Currently, most of the ANC experiments conducted on real vehicles are based on a rapid prototype controller. To comprehensively validate the effectiveness of the proposed multi-channel ANC system in the context of mass-production applications, real vehicle ANC experiments are performed using a DSP controller in this section.

Figure 9 illustrates the overall framework of the DSP controller-based multi-channel ANC system designed for the active control of vehicle interior engine noise. The multi-channel ANC system based on a DSP controller primarily consists of three parts. Firstly, the signal input part is designed with multiple signal inputs. The CAN bus transceiver module on the controller obtains engine speed information from the vehicle’s CAN bus, while the error microphones in the target control area collect noise signals, which are then fed into the main control chip for iterative calculations of the active noise control algorithm. Secondly, the algorithm implementation part involves executing the designed multi-channel ANC algorithm on the main control chip. This part utilizes an externally expanded flash memory chip to call the secondary path estimation data and control parameters. Finally, the signal output part processes the output signals from the main control chip and drives the secondary sources after amplified by the power amplifier, thus generating the anti-noise to reduce the main order noises in the target control area. Notably, the A/D and D/A conversions in the signal input and output parts are implemented by an audio chip.

The ADSP-2148x series DSP chip, based on Analog Devices’ Super Harvard architecture, is a high-speed processor specifically designed for audio signal processing. In this study, the ADSP-21489 from this series is employed as the main control chip. The chip features a maximum clock frequency of 450 MHz and 5 Mbits of on-chip RAM, providing the hardware resources required for real-time multi-channel ANC implementation. The main control chip is further integrated with the AD1938 audio chip, I2S serial interface, and CANFDCOM-100IE module. Additionally, it is equipped with a UART communication interface, microphone bias circuit, and power amplification circuit. This configuration enables the implementation of a four-channel ANC system with four analog input and four analog output channels. The structure of the DSP controller designed for the multi-channel ANC system is shown in Figure 10 and Figure 11 and presents the prototype of the self-developed DSP controller.

Before the real-vehicle experiments, the conventional, Zhang, and proposed four-channel ANC systems are individually implemented and executed on the same ADSP-21489 DSP controller using CrossCore Embedded Studio (CCES) software (2.11.1). Their actual operation times per iteration are measured under identical operating conditions and used as a direct metric for comparing the computational efficiency of the three systems.

Table 4. Comparison of the operation time per iteration for the conventional, Zhang, and proposed four-channel ANC systems based on the DSP controller.

	Conventional System	Zhang’s System	Proposed System
Model length	$\widehat{I}=256$	$\widehat{I}=256$	${\widehat{I}}^{\prime }=2$
Iteration time	0.35 ms	0.03 ms	0.03 ms

presents the corresponding comparison results. As evident from Table 4, the iteration times of both the proposed system and Zhang’s system are approximately 8.57% of that of the conventional system. This substantial reduction in computation time demonstrates the improved implementation efficiency of the proposed system and enables the use of either a higher sampling rate or lower-cost hardware, thereby enhancing its practicality for real-world applications.

4.2. ANC Experiments and Results Analysis

The ANC experimental platform based on the DSP controller in the real vehicle is shown in Figure 12. The test vehicle remains the same as described in Section 3, and the experiment is conducted on an asphalt road within an automobile proving ground. The four secondary sources, denoted as S1, S2, S3, and S4, are installed on the four car doors, while the four analog error microphones, E1, E2, E3, and E4, are securely fixed to the seat headrests, as illustrated in Figure 12b. Notably, all secondary sources and error microphones are externally mounted hardware, as the ANC system has not yet been fully integrated into the vehicle. To accurately assess the noise control effect, the four monitoring microphones, M1, M2, M3, and M4, are positioned approximately 10 cm to 15 cm away from their corresponding error microphones, outside the four seat headrests. The engine speed is obtained in real-time via the CAN protocol output signal. A computer is utilized to operate the CCES interface for ANC system management and the LMS Test.Lab (Version 17) interface for noise data acquisition. Additionally, all input and output signals of the active control system undergo low-pass filtering with a cut-off frequency of 500 Hz. The sampling frequency is set to 4096 Hz for both the secondary path identification stage and the active control stage.

To verify the effectiveness of the proposed multi-channel ANC system under typical accelerated operating conditions, a real-vehicle experiment is conducted in which the test vehicle rapidly accelerated from 2000 rpm to 3900 rpm over 8.2 s. For comparison, Zhang’s multi-channel ANC system is chosen as the reference benchmark, while the conventional multi-channel ANC system is excluded from consideration, as its noise attenuation performance has been demonstrated to be nearly identical to that of Zhang’s system in Section 3.

Figure 13, Figure 14, Figure 15 and Figure 16 illustrate the comparison of SPL curves before and after control at four error microphones for both multi-channel ANC systems under acceleration. For brevity, only the overall and 2nd-order SPL curves are presented here. As observed across all the four error microphones, the proposed system exhibits superior capability in tracking the acceleration noise, particularly at error microphones 3 and 4. Additionally, Zhang’s system tends to cause an undesirable increase in overall noise within certain speed intervals, whereas the proposed system maintains more stable noise attenuation. Furthermore, Figure 17 presents the noise time-frequency spectrums before and after control at four error microphones for both systems. The average A-weighted SPL reductions for overall noise as well as the 2nd-, 4th-, and 6th-order noise components over the entire speed range at the four error microphones are summarized in

Table 5. Average A-weighted SPL reductions (dB(A)) of overall, 2nd-, 4th- and 6th-order noises at four error microphones for two multi-channel ANC systems under accelerated work conditions.

	Zhang’s System				Proposed System
	Overall	2nd Order	4th Order	6th Order	Overall	2nd Order	4th Order	6th Order
E1	1.1	3.3	−0.1	1.9	1.3	4.4	0.1	1.6
E2	1.5	6.0	−0.7	1.7	2.1	6.2	0.1	1.9
E3	2.7	11.0	2.1	1.7	3.4	13.4	2.4	3.9
E4	1.4	7.7	1.5	−0.4	2.1	10.0	1.3	1.4

. The proposed system achieves average overall noise reductions of 1.3 dB(A), 2.1 dB(A), 3.4 dB(A), and 2.1 dB(A) at the four error microphones. Additionally, the average attenuation of the 2nd-order noise component is measured at 4.4 dB(A), 6.2 dB(A), 13.4 dB(A), and 10.0 dB(A) at the corresponding microphones. In comparison, Zhang’s system yields average overall noise reductions of 1.1 dB(A), 1.5 dB(A), 2.7 dB(A), and 1.4 dB(A), and the average reduction of the 2nd-order noise component is 3.3 dB(A), 6.0 dB(A), 11.0 dB(A), and 7.7 dB(A), respectively. It is worth noting that slight negative attenuation values are observed for some individual engine order components in Table 5, mainly for the 4th- and 6th-order components of Zhang’s system. This can be mainly attributed to the relatively low amplitude of the corresponding primary noise over most of the investigated speed range under the acceleration condition. For these weaker engine order components, the lower tracking capability of Zhang’s system makes accurate and stable noise suppression more difficult as the engine speed continuously varies, which may result in slight increases in the controlled noise level over some speed intervals.

The experimental results presented above demonstrate that the proposed multi-channel ANC system has an enhanced capability to track time-varying and nonstationary engine noise under accelerated operating conditions. Over the 2000–3900 rpm acceleration range, the proposed system achieves average overall noise reductions ranging from 1.3 to 3.4 dB(A) at the four error microphones, representing the mean attenuation over the entire acceleration process rather than at a specific steady-state operating condition. These results should be interpreted together with the attenuation of the individual engine order components. The overall A-weighted SPL includes not only the targeted engine order components but also other tonal components and broadband background noise that are not directly controlled by the narrowband ANC system. Therefore, the overall SPL reduction alone may not fully reflect the effectiveness of targeted engine order noise control. In addition, the extent of noise reduction is closely influenced by the acoustic characteristics of the vehicle cabin.

In particular, the 2nd-order engine noise is generally a major contributor to the low-frequency booming noise in vehicle cabins and is therefore selected as a primary control target. Over the same acceleration range, the proposed system achieves average attenuation of the 2nd-order component ranging from 4.4 to 13.4 dB(A) at the four error microphones. Such substantial attenuation can effectively reduce the prominent low-frequency booming characteristics associated with the engine order, thereby contributing to improved cabin acoustic comfort, even when the overall SPL reduction is relatively modest. This is consistent with the intended role of the narrowband ANC system, which focuses on selectively suppressing dominant tonal components rather than broadband cabin noise. In summary, the considerable attenuation of the dominant 2nd-order component, together with the general reduction in overall noise over the entire acceleration range, demonstrates the practical significance of the proposed system for targeted vehicle cabin noise control.

5. Conclusions

In this study, a computationally efficient and fast-converging multi-channel ANC system is developed for vehicle interior engine noise to achieve satisfactory noise attenuation and facilitate its practical application to vehicle cabin noise control. The proposed system is based on the adaptive notch FXLMS algorithm and incorporates an LSP equalization method to address the high computational burden and frequency-dependent convergence behavior of conventional multi-channel ANC systems. The main findings of this study are summarized as follows:

(1)

An LSP equalization method is developed by integrating LSP estimation with eigenvalue equalization to generate low-order equalized LSP models with normalized amplitude-frequency responses. The proposed method improves the convergence performance of the multi-channel ANC system while reducing the computational burden.

(2)

The proposed system substantially reduces computational complexity. Under the typical conditions considered, the proposed system requires only 1.36% of the multiplication operations of the conventional multi-channel ANC system, demonstrating the effectiveness of the proposed LSP equalization method in reducing computational cost.

(3)

Numerical simulations demonstrate improved convergence speed and noise attenuation performance. Compared with the conventional and Zhang’s multi-channel ANC systems, the proposed system achieves faster convergence and superior noise attenuation for both synthesized multi-tonal noise and real interior engine noise.

(4)

Real-vehicle experiments validate the practical noise control capability of the proposed system. Under an accelerated operating condition from 2000 rpm to 3900 rpm, the average attenuation of the 2nd-order noise component at the four error microphones reaches 4.4 dB(A), 6.2 dB(A), 13.4 dB(A), and 10.0 dB(A). These results demonstrate effective tracking and attenuation of time-varying engine noise and the potential of the proposed system for practical vehicle cabin noise control.

Overall, the proposed system achieves a favorable balance among computational efficiency, convergence performance, and noise attenuation, demonstrating its effectiveness for practical vehicle interior engine noise control. Future work will focus on further improving the adaptability and application range of the proposed approach. Specifically, an online adaptive LSP parameter estimation strategy will be investigated to address secondary path variations under changing operating conditions. Machine learning-based methods will also be explored to improve secondary path modeling accuracy and parameter optimization. In addition, the proposed approach will be modified and extended from narrowband ANC to broadband ANC systems and further investigated for emerging vehicle applications, including electric vehicles and other powertrain-related noise control scenarios.