# Microphone array design for the modal analysis of in-duct broadband noise

S. Fauqueux, Vincent Mons

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S. Fauqueux, Vincent Mons. Microphone array design for the modal analysis of in-duct broadband noise. Journal of Sound and Vibration, 2025, 618, pp.119356. 10.1016/j.jsv.2025.119356. hal-05242584

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Journal of Sound and Vibration

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# Microphone array design for the modal analysis of in-duct broadband noise

S. Fauqueux<sup>a ID,*</sup>, V. Mons<sup>b ID</sup>

## ARTICLE INFO

**Keywords:**

Broadband modes estimation

Microphone array design

In-duct acoustic

Condition number

Experimental design

Azimuthal and radial modes

Sensor placement

## ABSTRACT

The azimuthal and radial content of the acoustic field propagating in a duct can be experimentally characterized using an array of phased, wall-mounted microphones. Depending on the number of modes to reconstruct, the number of available microphones and their positions, such an inverse problem may be ill-conditioned, leading to an inaccurate estimation of the modal content of the acoustic signal. One solution involves introducing prior information about the unknown modal content, as achieved through regularization or compressive sensing methods, often based on empirical evidence. Another solution is to better choose the microphone positions. Following this latter approach, an automatic design procedure to identify the microphone array configuration that yields a well-posed inverse problem across the widest possible frequency range is proposed. The microphone array design problem is formulated as the minimization of the condition number of a Gram matrix, derived from the propagation functions between modes and microphones. Three other design experiment criteria are also investigated and compared. The proposed antenna design algorithm is both automatic and computationally efficient, thanks to a gradient-based solution method. The number of microphones is fixed and the frequency band within which the antenna is theoretically adequate is investigated. The present methodology is applied to the design of ONERA's test bench. Synthetic observations are considered to assess the antenna's reconstruction performances relative to a baseline configuration. Robustness with respect to frequency, flow presence and small microphone position shifts are investigated. The selected design is subsequently validated through an experimental campaign.

## 1. Introduction

The International Civil Aviation Organization (ICAO) identifies aircraft noise as the primary cause of adverse community reactions related to airport operations and expansion. To address this issue, one of the four recommended solutions is to reduce noise at its source, especially engine and airframe noises during take-off and landing phases. Acoustic wall liners in power plants and improved propulsion system integration are potential strategies for mitigating take-off phases noise [1], which necessitate characterizing engine acoustic sources. Acoustic imaging methods aim to experimentally determine the locations and strengths of these sources by using phased microphone arrays [2]. Their application in the context of in-duct propagation has been studied a while. Duct wall microphones are commonly employed to avoid disturbing the flow, even if the use of external measurements has also been investigated [3–7] and appear particularly relevant for short nacelle where microphones mounting is greatly constrained. A

<footnote><sup>a</sup> DAAA, ONERA, Institut Polytechnique de Paris, 92320, Châtillon, France</footnote>

<footnote><sup>b</sup> DAAA, ONERA, Institut Polytechnique de Paris, 92190, Meudon, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail addresses: Sandrine.Fauqueux@onera.fr (S. Fauqueux), Vincent.Mons@onera.fr (V. Mons).</footnote>

<footnote>https://doi.org/10.1016/j.jsv.2025.119356</footnote>

<footnote>Received 16 October 2024; Received in revised form 24 July 2025; Accepted 25 July 2025</footnote>

<footnote>Available online 5 August 2025</footnote>

<footnote>0022-460X/© 2025 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).</footnote>
primary strategy involves characterizing the fan's acoustic source distribution: focal points are predefined in the rotor and/or stator planes and the Doppler effect induced by rotating sources is accounted for in the Green's function. The source characterization problem, usually ill-conditioned as the number of potential sources exceeds the number of microphones, is typically solved using beamforming [8] or inverse methods [9]. Alternatively, leveraging the nacelle's axial symmetry allows expressing the in-duct acoustic field on the base of acoustic modes [10]. Azimuthal mode spectrum decomposition can be performed via a discrete azimuthal transform of pressure measurements from an in-duct ring [11]. It can also be formulated as an inverse problem and solved using suitable methods, including compressed sensing for tonal noise and non-negative least square algorithms for broadband noise [12,13], or a combination thereof [14]. Non-regular spacing between microphones is employed to extend to higher azimuthal orders [12]. Radial components of each azimuthal mode are required to fully characterize the sound field propagating in-duct, prompting the development of expanded imaging methods using azimuthally and axially extended antennas [15-20].

All these methods operate in the frequency domain and aim to solve a linear problem of the form $P = GA$, where $P$ is the Fourier transform of the pressure measured by the $N$ microphones, $A$ represents the $N_{mode}$ unknown modal amplitudes, and $G$ is the matrix containing the propagation function between the modes and the microphones. This initial formulation is typically applied for tonal noise characterization. In the case of broadband noise, cross-spectral matrices must be introduced, and the problem to solve becomes $\Gamma = GSG^H$, with $\Gamma$ and $S$ representing the cross-spectral matrices of the microphones and unknown modal amplitudes, respectively. Depending on the number of modes to determine and the number of available microphones, the problem may be over-determined or under-determined. However, even when $N = N_{mode}$, redundant equations (if they exist) can prevent the problem from being solved. In this case, even in an ideal context characterized by punctual and perfect microphones, no additional noise, and completely controlled and known propagation and sources, the inverse method may not converge to the injected modal content due to the high condition number of the matrix to be inverted [19], i.e. the propagation matrix $G$ for tonal noise or the Gram matrix $G^H G$ considered in this study. This issue can be addressed through singular value decomposition or Tikhonov regularization, which aims to pull the smallest eigenvalues up to enable numerical computation of the inverse matrix [16,20]. However, regularization methods impose additional a priori on the solution, which must be based on intrinsic properties of the researched quantities to improve the inverse method. Otherwise, the estimated solution will not be physical. Compressive sensing methods [5,14,18] reduce the dimension of the solution space by imposing that few modes explain the entire measurements. This hypothesis is particularly relevant in the case of fan tonal noise at subsonic rotor tip speeds, which appears at multiples of the blade passing frequency and the interaction frequencies of rotor and stator [10], even in modern architectures (UHBR - Ultra High Bypass Ratio) where dominant azimuthal modes are scattered into neighboring modes due to distortion and high variation of the flow for non-zero nacelle angle of incidence [21]. In the case of broadband noise, in-duct modal content must be evaluated with minimal additive assumptions. To this end, regularization methods or compressive sensing evaluation should be avoided. As noted by C. Hansen in the introduction of his book *Rank-deficient and discrete ill-posed problems*, it is advisable to avoid solving linear problem with large condition numbers caused by an incorrect mathematical model, which should therefore be modified [22]. In the context of in-duct modes estimation, the number of unknowns directly depends on the analysis frequency and cannot be adjusted. Consequently, an incorrect model formulation is caused by an incorrect microphone array design that fails to measure certain modal variations, thereby projecting this information onto lower-order modes. Thus, the only way to obtain well-posed mathematical problem is to select better microphone positions, if possible. The issue addressed in this article concerns the automatic design of a microphone array suitable for in-duct broadband mode estimation using wall-mounting microphones, over the widest possible frequency range.

The objective of the antenna design method explored in this study is not to reduce the number of microphones, as proposed in [11] or [14], but rather to ensure the validity of modes estimation. For this purpose, the authors aim to minimize the condition number of either the aforementioned propagation matrix or its associated Gram matrix. Additional criteria for sensor selection from predefined candidate positions will be discussed later. U. Tapken and L. Enghardt compared the condition number obtained for a limited set of sensor arrangements to distinguish between upstream and downstream modes [23]. A similar approach was adopted for estimating azimuthal and radial modes using external antennas [3,7] and in-duct microphone arrays [16,24]. The first automatic antenna design algorithm was proposed by S. Sack, M. Abom and G. Efraimsson, in [25], to optimize sensor configurations for acoustic multi-ports in ducts. The authors preliminary deduced so-called "conditions for non-singular sensor spacing" for every pair of microphones, ensuring well-conditioned system matrices. However, directly identifying microphone distributions that satisfy these conditions becomes increasingly challenging for more than two microphones. To address this, a genetic algorithm was introduced to minimize the averaged condition number across the entire frequency range. Unfortunately, this optimization procedure requires repetition with different initial populations, while a single instance of the genetic algorithm approach already demands a substantial number of condition number evaluations, which is expected to surge dramatically with an increasing number of microphones. Furthermore, S. Fauqueux defined an antenna by minimizing the condition number using Matlab's optimization functions for modal estimation on a realistic numerical model of UHBR tonal noise, but this study was limited to a single frequency [13]. Consequently, to the best of the authors' knowledge, no truly efficient approach for designing microphone arrays tailored to broadband in-duct propagation has been investigated thus far.

Beyond the domain of in-duct propagation, sensor array design is also a concern for localizing and characterizing un-ducted sources [2]. The initial selection of focal points at potential source positions, their discretization, and placement, all contribute to increased complexity. A common strategy involves minimizing both the beam width and the maximum side lobe level, which are related to the microphone array's response in the presence of a single source. The general shape of the antenna is often predetermined: it may be a single or multiple spiral, grid, circle or cross-shape, or it may be determined through an optimization process for either static [26-29] or moving [30] sources. Another key design criterion is ensuring that the distance between any two microphones is unique, thereby reducing spatial aliasing [31,32]. Furthermore, P.A. Nelson and S.H. Yoon investigated the
geometric parameters influencing the condition number of the propagation matrix in the context of a linear distribution of sources and microphones located in the same plane [33]. Their sensitivity study was later extended to estimate fan acoustic sources in duct, considering a microphone antenna comprising multiple rings [34].

Other criteria are considered in the field of experimental design [35–38], with a traditional motivation centered on achieving the most precise statistical inference possible. The most classic criteria aim to minimize either the determinant of the Fisher information matrix (D-optimal criterion), its trace (A-optimal criterion), or its largest eigenvalue (E-optimal criterion). The application of such methods in the acoustic field is devoted to source and/or sensor selection, from a set of predefine positions [39–41], but does not necessarily prevent the system from being ill-conditioned [39,42]. More recently, Koyama et al. provided a comprehensive overview of state-of-the-art source and sensor selection methods in sound field control [43], assessing their performance in two-dimensional broadband numerical scenarios.

In the first part of this article, an optimization strategy for in-duct antenna design is developed. Excluding brute force approaches, such as combinatorial searches among all possible designs [16,39–41,44] along with global optimization techniques whose computational cost dramatically increases with the number of microphones (e.g. genetic algorithm-based approaches), the use of gradient-based descent methods is here considered. This approach necessitates evaluating the gradient of the design criterion with respect to microphone positions. Various criteria are examined: the condition number, as well as A-, D-, and E-Optimal criteria derived from experimental designs. All these criteria can be expressed as functions of the (Gram) propagation matrix eigenvalues, which requires computing gradients of these eigenvalues. In non-linear inverse problems, the gradient of the condition number can be obtained through second-order adjoint techniques [45,46] or approximations in evaluating the condition number [47]. In the case of acoustic modes propagation in ducts, the analytical expression of the measured pressure field at the microphones positions is exploited to derive a straightforward and computationally inexpensive expression of the required gradient. This enables to design a computationally-efficient optimization approach that scales well with the number of in-duct modes to estimate and the number of microphones. The proposed microphone array design method is applied to the ONERA test bench for array processing validation, and its suitability is numerically and experimentally investigated in the second part of this article.

The paper is organized at follows. The antenna design methods are presented in part 2. The formulation of the microphone array design problems to be solved is detailed in Section 2.1, followed by their resolution in Section 2.2, and application to the ONERA test bench configuration. A comparison against design criteria leads to the selection of one microphone array. Stability analyses on microphone position errors, which may occur during manufacturing, and the impact of flow on the microphone array's performance are presented in Sections 2.3 and 2.4, respectively. Part 3 is dedicated to the numerical and experimental evaluation of the performances of the antenna designed in part 2. First, the modal estimation approach is introduced in Section 3.1. Subsequently, the antenna's performance and frequency validity range are assessed, under perfect conditions that are numerically simulated (Section 3.2) and on experimental measurements (Section 3.3). Finally, concluding remarks are provided in Section 4.

## 2. Microphone array design

### 2.1. Formulation of microphone array design problems

The test bench is assimilated to a rigid cylindrical duct, of constant radius R. A uniform and subsonic flow of Mach number $M_0$ along the revolution axis $(O, \bar{x})$ of the duct is considered. The Fourier transform<fnref n="1" /> $\hat{p}$ of the sound pressure field propagating inside the duct at frequency $\nu$ is expressed as a weighted sum of azimuthal (m) and radial (n) modes propagating upstream ($\epsilon = -$) and downstream ($\epsilon = +$) [10]. $(\bar{x}, \bar{r}, \bar{\theta})$ being the cylindrical coordinates of current point, one introduces the Helmholtz number $H_e = 2 \pi \nu R/c_0$, with $c_0$ the speed of sound, and:

$$x = \frac{\bar{x}}{\beta_0 R}; r = \frac{\bar{r}}{R}; \theta = \bar{\theta}; \xi_e = \frac{H_e}{\beta_0} \qquad (1)$$

where $\beta_0$ is defined by $\beta_0^2 = 1 - M_0^2$. The microphones being wall mounted, $r = 1$ and $\hat{p}$ is expressed as:

$$\hat{p}(x, \theta, \xi_e) = \sum_{m=-\infty}^{+\infty} \sum_{n=0}^{+\infty} \sum_{\epsilon=\pm} A_{m,n}^{\epsilon}(\xi_e) J_m(\mu_{m,n}) \exp(-i m \theta) \exp[-i k_{m,n}^{\epsilon}(\xi_e) x] \qquad (2)$$

$J_m$ denotes the Bessel function of the first kind, and $\mu_{m,n}$ is the $(n+1)$th root of its derivative, $J_m'$. The axial wave number, given by:

$$k_{m,n}^{\epsilon}(\xi_e) = -M_0 \xi_e + \epsilon \sqrt{\xi_e^2 - \mu_{m,n}^2} \qquad (3)$$

takes real value if and only if $\xi_e \ge \mu_{m,n}$. The mode is cut-on. Otherwise, the imaginary part of $k_{m,n}^{\epsilon}(\xi_e)$ takes a non-zero value, indicating that the mode is cut-off. Hereafter, only cut-on modes are retained. The number of these modes depends on $\xi_e$ and is denoted as $N_{mode}(\xi_e)$. Each ith cut-on mode, written as $[i]$, is associated to azimuthal order $m_i$ and radial order $n_i$ propagating on the direction defined by $\epsilon_i$. For conciseness, its wave number $k_{m_i,n_i}^{\epsilon}$ will be referred to as $k_{[i]}$ in the following discussions.

<footnote n="1">¹ The Fourier transform of function f at frequency v is defined by: $\hat{f}(x, v) = \int_{-\infty}^{+\infty} f(x, t) \exp(-2i \pi v t) dt$.</footnote>
The modal identification for broadband signal aims at determining the cross-spectral matrix of the mode amplitudes S(ξe) such that:

$$ \Gamma(\xi_e) = G(\xi_e) S(\xi_e) G^H(\xi_e) \quad (4) $$

with Γ(ξe) the power cross-spectral matrix of the measured acoustic pressure and G^H the conjugate transpose matrix of G. G(ξe) is the propagation matrix between the cut-on modes ([j])j=1,2,...,Nmode(ξe) and the N wall mounted microphones of coordinates (xᵢ, 1, θᵢ)ᵢ=1,2,...,N, and defined by:

$$ G_{[l,j]}(\xi_e) = J_{mj}(\mu_{mj,n_j}) \exp(-i m_j \theta_j) \exp[-i k_{[l]}(\xi_e) x_j] \quad (5) $$

All modes correlation are accounted and have to be estimated, as radial modes of same azimuthal order may be correlated by reflection at the duct-end and in the fan plan [17,19]. Furthermore, correlation between positive and negative value of the same azimuthal order has been highlighted on an aircraft air conditioning system composed of an air pump [24]. By multiplying equality (4) by G^H(ξe) on the left and by G(ξe) on the right, it can be written:

$$ G^H(\xi_e) \Gamma(\xi_e) G(\xi_e) = K(\xi_e) S(\xi_e) K(\xi_e) \quad (6) $$

where K(ξe) = G^H(ξe) G(ξe) is the Nmode(ξe) × Nmode(ξe) Hermitian matrix given by:

$$ K_{[p,l][q,\Theta,\xi_e]} = J_{mp}(\mu_{mp,n_p}) J_{mq}(\mu_{mq,n_q}) \sum_{j=1}^{N} \exp[i(m_p - m_q)\theta_j] \exp\{i[k_{[p]}(\xi_e) - k_{[q]}(\xi_e)]x_j\} \quad (7) $$

K represents both the Gram matrix of propagation between modes and microphones, as well as the inverse of the Fisher information matrix associated with the linear Eq. (4). Consequently, in cases where K(ξe) is invertible, the solution of Eq. (4) can be deduced:

$$ S(\xi_e) = G^+(\xi_e) \Gamma(\xi_e) G^{+H}(\xi_e) \quad (8) $$

where G^+(ξe) = K^−1(ξe) G^H(ξe) is the pseudo-inverse, or Moore-Penrose inverse, of G(ξe). By construction, the solution yielded by Eq. (8) is a positive semi-definite matrix.

The well-posed nature of the problem is directly linked to the condition number of the matrix K(ξe), which also quantifies the solution sensitivity of the linear problem (4) with respect to additional noise and propagation disturbances [48]. Let (λⱼ)ⱼ=1,2,...,Nmode(ξe) denotes its eigenvalues, which are real and positive due to K(ξe) being a non-negative Hermitian matrix. Since the matrix is square, its condition number is computed by dividing the largest eigenvalue, λmax, by the smallest, λmin. With X = (xᵢ)ᵢ=1,2,...,N and Θ = (θᵢ)ᵢ=1,2,...,N, the antenna design problem for modal estimation, focusing on minimizing the **condition number**, can be formulated as follows:

$$ (\mathbf{X}_{\text{opt}}(\xi_e), \mathbf{\Theta}_{\text{opt}}(\xi_e)) = \underset{\mathbf{X},\mathbf{\Theta}}{\operatorname{argmin}} \frac{\lambda_{\max}(\mathbf{X}, \mathbf{\Theta}, \xi_e)}{\lambda_{\min}(\mathbf{X}, \mathbf{\Theta}, \xi_e)} \quad (9) $$

The antenna design problems against A-, D- and E-optimal criteria are written as [37]:

* **A-optimal** design minimizes the trace of K^−1(ξe):

$$ (\mathbf{X}_{\text{opt}}(\xi_e), \mathbf{\Theta}_{\text{opt}}(\xi_e)) = \frac{1}{N_{\text{mode}}(\xi_e)} \underset{\mathbf{X},\mathbf{\Theta}}{\operatorname{argmin}} \sum_{j=1}^{N_{\text{mode}}(\xi_e)} \frac{1}{\lambda_j(\mathbf{X}, \mathbf{\Theta}, \xi_e)} \quad (10) $$

* **D-optimal** design minimizes the logarithm of K^−1(ξe) determinant:

$$ (\mathbf{X}_{\text{opt}}(\xi_e), \mathbf{\Theta}_{\text{opt}}(\xi_e)) = \frac{1}{N_{\text{mode}}(\xi_e)} \underset{\mathbf{X},\mathbf{\Theta}}{\operatorname{argmax}} \sum_{j=1}^{N_{\text{mode}}(\xi_e)} \log \lambda_j (\mathbf{X}, \mathbf{\Theta}, \xi_e) \quad (11) $$

* **E-optimal** design minimizes the largest eigenvalue of K^−1(ξe):

$$ (\mathbf{X}_{\text{opt}}(\xi_e), \mathbf{\Theta}_{\text{opt}}(\xi_e)) = \underset{\mathbf{X},\mathbf{\Theta}}{\operatorname{argmin}} \frac{1}{\lambda_{\min}(\mathbf{X}, \mathbf{\Theta}, \xi_e)} \quad (12) $$

One has:

$$ k_{[p]}(\xi_e) - k_{[q]}(\xi_e) = \epsilon_p \sqrt{\xi_e^2 - \mu_{mp,n_p}^2} - \epsilon_q \sqrt{\xi_e^2 - \mu_{mq,n_q}^2} \quad (13) $$

Thus, the flow Mach number M₀ does not appear explicitly in K expression but its effect is implicitly included in ξe and (xᵢ)ᵢ=1,2,...,N. Therefore, an antenna designed considering no flow and Helmholtz number Hₑ will be suitable, if compressed on the axial direction by a β₀ factor, at Helmholtz number β₀ × Hₑ.
<figure><img src="image_5.png" /><figcaption>Fig. 1. Microphone distribution in (a) 3-D and (b) 2-D dimensionless coordinate system for the DAAA test bench array.</figcaption></figure>

## 2.2. Resolution of the microphone array design problems

The objective is to design a microphone array, with a fixed number of microphones, suitable for modal estimation across a wide frequency range. In Section 2.1, the problem formulation is presented for a given value of $\xi_e$. Notably, the number of modes increases as the frequency rises. Two questions arise here:

* How many microphones are required to find a design that is suitable at $\xi_e = \xi_0$? Or, equivalently, what is the maximum value of $\xi_e$ that can theoretically be achieved with N microphones?

* Is a microphone array designed at $\xi_e = \xi_0$ still suitable for $\xi_e \le \xi_0$? If not, how to take account of $\xi_e$ variability?

Let us address the first question, omitting $\xi_e$ dependency for clarity. The matrix $G$ is an $N \times N_{mode}$ matrix, whose singular values are derived from the square root of $K$ eigenvalues. Notably, $G$ has at most $\min\{N, N_{mode}\}$ non-zero singular values. The Singular Value Decomposition (SVD) of $G$ is given by:

$$G = U\Sigma V^H \qquad (14)$$

where $U$ and $V$ are unitary matrices and $\Sigma$ is an $N \times N_{mode}$ matrix with zero off-diagonal terms. The eigen decomposition of $K$ becomes:

$$K = V\Sigma^H U^H U\Sigma V^H = V\Sigma^H \Sigma V^H = V\Lambda V^H \qquad (15)$$

If $N < N_{mode}$, then $\Lambda$ has at least $N_{mode} - N$ zero diagonal terms, rendering all design criteria infinite. Consequently, the number of microphones, $N$, must be greater than or equal to the number of modes, $N_{mode}$ [42]. Theoretically, the maximum achievable $\xi_e$ value with $N$ microphones is such that $N_{mode}(\xi_e) \le N$. A numerical validation followed by an experimental analysis will determine the effective frequency validity range of the antenna.

Practical insights can facilitate answering the second question. Let us examine the first antenna from DAAA test bench, depicted in Fig. 1, which comprises 113 microphones. The graphical representation of its associated design criteria (see Fig. 2-(a) and (b)) reveals notable peaks at approximately: $\xi_e = \mu_{1,0} \approx 1.84$, $\xi_e = \mu_{2,0} \approx 3.05$, $\xi_e = \mu_{0,1} \approx 3.83$, $\xi_e = \mu_{3,0} \approx 4.20$, $\xi_e = \mu_{4,0} \approx 5.32$ and $\xi_e = \mu_{1,1} \approx 5.33$. These peaks correspond to frequencies where modes transition from cut-off to cut-on, a phenomenon previously observed in condition number analyses by several authors [19,23,34,49]. The overall trend of the graphs as a function of frequency suggests that an antenna design suitable for a specific $\xi_e = \xi_0$ will remain efficient for all $\xi_e \le \xi_0$, regardless of the chosen design metric.

The various design metrics for the current microphone array explode slightly before $\xi_e \approx 5.35$, for $N_{mode} = 24$ (see Fig. 2-(b)). Notably, this occurs well before the emergence of the first radial mode with an azimuthal order of $m = 5$ ($\mu_{5,0} \approx 6.42$). Assuming a speed of sound $c_0 = 340$ m/s and no flow, this corresponds to a frequency of approximately $\nu \approx 1565$ Hz for the ONERA bench radius $R = 185$ mm. This frequency has been identified in [19] as the maximum theoretical valid frequency of the antenna. In theory, there is significant potential for enhancing the antenna's performance.

Prior to selecting the algorithm to solve problem (9), let us observe that the coefficients of the matrix $K(X, \Theta, \xi_e)$ are formulated as the sum of complex exponential functions in terms of microphones positions $(x_j, \theta_j)$. Consequently, the matrix can be expressed as an analytic function of each microphone coordinate $y(\xi_e)$ and other coordinates dependencies omitted for the sake of conciseness):

$$K(y) = \sum_{n=0}^{\infty} K^{(n)} y^n \qquad (16)$$

where each $K^{(n)}$ is a self-adjoint matrix (see Appendix A). According to P.D. Lax, relying on F. Rellich's theorem, its eigenvalues are regular analytical functions of $y$ [50] and are therefore infinitely differentiable. In particular:

$$\frac{\partial \lambda}{\partial y} = v^H \frac{\partial K}{\partial y} v \qquad (17)$$
<figure><img src="image_6.png" /><figcaption>Fig. 2. (a) K condition number, A- and E-criteria, and (b) D-criterion as functions of $\xi_e$ for the array design represented in Fig. 1; (c) number of cut-on modes as a function of $\xi_e$.</figcaption></figure>

where $\lambda$ and $\nu$ denote the eigenvalue and normal eigenvector respectively. The expression of the derivatives of $K(X, \Theta, \xi_e)$ with respect to $x_j$ and $\theta_j$ are given by:

$$ \left( \frac{\partial K_{[p],[q]}}{\partial x_j} \quad \frac{\partial K_{[p],[q]}}{\partial \theta_j} \right) = i \left( \begin{array}{c} k_{[p]}(\xi_e) - k_{[q]}(\xi_e) \\ m_p - m_q \end{array} \right) G_{j,[p]}^* G_{j,[q]} \quad (18) $$

with $G_{j,[p]}^*$ the complex conjugate of $G_{j,[p]}$, which enables us to explicitly express the gradient of the eigenvalues with respect to microphones coordinates. The gradients of the objective functions are expressed accordingly. Both condition number and E-optimal design rely on the smallest and/or largest eigenvalues, whose derivatives exhibit discontinuities near microphone coordinates where these eigenvalues are multiple. Due to “avoidance of crossing” phenomenon [50], analytic matrix-valued functions rarely possess multiple eigenvalues. However, if the spatial arrangement of microphones approaches a configuration that induces multiple smallest and/or largest eigenvalues at iteration $i$, the iterative algorithm may inaccurately estimate the objective function’s gradient. Nevertheless, it will re-evaluate these eigenvalues in the subsequent iteration $i+1$, potentially leading to a suboptimal optimization path. Ultimately, the optimization problems (9) to (12) are solved using Matlab’s `fminunc` function, employing a quasi-newton algorithm. In cases requiring an additional axial extension constraint, the `fmincon` function is utilized instead. The choice is made to keep $N = 113$ microphones in the new array design. All design algorithms are initialized with the microphone distribution chosen for the first antenna (see Fig. 1). In practice, the algorithm may encounter converge difficulties if the initial conditioning is poor, which typically occurs for $\xi_e$ values at the upper limit of the antenna’s theoretical validity range (i.e. $\xi_e$ such that $N_{mode}(\xi_e) \le N$). To mitigate this, it may be necessary to restart the algorithm multiple times, initializing iteration ($p$) with the microphone distribution obtained at iteration ($p-1$). Given the general trend of different design criteria to either increase (condition number and A- and E- criteria) or decrease (D-criterion) with frequency, a second effective strategy is to apply the algorithm iteratively for successive values of increasing $\xi_e^{(1)} < \xi_e^{(2)} < \dots < \xi_e^{(p)} < \dots < \xi_e^{(p)}$. Specifically, minimization for $\xi_e^{(p)}$ is initialized with the antenna design obtained at $\xi_e^{(p-1)}$.

Depending on the optimization criteria and successive minimizations, multiple microphone distributions can be obtained. Examples are provided in Fig. 3. The first design results from minimizing the condition number with $\xi_e = 13$, utilizing the first strategy (see Fig. 3-(b)). The remaining four designs were generated by iteratively applying the algorithm for increasing values of $\xi_e$: $\xi_e = 5; 7; 9; 11$ and $13$, optimizing the condition number for Design 2 (see Fig. 3-(c)), the A-criterion for Design 3 (see Fig. 3-(d)), D-criterion for Design 4 (see Fig. 3-(e)), and E-criterion for Design 5 (see Fig. 3-(f)). Additionally, a maximum axial length constraint of 1 m was imposed on the antenna. The calculation time to obtain the final design ranges from 1 to 2 minutes on a personal computer equipped with an Intel Xeon processor of 3.5 GHz. Notably, it proved impossible to design an antenna suitable for $\xi_e = 14$, regardless of the optimization path and design metric chosen: the algorithm fails to converge due to 114 modes being cut-on at this frequency, where $N < N_{mode}(\xi_e = 14)$. The condition numbers associated with each design are represented in Fig. 4. The graphical representation of E-criterion is remarkably similar, with magnitude values approximately two orders lower than those obtained for the condition number. For an equal number of microphones, the new designs exhibit equivalent condition numbers and largest eigenvalues of $K^{-1}$ (E-criterion), regardless of the design metric employed. This observation is even more pronounced concerning its trace (A-criterion) and determinant (D-criterion - Fig. 5). This trend generalizes when different microphone distributions are used for initialization and various increasing values of $\xi_e$ are considered, suggesting that optimizing against condition number and A-, D-, and E-criteria yields comparable results in this context.

For an equal number of microphones, the new microphone distributions significantly enhance the design metrics compared to the initial microphone array design, as evidenced in Figs. 4 and 5. The minimum dimensionless space between two microphones ranges from 0.1283 to 0.1808 across the various microphone distributions. Assuming a radius $R = 0.185$ m with no flow, this translates to an
<figure><img src="image_7.png" /><figcaption>Fig. 3. (x,θ) coordinates of microphone distribution, in the dimensionless coordinate system, for (a) initial antenna design, (b) Design 1 and (c) Design 2 obtained by condition number minimization, (d) Design 3 obtained by minimization of the A-criterion, (e) Design 4 obtained by minimization of the D-criterion and (f) Design 5 obtained by minimization of the E-criterion.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 4. Condition number, as a function of $ξ_e$, obtained for antenna designs represented in Fig. 3.</figcaption></figure>

effective spacing of more than 2.37 cm, which is compatible with the mounting of quarter-inch microphones. The antenna extensions for each design are: 81.8 cm, 81.2 cm, 82.5 cm, 84.6 cm and 94.8 cm. Notably, Design 1 exhibits a microphone distribution pattern that roughly resembles two azimuthal rings, facilitating the numbering of microphones after the antenna section's manufacture. This advantage leads to the selection of Design 1.

## 2.3. Stability analysis of microphone position errors

Once the design has been chosen, the next step is to manufacture and drill the section that will receive the microphones. Small shift on microphone position may append during the manufacturing, within the limits of the device's tolerance. The question is therefore whether the optimization criterion obtained with the theoretical design is sensitive to such small microphone position shifts. The focus is made on condition number.

In the following, given the microphone positions $(X, Θ)$, as obtained through the design algorithm and corresponding to Design 1, the objective is to identify the perturbations $(δX, δΘ)$ to these positions that induce the largest increase in the condition number of
<figure><img src="image_9.png" /><figcaption>Fig. 5. D-optimal criterion, as a function of $\xi_e$, obtained for antenna designs represented in Fig. 3.</figcaption></figure>

the antenna for a given mean level of perturbations. To this end, reliance is placed on the following second-order Taylor expansion, where $\xi_e$ dependency is omitted for the sake of notation:

$$ \begin{aligned} \mathrm{cond}\mathbf{K}(\mathbf{X} + \delta\mathbf{X}, \mathbf{\Theta} + \delta\mathbf{\Theta}) ={}& \mathrm{cond}\mathbf{K}(\mathbf{X}, \mathbf{\Theta}) + \nabla_{\mathbf{X}}\mathrm{cond}\mathbf{K}(\mathbf{X}, \mathbf{\Theta}) \cdot \delta\mathbf{X} + \nabla_{\mathbf{\Theta}}\mathrm{cond}\mathbf{K}(\mathbf{X}, \mathbf{\Theta}) \cdot \delta\mathbf{\Theta} + \frac{1}{2} \delta\mathbf{X} \cdot [\nabla_{\mathbf{X}\mathbf{X}}^2\mathrm{cond}\mathbf{K}(\mathbf{X}, \mathbf{\Theta}) \cdot \delta\mathbf{X}] \\ & + \frac{1}{2} \delta\mathbf{\Theta} \cdot [\nabla_{\mathbf{\Theta}\mathbf{\Theta}}^2\mathrm{cond}\mathbf{K}(\mathbf{X}, \mathbf{\Theta}) \cdot \delta\mathbf{\Theta}] + \delta\mathbf{\Theta} \cdot [\nabla_{\mathbf{\Theta}\mathbf{X}}^2\mathrm{cond}\mathbf{K}(\mathbf{X}, \mathbf{\Theta}) \cdot \delta\mathbf{X}] + \mathcal{O}\left[\|(\delta\mathbf{X}^T, \delta\mathbf{\Theta}^T)^T\|^{3/2}\right] \end{aligned} \quad (19) $$

The obtention of the second-order derivatives of the condition number with respect to the microphone positions as involved in the fourth to sixth term in the right-hand-side of Eq. (19) is detailed in Appendix B. It is worth emphasizing that considering both first- and second-order terms in this expansion is relevant here, as will now be explained. Design 1 has been obtained by minimizing the condition number at $\xi_e = 13$. At this frequency, first-order terms should theoretically vanish, leaving only second-order ones to consider. However, at frequencies significantly smaller or larger than $\xi_e = 13$, first-order terms may remain significant, and are therefore kept in this Taylor expansion. For a given mean intensity level $\delta$ of variations in microphone positions, the problem of identifying the perturbations that induce the largest increase in the condition number can thus be formulated as follows, based on Eq. (19):

$$ \max_{\delta\mathbf{X}, \delta\mathbf{\Theta}} \mathrm{cond}\mathbf{K}(\mathbf{X} + \delta\mathbf{X}, \mathbf{\Theta} + \delta\mathbf{\Theta}) \quad \text{subject to} \quad \|(\delta\mathbf{X}^T, \delta\mathbf{\Theta}^T)^T\| = \sqrt{2N\delta} \quad (20) $$

The solution to this optimization problem can be straightforwardly obtained by inverting the following system:

$$ [\mathbf{H}(\mathbf{X}, \mathbf{\Theta}) + \gamma \mathbf{I}] \begin{pmatrix} \delta\mathbf{X} \\ \delta\mathbf{\Theta} \end{pmatrix} = - \begin{pmatrix} \nabla_{\mathbf{X}}\mathrm{cond}\mathbf{K}(\mathbf{X}, \mathbf{\Theta}) \\ \nabla_{\mathbf{\Theta}}\mathrm{cond}\mathbf{K}(\mathbf{X}, \mathbf{\Theta}) \end{pmatrix} \quad (21) $$

where $\mathbf{I}$ here refers to the identity matrix of size $2N$ and $\gamma$ is the Lagrange multiplier that is associated to the equality constraint $\|(\delta\mathbf{X}^T, \delta\mathbf{\Theta}^T)^T\| = \sqrt{2N\delta}$ and

$$ \mathbf{H}(\mathbf{X}, \mathbf{\Theta}) = \begin{pmatrix} \nabla_{\mathbf{X}\mathbf{X}}^2 \mathrm{cond}\mathbf{K}(\mathbf{X}, \mathbf{\Theta}) & \nabla_{\mathbf{\Theta}\mathbf{X}}^2 \mathrm{cond}\mathbf{K}(\mathbf{X}, \mathbf{\Theta}) \\ \nabla_{\mathbf{\Theta}\mathbf{X}}^2 \mathrm{cond}\mathbf{K}(\mathbf{X}, \mathbf{\Theta})^T & \nabla_{\mathbf{\Theta}\mathbf{\Theta}}^2 \mathrm{cond}\mathbf{K}(\mathbf{X}, \mathbf{\Theta}) \end{pmatrix} \quad (22) $$

The evaluation of this Lagrange multiplier is detailed in Appendix C.

The company selected to manufacture the array section guarantees an angular tolerance of $\pm 0.1$ deg and an axial tolerance of $\pm 0.05$ mm, corresponding to a perturbation level of approximately $\delta \approx 1.7 \times 10^{-3}$ (with $\rho_0 = 1$), which is rounded up $\delta = 2 \times 10^{-3}$. Accordingly, at each frequency, the most critical variations in microphone positions are identified using Eq. (21) with $\delta = 2 \times 10^{-3}$, and the resulting condition number is plotted as a dash-dot line in Fig. 6, allowing comparison with the initial value for Design 1 (solid line in Fig. 6). Fig. 6 indicates that the condition number exhibits overall robustness to the considered microphone position variations, in particular for $\xi_e < 13$. One may thus be confident in the preservation of the antenna performances through the manufacturing process.

## 2.4. Impact of flow on microphone array performance

Although our test bench is currently not equipped with a flow generator, it is intriguing to investigate whether the new antenna design remains suitable for non-zero Mach numbers. As highlighted by Eq. (13), the flow Mach number $M_0$ does not explicitly appear in the $\mathbf{K}$ expression; however, its influence is implicitly embedded within the dimensionless variables $\xi_e$ and $(x_i)_{i=1,2,...,N}$. Assuming
<figure><img src="image_10.png" /><figcaption>Fig. 6. Condition number as a function of $\xi_e$: baseline (Design 1, solid line) and for the most critical variation in microphone positions ($\delta = 2 \times 10^{-3}$, dash-dot line).</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 7. Condition number obtained for Design 1 and different Mach numbers.</figcaption></figure>

no flow conditions, the antenna design yields microphone coordinates $\tilde{x}_i = R x_i$ and $\tilde{\theta}_i = \theta_i$, which are suitable for frequencies down to $\nu^{\max} = c_0 H_e^{\max} / (2\pi R)$, where $H_e^{\max} = \xi_e^{\max}$ corresponds to the maximum value of $\xi_e$, such that $N_{\text{mode}}(\xi_e) \le N$.

Interestingly, the same condition number can be achieved with flow for frequencies below $\beta_0 \times \nu^{\max}$, provided that the microphone distribution is compressed axially by a factor of $\beta_0$, which is not feasible in practice. With the microphone positions $(\tilde{x}_i, \tilde{\theta}_i)_{i=1,2,...,N}$ established, Fig. 7 illustrates the condition number of Design 1 for various Mach numbers $M_0$: 0.2, 0.3 and 0.4. For flows with Mach numbers exceeding 0.4, the modal basis should account for potential shear effects, particularly for upstream modes [24], which is not captured by the axial wave numbers given by Eq. (3). The sole impact of flow presence is a reduction in the maximum frequency at which the antenna remains suitable. Regardless of the $M_0$ value, the maximum suitable frequency corresponds to the one such that $N_{\text{mode}}(\nu) \le N$.

### 3. Evaluation of antenna performance: numerical and experimental assessments

The article's first part established a framework to define a microphone array theoretically suitable for azimuthal and radial modes estimation in-duct, considering all potential correlations between modes. The objective of this second part is to validate the antenna's performance and frequency validity range through a two-part approach: firstly, under idealized conditions via numerical simulations (Section 3.2), and subsequently, through experimental measurements (Section 3.3). As a preliminary step, the inverse method is presented.

#### 3.1. Estimation method

The inverse method aims to resolve Eq. (4), subject to the constraint that the matrix $S(\xi_e)$ is semi-definite and positive. All correlations between cut-on modes must be estimated. Our computational approach, dubbed ARMADA (Azimuthal and Radial Modal Analysis for In-Duct Acoustics), minimizes the discrepancy, in the least square sense, between $\Gamma(\xi_e)$ and its theoretical form:

$$ \tilde{S}(\xi_e) = \underset{S(\xi_e)}{\operatorname{argmin}} \|\Gamma(\xi_e) - G(\xi_e) S(\xi_e) G^H(\xi_e)\|_F^2 \quad (23) $$
with $\|\cdot\|_F$ the Frobenius matrix norm. The minimum mean square error estimator is the best linear unbiased estimates of $S(\xi_e)$ (Gauss–Markov theorem). The quadratic minimization problem (23) is resolved under the constraint that $S(\xi_e) \in \mathcal{H}_+^{N_{\text{mode}}(\xi_e)}$, $\forall \xi_e$, with:

$$ \mathcal{H}_+^N = \{S \in \mathcal{M}_{N \times N}(\mathbb{C}) \text{ such that } S = S^\text{H} \text{ and } S \ge 0\} \qquad (24) $$

The objective function, given by Eq. (23), is a convex function. The set $\mathcal{H}_+^N$, complex Hermitian version of the set of real symmetric positive semi-definite matrices, is a convex cone. Thus, any locally optimal solution of our problem is also (globally) optimal [51].

The solution of Eq. (23), subject to the constraint that the matrix S is a non-negative Hermitian matrix, is provided by Eq. (8). Notably, the condition number deteriorates significantly near frequencies where modes transition from cut-off to cut-on. As a result, solving the inverse problem using gradient-based methods is recommended over pseudo-inversion: whereas iterative methods yield satisfactory solutions whose accuracy depends on the chosen stopping criteria, the stability of pseudo-inverse computations cannot be guaranteed when the matrix approaches singularity. Furthermore, regularization or penalization techniques can be easily integrated in future developments to incorporate physical a priori knowledge, such as azimuthal mode intensity constraints, thereby enhancing solution reliability.

For a given value of $\xi_e$, the problem is iteratively resolved using a gradient descent algorithm with projection on $\mathcal{H}_+^{N_{\text{mode}}(\xi_e)}$, denoted $\mathcal{P}_{\mathcal{H}_+^{N_{\text{mode}}(\xi_e)}}$. In the following, $\xi_e$ dependency is omitted for clarity. The iterative algorithm is:

$$ \left\{ \begin{array}{l} \tilde{S}^{(p)} = S^{(p-1)} + \alpha [\mathbf{B} - \mathbf{K} S^{(p-1)} \mathbf{K}] \\ S^{(p)} = \mathcal{P}_{\mathcal{H}_+^{N_{\text{mode}}(\xi_e)}}[\tilde{S}^{(p)}] \end{array} \right. \qquad (25) $$

with $\mathbf{B} = \mathbf{G}^\text{H} \mathbf{\Gamma} \mathbf{G}$ and $\alpha$ such that application:

$$ \mathbf{S} \to \mathbf{S} + \alpha [\mathbf{B} - \mathbf{K} \mathbf{S}] \qquad (26) $$

is a contraction for the Frobenius norm, in order to ensure the iterative algorithm convergence. $\alpha$ should verify [52]:

$$ 0 < \alpha < \frac{2}{\lambda_{\max}^2} \qquad (27) $$

with $\lambda_{\max}$ the maximum eigenvalue of K. In practice, its value is set to the optimum, enabling the best convergence rate:

$$ \alpha = \frac{2}{\lambda_{\min}^2 + \lambda_{\max}^2} \qquad (28) $$

The algorithm is initialized by using:

$$ \mathbf{S}^{(0)} = \mathbf{Id} \times \frac{\|\mathbf{\Gamma}\|_F}{\|\mathbf{G}\|_F} \qquad (29) $$

where $\mathbf{Id}$ denotes the identity matrix of size $N_{\text{mode}} \times N_{\text{mode}}$. The process stops at the first iteration ($p$) such that:

$$ \varepsilon^{(p)} = \frac{\|\mathbf{\Gamma} - \mathbf{G} \mathbf{S}^{(p)} \mathbf{G}^{\text{H}}\|_{\text{F}}}{\|\mathbf{\Gamma}\|_{\text{F}}} < \varepsilon \qquad (30) $$

or if the algorithm stagnates, that is: the residual error $\varepsilon^{(p)}$ has not changed since more than 50 iterations, considering a given precision. Such algorithm requires a large number of iterations when the Hessian of the objective function, $\mathbf{K}^2$, has a large condition number [51]. The low condition number of the array design thus accelerates ARMADA convergence, as well as reduces its sensibility to extraneous noise.

## 3.2. Numerical validation

This section focusses on numerical evaluation of the antenna's suitability under idealized conditions, specifically: (i) absence of additional noise and (ii) in-duct propagation that aligns with the theoretical model described by Eq. (2). In this case, the power cross-spectral matrix of the acoustic pressure field measured by the antenna can be mathematically expressed as:

$$ \mathbf{\Gamma}(\xi_e) = \mathbf{G}(\xi_e) \mathbf{S}^{\text{sim}}(\xi_e) \mathbf{G}^{\text{H}}(\xi_e) \qquad (31) $$

Let $\tilde{S}$ denote the cross-spectral matrix of the mode amplitudes estimated by the inverse method. Do the microphone positions ensure accurate estimation of in-duct modes? Additionally, what is the discrepancy between the simulated matrix $\mathbf{S}^{\text{sim}}$ and the estimated matrix $\tilde{S}$? The error is defined as the Frobenius norm of $\tilde{S} - \mathbf{S}^{\text{sim}}$, giving equally importance to the errors in both auto-spectrum and cross-spectrum, normalized by the Frobenius norm of $\mathbf{S}^{\text{sim}}$. $\mathbf{S}^{\text{sim}}$ is randomly simulated, while ensuring its semi-definite positivity property. 100 matrices are simulated for each investigated frequency value, to obtain statistics on reconstruction error. The algorithm consistently converges when $\xi_e < 11.3$, meeting the criterion outlined in Eq. (30), except near cut-off frequencies where stagnation occurs due to an exploding condition number, and for $\xi_e$ from 11.3. Figs. 8-(a) and 8-(b) represent the averaged and maximum error as functions of $\xi_e$, alongside the number of iterations, respectively. These plots exhibit similar trends to the condition number's evolution with respect to $\xi_e$. The cross spectral matrix of mode amplitudes is retrieved with an error of less than 8% within the antenna's valid frequency range, and typically below 1%. In stark contrast, the previous antenna design yielded errors exceeding 30% from 1500 Hz ($\xi_e \approx 5.35$), with modal amplitude gaps surpassing 10 dB, despite the iterative algorithm's convergence [19]. These results demonstrate the superior reconstruction performance of our present array compared to the initial design.
<figure><img src="image_12.png" /><figcaption>Fig. 8. (a) Averaged and maximum error and (b) iteration number obtained with 100 random simulations of $S^{sim}$ for every $ξ_e$, new antenna ATHAC (black) and previous antenna (gray).</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 9. (a) Test bench equipped with ATHAC antenna and installed in the anechoic room and (b) scheme.</figcaption></figure>

## 3.3. Experimental evaluation

### 3.3.1. Experimental setup

The antenna defined by the first design, and subsequently named “ATHAC antenna”, has been manufactured and underwent experimental testing at the end of 2022 to validate its performance. The ONERA test bench, with a radius of 0.185 m and equipped with the new microphone array, is set up in an anechoic room (see Fig. 9-(a)). A detailed schematic of the bench, comprising multiple cylindrical components, is provided in Fig. 9-(b). The setup featured no flow and untreated duct terminations. One end of the duct is baffled for a separate study [7]. To attenuate cut-off modes generated by reflections at the duct ends, rigid ducts of lengths 1 m and 0.5 m are positioned upstream of the generation unit and downstream of the antenna. An axial gap of approximately 0.28 m exists between the generation unit and the first microphone of the ATHAC antenna. The generation unit is equipped with 16 loudspeakers, evenly spaced in azimuth at the same axial position, allowing for the independent production of azimuthal modes $m_0$ from -8 to 8 by sending a phase-shifted sinusoidal function to each speaker. However, this set-up also generates aliased azimuthal modes of the form $m_0 + 16i, \forall i \in Z^*$, meaning that azimuthal mode $\pm 8$ could be produced simultaneously. For frequencies where azimuthal modes $m = m_0 \pm 16$ are cut-off, single azimuthal mode $m_0 = -7, -6, ..., 7$ can be excited alone. Each loudspeaker has a nearby 1/4" control microphone, enabling the evaluation of the azimuthal power spectral density of modes from -7 to 7 without aliased modes. A temperature probe, located near the baffled duct exit, provides the sound velocity: $c = (1.4 \times 287 T)^{1/2}$, with T being the temperature in Kelvin. During the three-day measurement period, the anechoic chamber's temperature varied between 295.4 and 301.2 K, resulting in sound velocities ranging from 344.5 to 347.9 m/s. Each cut-on azimuthal mode from 0 to 8 are generated independently for frequencies from 300 to 4500 Hz ($H_e$ from $\approx 1.67$ to $\approx 15.14$), at intervals of 100 Hz, yielding to a total of 266 measurements. The sampling frequency is set to 16.384 kHz, and each measurement point is acquired over a duration of 30 seconds, ensuring the convergence of Welch's method for estimating cross-spectral matrices.

### 3.3.2. Experimental validation procedure

A suitable design enables precise retrieval of the modal content within the duct in experimental set-ups. Contrary to what was done in Section 3.2, the modal content is here unknown. Nevertheless, each measurement point is specifically tailored to target a
particular frequency and azimuthal mode, as dictated by the generation module. The strategy leverages the single azimuthal content of the generated acoustic field: by restricting the modal basis to the cut-on radial modes of the specified azimuthal order $m_0$, the problem dimension is significantly reduced. Indeed, assuming the azimuthal generation unit operates perfectly, $\hat{p}$ can be expressed as:

$$ \hat{p}(x, r, \theta, \xi_e) = \sum_{n=0}^{+\infty} \sum_{\epsilon=\pm} A_{m_0, n}^{\epsilon}(\xi_e) J_{m_0}(\mu_{m_0, n} r) \exp(-i m_0 \theta) \exp[-i k_{m_0, n}^{\epsilon}(\xi_e) x] \quad (32) $$

and one axial array is sufficient to estimate the cross-spectral matrix of cut-on mode amplitudes $A_{m_0, n}^{\epsilon}$, which can be compared to the one obtained considering the entire modal basis and the whole antenna.

The first step involves verifying that the targeted azimuthal mode $m_0$ from the generation module is indeed dominant. The second step entails extracting a sub-array from the ATHAC antenna for estimation on the reduced modal basis, followed by validation that the azimuthal modes $m \neq m_0$ are effectively filtered out by this method. Finally, the radial content obtained using the entire modal basis with ATHAC antenna and the one obtained using the reduced modal basis with sub-antenna are compared. Hereafter, only measurement points within the antenna's validity range are considered (i.e., those where the azimuthal mode is generated at frequencies strictly below 4200 Hz).

### 3.3.3. Is targeted azimuthal mode $m_0$ dominant?

After evaluating the measured spectrum, it is confirmed that the targeted generation frequency is systematically reached. An azimuthal mode analysis by Fourier transform is applied to the data from the ring of control microphones [13]. The generated azimuthal mode $m_0$ is found to be fully predominant, except in approximately ten cases where the generation frequency is in the vicinity of cut-off radial modes of a different azimuthal order $m \neq m_0$. These exceptions can be attributed to two factors: firstly, the control microphones are positioned too close to the loudspeakers, which hinders effective attenuation of these modes; secondly, radial modes associated with the same azimuthal order interfere constructively or destructively, depending on the axial position of the microphones ring.

Consequently, the ARMADA estimation method, assuming correlated modes, is applied to the entire modal basis comprising all cut-on modes. The azimuthal mode intensity $I_{m_0}$ is then evaluated by summing the individual modal intensities of all cut-on modes ($m_0, n, \epsilon$):

$$ I_{m_0}(H_e) = \sum_{n \in \mathbb{N}; \mu_{m_0, n} \le H_e} \sum_{\epsilon=\pm} I_{m_0, n}^{\epsilon}(H_e) \quad (33) $$

given by [53] (no flow):

$$ I_{m,n}^{\epsilon}(H_e) = \frac{1}{\rho_0 c_0} \left| A_{m,n}^{\epsilon}(H_e) \right|^2 Y_{m,n} \frac{\sqrt{H_e^2 - \mu_{m,n}^2}}{H_e} \quad (34) $$

with:

$$ Y_{m,n} = \frac{1}{2} \left( 1 - \frac{m^2}{\mu_{m,n}^2} \right) J_m^2(\mu_{m,n}) \quad \forall (m, n) \neq (0, 0) \quad (35) $$

$$ Y_{0,0} = \frac{1}{2} \quad (36) $$

The difference in intensity between the targeted azimuthal mode $m_0$ and non-targeted modes $m \neq m_0$ is represented separately for each target azimuthal mode $m_0$ in Fig. 10. The Helmholtz numbers corresponding to the cut-off frequencies of both the azimuthal $m_0$ and its aliased modes $m_0 + 16i, \forall i \in \mathbb{Z}^*$, are depicted by black dashed vertical lines. The targeted azimuthal mode $m_0$ remains fully dominant for $H_e < \mu_{m_0-16,0}$, until its aliased mode $m_0 - 16$ reaches cut-on. Its predominance slightly decreases when its radial content diversifies. Upon reaching cut-on, the aliased azimuthal mode $m_0 - 16$ achieves a similar importance to that of the targeted mode $m_0$. Consequently, the azimuthal mode generation is deemed satisfactory, even in cases where the azimuthal Fourier transform performed with the control ring reveals unwanted azimuthal modes. In the following, configurations featuring cut-on aliased azimuthal modes will be excluded from consideration. As a result, the cases where $m_0 = 8$ are hereby eliminated.

### 3.3.4. Extraction of sub-antenna and verification of azimuthal mode filtering $m \neq m_0$

In practice, even when the targeted azimuthal mode $m_0$ is dominant, the generation of azimuthal modes is not perfect, and a small amount of energy always exists in non-targeted azimuthal modes $m \neq m_0$, which are inevitably correlated with $m_0$ since all modes are induced by the same generation unit. The energy associated with these non-targeted modes should not be projected onto the reduced modal basis, comprising only the cut-on radial modes of azimuthal order $m_0$, but rather identified as additive noise. As frequency increases, more modes become cut-on, raising the probability that a radial mode associated with $m \neq m_0$ will have an axial wave number close to that of a radial mode associated with $m_0$.

To address this, a sub-antenna is coherently extracted from the ATHAC antenna, consisting of a near-axial array for identifying radial modes, along with microphones covering a wide range of azimuthal positions (potentially at different axial positions) to filter out non-targeted azimuthal modes $m \neq m_0$ (see Fig. 11). The inverse method ARMADA, assuming correlated modes, is applied using the reduced modal basis composed of cut-on modes ($m_0, n$) and the sub-antenna. Notably, the condition numbers for all problems are low across frequency range studied for every $m_0 = 0, 1, ..., 7$.
<figure><img src="image_14.png" /><figcaption>**Fig. 10.** Difference between intensity of targeted azimuthal mode $m_0$ and azimuthal mode $m \neq m_0$, separately for each azimuthal mode (a) $m_0 = 0$, (b) $m_0 = 1$, (c) $m_0 = 2$, (d) $m_0 = 3$, (e) $m_0 = 4$, (f) $m_0 = 5$, (g) $m_0 = 6$, (h) $m_0 = 7$ and (i) $m_0 = 8$.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>**Fig. 11.** Microphone distribution in (a) 3-D (b) and 2-D dimensionless coordinate system of ATHAC antenna and microphones selected in the sub-array (filled circles).</figcaption></figure>

Let us denote $\tilde{S}$, $\tilde{S}^{m_0}$, and $\tilde{S}_{m_0:m_0}$ as the cross-spectral matrices of mode amplitudes estimated by the inverse method when considering, respectively, the complete basis with the ATHAC antenna, the reduced basis with the sub-antenna, and the sub-matrix from $\tilde{S}$ containing only azimuthal modes of order $m_0$. The unexplained parts of measurements acquired on the whole antenna, the sub antenna, and its complement (i.e: microphones belonging to the ATHAC antenna but not to the sub-antenna, denoted “ATHAC
<figure><img src="image_16.png" /><figcaption>Fig. 12. Part of measurements of the whole (plain), sub (star) and complement (circle) antennas that are not explained by $\tilde{S}$ (Complete basis), $\tilde{S}_{m_0,m_0}$ (Complete basis - $m_0$ only) and $\tilde{S}^{m_0}$ (Reduced basis), separately for each azimuthal mode (a) $m_0 = 0$, (b) $m_0 = 1$, (c) $m_0 = 2$, (d) $m_0 = 3$, (e) $m_0 = 4$, (f) $m_0 = 5$, (g) $m_0 = 6$ and (h) $m_0 = 7$.</figcaption></figure>

\Sub antenna") by $\tilde{S}$ are identical for all cases (see Fig. 12 - plain, star and circle markers). This corresponds to convergence error: $\epsilon^{(p)}$, given by Eq. (30) at the last algorithm iteration $p$, which is consistently below 10% except near Helmholtz numbers where radial modes ($m_0, n$) transition from cut-off to cut-on (previously observed in Fig. 5 regarding the condition number of the ATHAC antenna). Similarly, the part of measurements acquired on the whole antenna, the sub antenna, and its complement, and unexplained by $\tilde{S}^{m_0}$ and $\tilde{S}_{m_0,m_0}$, are also identical (see Fig. 12 - red and blue curves). However, reducing the modal basis to only azimuthal modes $m_0$ does not entirely explain the measurements. Thus, it appears that azimuthal modes of orders different from $m_0$ are effectively filtered, despite their non-negligible contribution to the measurements.

### 3.3.5. Comparison of radial content obtained with the two methods

The absolute difference, in dB, between the intensity of the azimuthal mode $m_0$ obtained using the entire modal basis with the ATHAC antenna and that obtained using the reduced modal basis with the sub-antenna is illustrated in Fig. 13. Outside the vicinity of cut-off frequencies, this difference remains consistently below 0.3 dB across the entire frequency range studied for targeted mode $m_0 \ge 3$, and for $\epsilon_e \le 10$ when $m_0 \le 2$. In other cases, the intensity difference occasionally reaches up to 0.6 dB. Near cut-off frequencies, the gaps are always less than 1 dB, except for a single instance when $m_0 = 0$ is generated at 4000 Hz ($H_e \approx 13.49$), close to the cut-off frequency of mode (0,4), resulting in a difference of 4.5 dB. Azimuthal modes $m \ne m_0$ have been successfully filtered out, with a tolerance of either 0.3 or 0.6 dB on the modal intensity, depending on both the generated azimuthal mode and the frequency.

Only the comparison of significant modes is of interest. Significant modes ($m_0, n, \epsilon$) are defined as those where $I_{m_0,n}^{\epsilon} \ge I_{m_0} - \delta$ dB, with two threshold values: $\delta = 7$ (i.e., the modal intensity represents more than 20% of $I_{m_0}$) and $\delta = 10$ (i.e., the modal intensity represents more than 10% of $I_{m_0}$). The absolute difference, in dB, between the modal intensity obtained using the entire modal basis with the ATHAC antenna and that obtained using the reduced modal basis with the sub-antenna is illustrated in Fig. 14, with
<figure><img src="image_17.png" /><figcaption>**Fig. 13.** Absolute difference, in dB, between $I_{m0}$ estimated using the entire modal basis with ATHAC antenna, and the reduced modal basis with sub-antenna, separately for each azimuthal mode (a) $m_0 = 0$, with zoom in (b), (c) $m_0 = 1$, (d) $m_0 = 2$, (e) $m_0 = 3$, (f) $m_0 = 4$, (g) $m_0 = 5$, (h) $m_0 = 6$ and (i) $m_0 = 7$.</figcaption></figure>

a scale determined based on results outside the vicinity of cut-off frequencies. Note that this difference corresponds to the gap, in dB, between the squared amplitudes of modes estimated by each method. The overall trend is consistent with that observed for $I_{m0}$ (see Fig. 13). Most gaps are smaller than 0.5 dB at frequencies where only three radial modes are cut-on, and less than 1 dB near cut-on frequencies. However, the difference can occasionally reach up to 1.6 dB or even 2.5 dB for less significant modes. The comparison yields highly satisfactory results in light of the uncertainties associated with the inverse method on the reduced azimuthal basis. These findings are consistent with theoretical expectations: results obtained near cut-off frequencies exhibit increased sensitivity to additional noise (i.e., the presence of azimuthal modes different than $m_0$ on the reduced basis), and error may grow with frequency. Notably, the modal amplitudes estimated using the ATHAC antenna generally achieve an accuracy of 0.25 dB.

## 4. Conclusion

This article focuses on characterizing acoustic fields propagating in ducts, with a particular emphasis on the automatic design of a microphone array suitable for in-duct mode estimation using wall-mounting microphones over the widest possible frequency range. The primary objective of the antenna design method investigated here is to ensure the validity of mode estimations for both tonal and broadband noise propagating within a rigid cylindrical duct of constant radius, subjected to uniform and subsonic flow.

The antenna array design problem relies on optimizing various metrics of the Gram matrix $K = G^H G$, which is engendered by the family of vectors $(G_{[j]})_{j=1,2,...,N_{\text{mode}}}$ representing the propagation between each mode $[j]$ and all microphones. These metrics include minimizing the condition number, trace (A-optimal criterion), determinant (D-optimal criterion), or largest eigenvalue (E-optimal criterion) of its inverse matrix. The ultimate goal is to determine a microphones distribution that induces a family of vectors as orthogonal as possible. The study revealed that the number of microphones to consider must correspond to the maximum number of cut-on modes within the frequency bandwidth of interest. Moreover, it was found that an antenna designed for optimal performance at a given frequency remains suitable for lower frequencies. The proposed design algorithm leveraged the differentiability of Gram
<figure><img src="image_18.png" /><figcaption>**Fig. 14.** Absolute difference, in dB, between $I_{m_0,n}^\varepsilon$ estimated using the entire modal basis with ATHAC antenna, and the reduced modal basis with sub-antenna, for significant modes only, separately for each azimuthal mode (a) $m_0 = 0$, (b) $m_0 = 1$, (c) $m_0 = 2$, (d) $m_0 = 3$, (e) $m_0 = 4$, (f) $m_0 = 5$, (g) $m_0 = 6$ and (h) $m_0 = 7$.</figcaption></figure>

matrix K eigenvalues and their analytical derivative with respect to microphone positions, which is straightforward and inexpensive to evaluate, and allows the use of efficient gradient based descent methods. However, the algorithm may struggle to converge if the initialization criteria are poorly chosen, particularly at the upper limit of the theoretical validity range of the antenna. To address this, two strategies are proposed: (1) applying the gradient-based descent method multiple times near the maximum frequency of interest, with each iteration initialized using the previous microphones distribution; and (2) iteratively applying the gradient-based descent method for increasingly higher frequencies, with each iteration initialized using the microphone distribution obtained for the immediately lower frequency. The algorithm is computationally efficient, requiring approximately 1 – 2 minutes on a personal computer when applied to the frequency bandwidth $H_e \in [0, \mu_{12,0}]$ ($\mu_{12,0} \approx 13.87$), where up to 110 modes are cut-on. Antenna designs obtained through optimization for the various design metrics (condition number, A-, D- and E- metrics) exhibit significantly improved metrics compared to the initial test bench microphone array design. Notably, optimizing these design metrics yields similar results to minimizing the condition number of K. The condition number remains stable against small microphone position shifts. Furthermore, a microphone array designed considering no flow remains suitable with flow in a frequency range where the maximum number of cut-on modes is lower than or equal to the number of microphones.

One of the designs has been selected for further evaluation. Initially, the antenna's performance and frequency validity range have been assessed under ideal conditions: perfect, punctual microphones; no additional noise; and in-duct propagation matching the theoretical model. The cross-spectral matrix of mode amplitudes could be retrieved with less than 1% error (always less than 8%) within the antenna's validity frequency range, which was substantially extended compared to the initial antenna. The microphone array has been manufactured and employed in an experimental campaign to verify its performance. The generation unit, equipped with 16 loudspeakers regularly arranged in azimuth on the same axial position, produces azimuthal modes from –8 to 8 alone, including aliased azimuthal modes when cut-on. An estimation method assuming correlated modes and considering the entire modal base composed of cut-on modes has been applied. Estimated modes explain more than 90% of the measured cross-spectral matrix across the theoretical frequency validity range of the antenna, except near cut-off frequencies of radial modes associated with
the generated azimuthal order $m_0$, where this percentage decrease to 65%. Mode intensities are compared with the one obtained using a modal basis restricted to cut-on radial modes of azimuthal order $m_0$ and a sub-antenna. Azimuthal modes different from $m_0$, potentially generated by the generation unit with low energy, are filtered out by the sub-antenna with a tolerance of 0.3 to 0.6 dB on the modal intensity, depending on the azimuthal mode and frequency. The modal amplitudes of significant modes (i.e., those representing more than 20% of the azimuthal intensity $I_{m0}$) are obtained with an accuracy of 0.25 dB, which is highly satisfactory. Generally, errors increase near cut-off frequencies, coinciding with an increase in the condition number. These findings experimentally confirm the theoretical frequency validity range of the antenna design.

### CRediT authorship contribution statement

**S. Fauqueux:** Writing – original draft, Validation, Software, Methodology, Conceptualization. **V. Mons:** Writing – review & editing, Software, Methodology, Conceptualization.

### Declaration of competing interest

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

### Acknowledgment

The authors would like to thank Fabrice Desmerger, Laurent Coste and Benjamin Etchebarne, from ONERA, for their involvement during the experimental campaign.

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

## Appendix A. Proof of Eq. (16)

Let us consider $y = \theta_j$ and denote $K^{(0)}$ the $K$ matrix estimated when $\theta_j = 0$. According to Eq. (7) :

$$K_{[p],[q]}(y) = K_{[p],[q]}^{(0)} + J_{m_p}(\mu_{m_p,n_p}) J_{m_q}(\mu_{m_q,n_q}) \exp[i(k_{[p]} - k_{[q]})x_j] \sum_{n \ge 1} i^n (m_p - m_q)^n y^n \quad (A.1)$$

$K^{(0)}$ is a self-adjoint matrix, as is $K^{(n)}$ defined by :

$$K_{[p],[q]}^{(n)}(y) = J_{m_p}(\mu_{m_p,n_p}) J_{m_q}(\mu_{m_q,n_q}) \exp[i(k_{[p]} - k_{[q]})x_j] i^n (m_p - m_q)^n \quad (A.2)$$

Let us now consider $y = x_j$ and denote $K^{(0)}$ the $K$ matrix estimated when $x_j = 0$. According to Eq. (7) :

$$K_{[p],[q]}(y) = K_{[p],[q]}^{(0)} + J_{m_p}(\mu_{m_p,n_p}) J_{m_q}(\mu_{m_q,n_q}) \exp[i(m_p - m_q)\theta_j] \sum_{n \ge 1} i^n (k_{[p]} - k_{[q]})^n y^n \quad (A.3)$$

$K^{(0)}$ is a self-adjoint matrix, as is $K^{(n)}$ defined by :

$$K_{[p],[q]}^{(n)}(y) = J_{m_p}(\mu_{m_p,n_p}) J_{m_q}(\mu_{m_q,n_q}) \exp[i(m_p - m_q)\theta_j] i^n (k_{[p]} - k_{[q]})^n \quad (A.4)$$

## Appendix B. Second-order derivatives of the condition number with respect to the microphone positions

The first- and second-order derivatives of the condition number of the matrix $K$ with respect to the microphone positions are given by

$$\frac{\partial \mathrm{cond} K}{\partial y} = \mathrm{cond} K \left( \frac{1}{\lambda_{\max}} \frac{\partial \lambda_{\max}}{\partial y} - \frac{1}{\lambda_{\min}} \frac{\partial \lambda_{\min}}{\partial y} \right) \quad (B.1)$$

and

$$\frac{\partial^2 \mathrm{cond} K}{\partial y \partial z} = \frac{1}{\mathrm{cond} K} \frac{\partial \mathrm{cond} K}{\partial y} \frac{\partial \mathrm{cond} K}{\partial z} + \mathrm{cond} K \left( \frac{1}{\lambda_{\max}} \frac{\partial^2 \lambda_{\max}}{\partial y \partial z} - \frac{1}{\lambda_{\max}^2} \frac{\partial \lambda_{\max}}{\partial y} \frac{\partial \lambda_{\max}}{\partial z} + \frac{1}{\lambda_{\min}^2} \frac{\partial \lambda_{\min}}{\partial y} \frac{\partial \lambda_{\min}}{\partial z} - \frac{1}{\lambda_{\min}} \frac{\partial^2 \lambda_{\min}}{\partial y \partial z} \right) \quad (B.2)$$

respectively, where $y$ and $z$ refer to any azimuthal or axial coordinate of the microphones. The second-order derivative (B.2) involves both first-order derivatives of the eigenvalues of $K$ with respect to the microphone positions, as given by Eqs. (17)–(18), and second-order ones. The latter may be evaluated through

$$\frac{\partial^2 \lambda}{\partial z \partial y} = 2 \Re \left( \mathbf{v}^H \frac{\partial \mathbf{K}}{\partial y} \frac{\partial \mathbf{v}}{\partial z} \right) + \mathbf{v}^H \frac{\partial^2 \mathbf{K}}{\partial z \partial y} \mathbf{v} \quad (B.3)$$
where, as in Eq. (17), **v** refers to the eigenvector with unitary norm that is associated to the eigenvalue **λ**. In Eq. (B.3), the expression of ∂**K**/∂**y** is given by (18), while, still relying on the assumption that **λ** has unitary geometric multiplicity, the derivative of the eigenvector ∂**v**/∂**z** may be obtained through inverting:

$$ (\mathbf{K} - \lambda \mathbf{I} \mathbf{d}) \frac{\partial \mathbf{v}}{\partial \mathbf{z}} = - \left( \frac{\partial \mathbf{K}}{\partial \mathbf{z}} - \frac{\partial \lambda}{\partial \mathbf{z}} \mathbf{I} \mathbf{d} \right) \mathbf{v} \quad (B.4) $$

where **I**d here refers to the identity matrix of size N<sub>mode</sub>. Finally, Eq. (B.3) also requires the second-order derivatives of **K** with respect to the microphone positions, which are given by:

$$ \left( \begin{array}{c} \frac{\partial^2 \mathbf{K}_{[p], [q]}}{\partial x_l \partial x_j} \\ \frac{\partial^2 \mathbf{K}_{[p], [q]}}{\partial \theta_l \partial \theta_j} \\ \frac{\partial^2 \mathbf{K}_{[p], [q]}}{\partial x_l \partial \theta_j} \end{array} \right) = - \left( \begin{array}{c} [k_{[p]}(\xi_e) - k_{[q]}(\xi_e)]^2 \\ (m_p - m_q)^2 \\ [k_{[p]}(\xi_e) - k_{[q]}(\xi_e)] (m_p - m_q) \end{array} \right) \mathbf{G}_{j,[p]}^* \mathbf{G}_{j,[q]} \delta_{lj} \quad (B.5) $$

## Appendix C. Evaluation of the Lagrange multiplier for the optimization problem (20)

In this appendix, we discuss the identification of the Lagrange multiplier **γ** that is associated to the equality constraint in the optimization problem (20) to determine optimal perturbations in the microphone positions around selected designs. Based on second-order optimality conditions and finding a lower bound for the norm of the right-hand-side in Eq. (21), one may find:

$$ -\tilde{\lambda}_{\max} \left( 1 + \frac{\| \nabla \text{cond} \mathbf{K} \|}{\tilde{\lambda}_{\max} \sqrt{2N\delta}} \right) \le \gamma \le -\tilde{\lambda}_{\max} \quad (C.1) $$

where $\tilde{\lambda}_{\max}$ is the largest eigenvalue of the Hessian matrix **H**(**X**, **Θ**) as defined in (22), while $\nabla \text{cond} \mathbf{K}$ gathers the first-order derivatives of $\text{cond} \mathbf{K}$ with respect to the axial and azimuthal microphone positions. In practice, it is found that the value of **γ** that allows to fulfill the equality constraint in Eq. (20) is close to the lower bound in (C.1), so that a proper value of **γ** may be straightforwardly found through tuning and starting from this lower bound. Incidentally, two limiting cases may be identified from Eq. (C.1). When $\|\nabla \text{cond} \mathbf{K}\|/(\tilde{\lambda}_{\max} \sqrt{2N\delta}) \ll 1$, which corresponds to vanishing gradient terms compared to Hessian ones in the Taylor expansion (19), $\gamma \sim -\tilde{\lambda}_{\max}$ and the optimal perturbations in the microphone perturbations as given by Eq. (21) are aligned with the eigenmode of **H**(**X**, **Θ**) that is associated to $\tilde{\lambda}_{\max}$. Conversely, when $\|\nabla \text{cond} \mathbf{K}\|/(\tilde{\lambda}_{\max} \sqrt{2N\delta}) \gg 1$, $\gamma \sim -\|\nabla \text{cond} \mathbf{K}\|/(\sqrt{2N\delta})$ and the optimal perturbations are aligned with $\nabla \text{cond} \mathbf{K}$.

## Data availability

Data will be made available on request.

## References

[1] Icao environmental report 2019 (chapter 1 – aviation and environment outlook). https://www.icao.int/environmental-protection/Documents/ICAO-ENV-Report2019-F1-WEB%20(1).pdf.

[2] R. Merino Martínez, P. Sijtsma, M. Snellen, T. Ahlefeldt, J. Antoni, C.J. Bahr, D. Blacodon, D. Ernst, A. Finez, S. Funke, T.F. Geyer, S. Baxter, G. Herold, X. Huang, W.M. Humphreys, Q. Leclère, A. Malgoezar, U. Michel, T. Padois, A. Pereira, C. Picard, E. Sarradj, H. Siller, D.G. Simons, C. Spehr, A review of acoustic imaging methods using phased microphone arrays, CEAS Aeronaut. J. 10 (1) (2019) 197–230, http://dx.doi.org/10.1007/s13272-019-00383-4.

[3] F. Castres, P. Joseph, Experimental investigation of an inversion technique for the determination of broadband duct mode amplitudes by the use of near-field sensor arrays, J. Acoust. Soc. Am. 122 (2007) 848–859, http://dx.doi.org/10.1121/1.2747166.

[4] S. Léwy, Numerical inverse method predicting acoustic spinning modes radiated by a ducted fan from free-field test data, J. Acoust. Soc. Am. 124 (2008) 247–256, http://dx.doi.org/10.1121/1.2931952.

[5] W. Yu, Z. Ma, A.S.H. Lau, X. Huang, Analysis and experiment of the compressive sensing approach for duct mode detection, AIAA J. 6 (2018) 648–657, http://dx.doi.org/10.2514/1.J056347.

[6] C. Ford, A. Pereira, C. Bailly, Preliminary results of modal analysis for a ducted axial fan from external pressure measurements, in: 28th AIAA/CEAS Aeroacoustics Conference, (2022-3097) 2022, http://dx.doi.org/10.2514/6.2022-3097.

[7] B. Etchebarne, S. Fauqueux, M. Lorteau, D. Marx, V. Valeau, Feasibility of in-duct aeroacoustic source characterization with external antenna: A flanged duct configuration, in: 30th AIAA/CEAS Aeroacoustics Conference, (2024-3135) 2024, http://dx.doi.org/10.2514/6.2024-3135.

[8] P. Sijtsma, Feasibility of in-duct beamforming, in: 13th AIAA/CEAS Aeroacoustics Conference, (2007-3696) 2007, http://dx.doi.org/10.2514/6.2007-3696.

[9] C.R. Lowis, P. Joseph, Determining the strength of rotating broadband sources in ducts by inverse methods, J. Sound Vib. 295 (2006) 614–632, http://dx.doi.org/10.1016/j.jsv.2006.01.031.

[10] J.M. Tyler, T. Sofrin, Axial flow compressor noise studies, SAE Trans. (1962) 309–332, http://dx.doi.org/10.4271/620532.

[11] E.R. Rademaker, P. Sijtsma, B.J. Tester, Mode detection with an optimized array in a model turbofan engine intake at varying shaft speeds, in: 7th AIAA/CEAS Aeroacoustics Conference and Exhibit, (2001–2181) 2001, http://dx.doi.org/10.2514/6.2001-2181.

[12] P. Sijtsma, H. Brouwer, Deconvolution of azimuthal mode detection measurements, J. Sound Vib. 422 (2018) 1–14, http://dx.doi.org/10.1016/j.jsv.2018.02.029.

[13] S. Fauqueux, Duct azimuthal and radial modal detection on the cfd modeling of uhbr engine tonal noise, in: 25th AIAA/CEAS Aeroacoustics Conference, (2019–2417) 2019, http://dx.doi.org/10.2514/6.2019-2417.
[14] M. Behn, R. Kisler, U. Tapken, Efficient azimuthal mode analysis using compressed sensing, in: 22nd AIAA/CEAS Aeroacoustics Conference, (2016–3038) 2016, http://dx.doi.org/10.2514/6.2016-3038.

[15] W. Jürgens, U. Tapken, B. Pardowitz, P. Kausche, G.J. Bennett, L. Enghardt, Technique to analyse characteristics of turbomachinery broadband noise sources, in: 16th AIAA/CEAS Aeroacoustics Conference, (2010–3979) 2010, http://dx.doi.org/10.2514/6.2010-3979.

[16] A. Pereira, A. Finez, Q. Leclère, E. Salze, P. Souchotte, Modal identification of a small-scale ducted fan, in: 22nd AIAA/CEAS Aeroacoustics Conference, (2016–3063) 2016, http://dx.doi.org/10.2514/6.2016-3063.

[17] U. Tapken, B. Pardowitz, M. Behn, Radial mode analysis of fan broadband noise, in: 23rd AIAA/CEAS Aeroacoustics Conference, (2017–3715) 2017, http://dx.doi.org/10.2514/6.2017-3715.

[18] M. Behn, B. Pardowitz, U. Tapken, Compressed sensing based radial mode analysis of the broadband sound field in a low-speed fan test rig, in: 7th Berlin Beamforming Conference, 2018.

[19] S. Fauqueux, R. Davy, Modal deconvolution method in a finite circular duct, using flush-mounted microphones, in: 2018 AIAA/CEAS Aeroacoustics Conference, (2018–3927) 2018, http://dx.doi.org/10.2514/6.2018-3927.

[20] A. Pereira, M.C. Jacob, Modal analysis of in-duct fan broadband noise via an iterative bayesian inverse approach, J. Sound Vib. 520 (2022) http://dx.doi.org/10.1016/j.jsv.2021.116633.

[21] M. Laban, J.C. Kok, H.H. Brouwer, Analysis of uhrb engine tonal noise, in: Analysis of UHBR Engine Tonal Noise, (2018–3780) 2018, http://dx.doi.org/10.2514/6.2018-3780.

[22] C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems, Society for Industrial and Applied Mathematics (SIAM), 1998.

[23] U. Tapken, L. Enghardt, Optimization of sensor arrays for radial mode analysis in flow ducts, in: 12th AIAA/CEAS Aeroacoustics Conference, (2006–2638) 2006, http://dx.doi.org/10.2514/6.2006-2638.

[24] S. Fauqueux, R. Davy, F. Mery, Duct modal detection tool to characterize the noise source generated by an air pump, in: 25th AIAA/CEAS Aeroacoustics Conference, (2019–2419) 2019, http://dx.doi.org/10.2514/6.2019-2419.

[25] S. Sack, M. Abom, G. Efraimsson, On acoustic multiport characterisation including higher order modes, Acta Acoust. United Acust. 192 (5) (2016) 834–850, http://dx.doi.org/10.3813/AAA.918998.

[26] J. Hald, J.J. Christensen, A Class of Optimal Broadband Phased Array Geometries Designed for Easy Contraction, Inter-noise, 2002.

[27] M. Bai, J.-H. Lin, K.-L. Liu, Optimized microphone deployment for near-field acoustic holography: To be, or not to be random, that is the question, J. Sound Vib. 329 (2010) 2809–2824, http://dx.doi.org/10.1016/j.jsv.2010.01.027.

[28] A. Malgoezar, M. Snellen, P. Sijtsma, D. Simons, Improving beamforming by optimization of acoustic array microphone positions, in: Berlin Beamforming Conference, 2016.

[29] E. Sarradj, A generic approach to synthesize optimal array microphone arrangements, in: Berlin Beamforming Conference, 2016.

[30] F. Le Courtois, J.-H. Thomas, F. Poisson, J.-C. Pascal, Genetic optimisation of a plane array geometry for beamforming. Application to source localisation in a high speed train, J. Sound Vib. 371 (2016) 78–93, http://dx.doi.org/10.1016/j.jsv.2016.02.004.

[31] D. Bush, N. Xiang, Broadband implementation of coprime linear microphone arrays for direction of arrival estimation, J. Acoust. Soc. Am. 138 (1) (2015) 447–456, http://dx.doi.org/10.1121/1.4923159.

[32] W. Wang, S. Chen, R. Wang, A fast irregular microphone array design method based on acoustic beamforming, IEEE Sensors J. 23 (6) (2023) 6156–6168, http://dx.doi.org/10.1109/JSEN.2023.3240888.

[33] P.A. Nelson, S.H. Yoon, Estimation of acoustic source strength by inverse methods: Part i, conditioning of the inverse problem, J. Sound Vib. 233 (4) (2000) 643–668, http://dx.doi.org/10.1006/jsvi.1999.2837.

[34] K. Kim, P.A. Nelson, Estimation of acoustic source strength within a cylindrical duct by inverse methods, J. Sound Vib. 275 (2004) 391–413, http://dx.doi.org/10.1016/j.jsv.2003.06.032.

[35] D.M. Steinberg, W.G. Hunter, Experimental design: Review and comment, Technometrics 26 (1984) 71–97.

[36] Y. Dodge, V.V. Fedorov, H.P. Wynn (Eds.), Optimal Design and Analysis of Experiments, Elsevier, Science, 1988.

[37] F. Pukelsheim, Optimal Design of Experiments, Society for Industrial and Applied Mathematics, 2006.

[38] J. Ranieri, A. Chebira, M. Vetterli, Near-optimal sensor placement for linear inverse problems, IEEE Trans. Signal Process. 62 (5) (2014) 1135–1146, http://dx.doi.org/10.1109/TSP.2014.2299518.

[39] F. Asano, Y. Suzuki, D.C. Swanson, Optimization of control source configuration in active control systems using Gram–Schmidt orthogonalization, IEEE Trans. Speech Audio Process. 7 (1999) 213–220.

[40] S. Joshi, S. Boyd, Sensor selection via convex optimization, IEEE Trans. Signal Process. 57 (2009) 451–462, http://dx.doi.org/10.1109/TSP.2008.2007095.

[41] S.A. Verburg, F. Elvander, T. van Waterschoot, E. Fernandez-Grande, Optimal sensor placement for sound field reconstruction, in: Proceedings of the 24th International Congress on Acoustics, 2022.

[42] K. Liu, R.-J. Yan, G. Guides Soares, Optimal sensor placement and assessment for modal identification, Ocean Eng. 165 (2018) 209–220, http://dx.doi.org/10.1016/j.oceaneng.2018.07.034.

[43] S. Koyama, G. Chardon, L. Daudet, Optimizing source and sensor placement for sound field control: An overview, IEEE/ ACM Trans. Audio, Speech Lang. Process. (2020) http://dx.doi.org/10.1109/TASLP.2020.2964958.

[44] K. Manohar, B.W. Brunton, J.N. Kutz, S.L. Brunton, Data-driven sparse sensor placement for reconstruction: Demonstrating the benefits of exploiting known patterns, IEEE Control Syst. Mag. 38 (2018) 63–86, http://dx.doi.org/10.1109/MCS.2018.2810460.

[45] I. Yu. Gejadze, V. Shutyavc, On computation of the design function gradient for the sensor-location problem in variational data assimilation, SIAM J. Sci. Comput. 34 (2012) B127–B147, http://dx.doi.org/10.1137/110825121.

[46] V. Mons, O. Marquet, Linear and nonlinear sensor placement strategies for mean-flow reconstruction via data assimilation, J. Fluid Mech. 923 (2021) A1, http://dx.doi.org/10.1017/jfm.2021.488.

[47] W. Kang, L. Xu, Optimal placement of mobile sensors for data assimilations, Tellus A 64 (2012) 17133, http://dx.doi.org/10.3402/tellusa.v64i0.17133.

[48] G.H. Golub, C.F. Van Loan, Matrix Computations, The Johns Hopkins University Press, 2013.

[49] F. Castres, P. Joseph, Mode detection in turbofan inlets from near field sensor arrays, J. Acoust. Soc. Am. 121 (2007) 796–807, http://dx.doi.org/10.1121/1.2427124.

[50] P.D. Lax, Linear Algebra and Its Applications, second ed., Wiley-Interscience, 2007.

[51] S. Boyd, L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004.

[52] H.L. Van Trees, Optimum Array Processing: Part IV of Detection, Estimation, and Modulation Theory, Wiley-Interscience, 2002.

[53] C.L. Morfey, Acoustic energy in non-uniform flows, J. Sound Vib. 14 (2) (1971) 159–187, http://dx.doi.org/10.1016/0022-460X(71)90381-6.