Differentiable Optimization of Linear Differential Microphone Arrays: A Joint Geometry and Filter Design Framework

Ramani Duraiswami; Siminfar Samakoush Galougah

REVIEW 5 major objections 5 minor 17 references

Differentiable Optimization of Linear Differential Microphone Arrays: A Joint Geometry and Filter Design Framework

T0 review · 5 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper claims that one differentiable optimization over microphone positions and filter weights can reproduce desired Nth-order linear differential microphone array beampatterns with the minimum number of microphones.

desk verdict The core idea—jointly optimizing LDMA geometry and filters with automatic differentiation—is new and plausible, but the paper overclaims and never clarifies whether the optimized geometry is fixed across frequency, which undermines the wideband design claim. read the letter →

arxiv 2412.05123 v2 pith:5HTWHU3R submitted 2024-12-06 cs.SD eess.AS

Siminfar Samakoush Galougah , Ramani Duraiswami This is my paper · ORCID

classification cs.SDeess.AS

keywords lineardifferentialmicrophonearraydifferentiableprogrammingbeamforminggeometryoptimizationdirectivityfactorwhitenoisegainconstrainedautomaticdifferentiation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper sets out to show that the multistage design of linear differential microphone arrays can be collapsed into a single constrained optimization. It uses a nonuniform delay-and-sum beamformer as the base model and optimizes its filter weights and microphone spacings together with automatic differentiation, under a distortionless constraint at the target direction and spacing limits that prevent spatial aliasing. The authors claim this joint geometry-and-filter optimization reproduces desired Nth-order beampatterns even with the minimum $M=N+1$ microphones, for any tested steering direction, without a separate geometry-design stage. They support this with second- and third-order examples at steering angles $0$, $\pi/3$, and $\pi$, reporting pattern mean-squared error below $0.005$ and directivity factor within about $0.01$ of target.

What carries the argument

The object that carries the argument is the beampattern-matching cost with two constraints: the MSE functional $\epsilon_{BM} = \frac{1}{2\pi}\int_0^{2\pi} |B_M[h(w),\theta] - B_{d,N}[\theta]|^2 \, d\theta$, evaluated through the steering vector $d(w,\theta)$, together with the distortionless constraint at $\theta_d$ and the interelement-spacing bounds. Automatic differentiation gives the gradient of this cost with respect to both the filter vector $h(w)$ and the delay variables $\tau_m = \delta_m/c$, so a single constrained gradient-based solve returns geometry and filter at once. The $\delta_{\max} < \lambda/2$ bound is the spatial-aliasing safeguard, and setting $\delta_{\min}=0$ lets the optimizer collapse or expand microphone spacings as needed.

What would settle it

For the paper's second-order setup ($M=5$, $f=1$ kHz, $\theta_d=0$), repeat the optimization from many random initial guesses for $H_m(w)$ and $\tau_m$; finding even one initialization that yields a beampattern far from the desired one, with MSE above $0.005$ or DF error above $0.01$, would contradict the claim that the method reliably achieves the target LDMA beampattern.

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Extended reading notes

Core claim

The central claim is that a single differentiable optimization over both the complex filter weights $H_m(w)$ and the inter-microphone delays $\tau_m = \delta_m/c$ can drive the beampattern of a nonuniform delay-and-sum linear array onto the desired Nth-order DMA pattern $B_{d,N}[\theta] = \sum_{n=0}^N a_n \cos^n(\theta - \theta_d)$. The optimization minimizes the mean-squared error between achieved and desired beampatterns subject to the distortionless constraint $d^H(w,\theta_d) h(w)=1$ and spacing bounds $0 \le \delta_m \le \delta_{\max} < \lambda/2$. The reported experiments for a second-order array with $M=5$ microphones and a third-order array with the minimum $M=N+1=4$ microphones, at steering angles $0$, $\pi/3$, and $\pi$, show pattern MSE below $0.005$ and directivity factor within roughly $0.01$ of the target across frequencies up to 4 kHz. The authors take these results to show that the multistage LDMA design pipeline can be replaced by one joint geometry-and-filter optimization.

Load-bearing premise

The method's success rests on the optimizer converging from whatever initial guess is chosen for the filter weights and microphone delays, yet the paper reports the fit is highly dependent on that guess and provides no procedure for choosing it.

Editorial extensions

If this is right

A one-step constrained optimization can replace the conventional multistage LDMA design procedure, removing the need to fix geometry before designing filters.
Minimum-size arrays ($M = N+1$) suffice for Nth-order patterns, so directivity can be obtained with fewer microphones and lower implementation cost.
The formulation is steerable: the same optimization succeeds for target directions $0$, $\pi/3$, and $\pi$, so the design does not need separate steering machinery.
The optimized arrays keep their directivity factor close to the desired value and pattern MSE below $0.005$ over the tested band up to 4 kHz, suggesting wideband usability.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The initialization sensitivity the paper acknowledges suggests a natural extension: a multi-start or warm-start strategy would turn the method into a deterministic design tool, but the paper does not test this.
Because the optimization returns microphone positions explicitly, the same differentiable formulation could be extended to 2D or 3D layouts and to time-varying geometries for moving sources; only linear arrays are treated here.
The paper evaluates white noise gain but does not constrain it, so adding a WNG penalty or constraint to the same differentiable cost would likely trade a little pattern fidelity for resilience to sensor noise.
The reported design times would make the framework suitable for online re-optimization whenever the desired beampattern or steering direction changes, an adaptive capability the authors mention but do not demonstrate.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The core idea here is genuinely new and reasonable: instead of the usual two-stage LDMA design pipeline, set up a single differentiable optimization over both microphone spacings δ_m and filter weights H_m(w), and solve it with JAX and SLSQP. That is a sensible engineering contribution, and the reported MSE (< 0.005) and DF error (~0.01) are consistent with a converged fit. If the method works as described, it could give acoustic engineers a one-shot design tool for voice-capture arrays.

The paper's weaknesses are mostly presentation and a load-bearing ambiguity. First, the abstract promises WNG, computation time, and state-of-the-art comparisons; the body delivers only MSE and DF. That mismatch should be fixed. Second, the phrase "proving its ability to achieve the optimal beampattern" is not backed by any derivation; the numerical matches are a property of the optimization, not a proof. Third, Section 3 admits the results are "highly dependent on the initial guess," but there is no initialization procedure, no multi-start, and no landscape analysis. That alone is a reproducibility problem.

The more serious issue is the frequency-dependent geometry. The optimization problem in Eq. 5 treats δ_m as a plain variable, not a function of frequency, but the frequency sweeps in Figs. 5–6 are presented without saying whether δ_m was re-optimized at each frequency. If it was, then the optimized spacings differ from frequency to frequency, and the MSE/DF curves do not describe any single physical array. The paper never states that δ_m is shared across frequencies, and it never reports the converged δ_m values. This directly undermines the claimed "joint geometry and filter design" framework for wideband LDMAs. If the authors actually fixed δ_m after a reference-frequency optimization and then only adjusted H_m(w), they need to say so explicitly and show the geometry.

I agree with the reader's reject verdict, though I would soften the soundness score slightly: the optimization is correctly posed, and with the frequency coupling clarified and the overclaims trimmed, the method could be salvaged. The idea deserves referee time, but only if the authors are willing to address the ambiguity and provide code or at least the converged parameters. As is, it is not a reliable design tool.

For peer review: yes, send it out—the novelty is real and the flaws are fixable—but expect major revision. I would not cite it in its current form.

Referee Report

5 major / 5 minor

Summary. The manuscript proposes a differentiable-optimization framework for designing nonuniform linear differential microphone arrays (LDMAs). The optimization variable includes both the complex filter weights H_m(w) and the interelement spacings delta_m; the objective (Eq. 5) minimizes the mean squared error between the array beampattern and an Nth-order target beampattern (Eq. 4), subject to a distortionless-response constraint and spacing bounds. The authors use JAX with SLSQP and report close beampattern matches for N=2 and N=3 at steering angles 0, pi/3, and pi, with MSE below 0.005 and DF error around 0.01. They conclude that the method can replace the multistage LDMA design process and realize desired patterns even with M=N+1 microphones.

Significance. If the central claim held, the paper would offer a simple, one-stage alternative to current LDMA design procedures, which is a genuinely useful idea and the problem formulation is clearly stated. Credit is due for using standard performance metrics, explicitly constraining the distortionless response, and setting physically motivated spacing limits. However, the manuscript as written does not establish that the reported broadband results correspond to a single realizable physical array, does not provide a reproducible initialization or convergence analysis, and promises evaluations (WNG, computation time, comparisons with state-of-the-art methods) that do not appear in the results section. The word 'proving' in the abstract overstates what an MSE-minimization against the target beampattern can establish.

major comments (5)

[Section 3 / Eq. (5), Figs. 5-6] The realizability of the broadband design is not established. In Eq. (5), the spacing variables delta_m are not indexed by frequency, but Figs. 5 and 6 plot MSE and DF against frequency from 0 to 4 kHz. The manuscript never states that a single set of delta_m values is shared across the frequency grid, and it never reports the optimized spacings. If Eq. (5) was solved independently at each frequency, which is the literal reading of Section 3, then the geometry changes with frequency and the broadband MSE/DF curves do not describe any physical microphone array. The authors must either (i) reformulate the problem so that delta_m is optimized once over the whole frequency band and report the resulting spacings, or (ii) explicitly restrict all claims to per-frequency designs, in which case the 'joint geometry and filter design' and 'hardware-efficient' claims do not follow.
[Section 3, initial-guess discussion] The results are not reproducible as reported. The text states that 'a few misfits' are 'highly dependent on the initial guess for the optimization parameters, H_m(w) and tau_m', and that proper adjustment fixes them, but no initialization procedure is given. Since SLSQP is a local optimizer, the reader cannot know whether the reported matches are generic or the result of hand-tuned initialization. Please provide the exact initialization used, add a multi-start or random-seed study, and report the success rate or the distribution of final MSE/DF values over restarts.
[Section 2.3 and Section 3, delta_max setting] The spatial-aliasing constraint is internally inconsistent with the frequency sweep. The text says 'we constrain the maximum interelement spacing delta_max to be less than lambda/2', and Section 3 sets delta_max = 15 cm. At 4 kHz, lambda/2 is 4.25 cm, so 15 cm is almost four times the aliasing-free limit. If delta_max is held at 15 cm for the entire 0-4 kHz sweep, the high-frequency results are invalid; if delta_max is recomputed at each frequency, this again makes the geometry frequency-dependent. Please clarify how delta_max was set for each frequency and recompute the affected results.
[Abstract and Section 4] The abstract promises evaluation 'including Mean Squared Error (MSE), Directivity Index (DI), White Noise Gain (WNG), and computation time, and comparison with state-of-the-art methods', and the introduction claims the method 'outperforms existing methods in terms of cost and implementation time'. The results section contains none of these: WNG is never defined or computed, no computation-time table is given, and no quantitative comparison with the cited LDMA methods [8]-[13] appears. Either add the missing experiments and comparisons or revise the claims to match what is actually reported.
[Abstract and Section 2.3] The phrase 'proving its ability to achieve the optimal beampattern of LDMAs' is not supported by the analysis. Equation (5) directly minimizes the MSE against Bd,N[theta], so a close match between the optimized beampattern and the target is a property of the optimization objective rather than an independent proof. The word 'proving' should be replaced by 'demonstrating' or 'approximating', unless a theoretical guarantee of global optimality or a comparison against a known optimum is supplied.

minor comments (5)

[Eq. (5)] In the objective of Eq. (5), the first beampattern is written as 'Bd,N[h(w), theta]', but Eq. (4) defines Bd,N[theta] as independent of h; drop the h(w) argument to avoid confusion.
[Eq. (3)] The summation in Eq. (3) is missing the lower limit 'm=1' on the sum symbol; the exponent 'Pm-1 i=1 w tau_i cos theta' is also not rendered properly and should be written as a subexpression 'Σ_{i=1}^{m-1} w τ_i cos θ'.
[Section 3] The text says 'f = 1kHz, ω = 2πf = 2π × 1000 ≈ 6, 283', but Fig. 5 and Fig. 6 sweep frequency up to 4 kHz; the initial f = 1 kHz, ω value is only relevant to the beampattern plots. Please state this explicitly so the reader knows which figure uses which frequency setting.
[Figures 5 and 6] The captions say 'Minimum mean square error' and 'Directivity Factor ... compared with the desired one', but the axis labels, units, and the definition of the plotted 'DF error' (stated in the text as around 0.01) are not given; please add axis labels and define the error metric in the figure captions.
[References] The reference list omits page ranges or article numbers for some entries (e.g., [3] and [6] are arXiv preprints without version identifiers); please standardize the bibliography format.

Circularity Check

1 steps flagged · score 6.0 of 10

The central 'matching the desired Nth-order beampattern' result is the optimization objective itself; reported MSE/DF closeness is training loss, not an independent prediction.

fitted input called prediction [Section 2.3, Eq. (5); Section 3, Figs. 5–6 and conclusion]
"The objective is to find the optimum weights and positions of the microphones in our beamforming system to minimize the mean squared error (MSE) between the desired beampattern and the proposed one. … Fig. 5 and Fig. 6 show the MSE and the DF for the second order LDMA versus frequency … the DF for the proposed method is close to the DF of the desired beampattern with the error rate around 0.01 and the MSE is less than 0.005."

The variables H_m(w) and δ_m are fitted by minimizing exactly the MSE defined between B_M and B_{d,N} in Eq. (5). The paper then reports that same MSE (<0.005) and the resulting DF closeness as evidence that the method achieves the desired pattern. DF is a functional of B_M, so any B_M close to B_{d,N} in the optimized objective automatically has DF close to the desired DF; this is a continuous consequence of the fit, not a separate validated prediction. The abstract's claim of 'proving its ability to achieve the optimal beampattern' therefore restates the objective function rather than deriving an independent property.

full rationale

The only circular step is the identity between the optimization objective and the reported validation metric: Eq. (5) minimizes MSE(B_M, B_{d,N}), and Figs. 5–6 plus the conclusion report that this MSE is small and that DF is close. That is a training-loss report, not an independent first-principles result. No load-bearing self-citation occurs (the single co-authored reference [7] is background related work). Two non-circular correctness concerns are explicitly noted in the text and are weighed separately: Section 3 admits the fit has 'a few misfits' that are 'highly dependent on the initial guess' with no initialization or multi-start protocol, and the frequency sweep (Figs. 5–6) is ambiguous as to whether δ_m is re-optimized per frequency; if it is, the reported broadband curves do not describe a single physical array. Neither of these is a self-referential reduction, so they do not raise the circularity score above 6. The genuine non-circular content—the differentiable SLSQP implementation, constraint handling, and the M=N+1 feasibility experiment—does not rescue the central 'match' claim from being the fitted objective.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the standard far-field steering model and the Nth-order beampattern parameterization, both from prior DMA literature. The load-bearing unproven premises are the existence of a minimal-microphone configuration realizing the target pattern and the reliability of a local optimizer from unspecified initial guesses. These are not derived or characterized, which is why the correctness risk is medium and the soundness score is low.

free parameters (3)

Desired beampattern coefficients a_n (Eq. 4)
The target pattern is the optimization objective; the paper never specifies the numerical values used for N=2 and N=3, so the reported matches depend on an undisclosed design choice.
Initial guess for filter weights H_m(w) and delays tau_m
Section 3 states results are highly dependent on the initial guess and that misfits are corrected by adjusting it; no initialization scheme is given.
Spacing bounds delta_min and delta_max = 0 to 0.15 m
Chosen for the simulation to enforce the half-wavelength anti-aliasing bound; this constraint set affects which beampatterns are feasible.

assumptions (5)

domain assumption Far-field plane-wave propagation with speed c=340 m/s and free-field steering vector d(w,theta) as in Eq. 1
All beampattern and DF calculations assume anechoic far-field conditions and no reverberation or sensor noise; this is standard in DMA design but unstated.
domain assumption An Nth-order DMA desired beampattern is represented as Bd,N[theta] = sum_n a_n cos^n(theta - theta_d) with sum a_n = 1 (Eq. 4)
Standard parameterization from the DMA literature; the actual coefficient values used in the simulations are not given.
ad hoc to paper A nonuniform delay-and-sum beamformer with M=N+1 microphones and spacing within [delta_min, delta_max] can realize the desired beampattern
The paper asserts in Section 3 that M=N+1 gives desired performance, but supplies no existence proof or sufficient condition; this feasibility is the premise the optimizer tries to satisfy.
ad hoc to paper SLSQP, a local constrained optimizer, converges to an acceptable solution from the chosen initial guess
Section 3 admits results are highly dependent on the initial guess for H_m and tau_m; no global optimality or robustness analysis is given.
domain assumption Interelement spacing below half a wavelength avoids spatial aliasing
Standard spatial-aliasing condition cited from [8] and imposed through delta_max.

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Cite this review

Pith. "Pith review of Differentiable Optimization of Linear Differential Microphone Arrays: A Joint Geometry and Filter Design Framework." pith.science (2026). https://pith.science/paper/5HTWHU3R

@misc{pith2026241205123,
  author       = {Pith},
  title        = {Pith review of: Differentiable Optimization of Linear Differential Microphone Arrays: A Joint Geometry and Filter Design Framework},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5HTWHU3R}},
  note         = {Machine review of arXiv:2412.05123}
}

read the original abstract

This paper presents a differentiable optimization framework for the design of constrained Linear Differential Microphone Arrays (LDMAs). The proposed method leverages a non-uniform delay-and-sum beamformer as a light-weight base system model, proving its ability to achieve the optimal beampattern of LDMAs by jointly optimizing microphone positions and filter weights. The formulation enables the optimized design of a filter with a distortion-free constraint in the desired sound direction, while also imposing constraints on microphone positioning to ensure consistent performance. Through evaluation on multiple metrics, including Mean Squared Error (MSE), Directivity Index (DI), White Noise Gain (WNG), and computation time, and comparison with state-of-the-art methods, this approach demonstrates a flexible, directive, robust, and hardware-efficient design.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

17 extracted references · 16 canonical work pages

[8]

Masuda and D

N. Masuda and D. Saito, ”Synthesizer sound matching with differentiable DSP,” in Proc. ISMIR, pp. 428-434, 2021

work page 2021
[13]

C. Pan, J. Chen, and J. Benesty, ”Theoretical analysis of differential microphone array beamforming and an improved solution,” IEEE/ACM Trans. Audio, Speech, Lang. Process., vol. 23, no. 11, pp. 2093-2105, Nov. 2015

work page 2015
[1]

INTRODUCTION Derivatives play a vital role in machine learning, especially in optimization processes. Automatic Differentiation (AD) has revolutionized the computation of these derivatives, and its incorporation into machine learning frameworks as differ- entiable programming has opened new avenues for optimiz- ing complex functions with greater efficienc...

work page Pith review arXiv 2024
[2]

SYSTEM MODEL AND PROBLEM DEFINITION 2.1. System Model of the Nonuniform LDMAs We consider a plane wave coming from a far-field sound source with the speed of c = 340 m/s encountering an adaptive nonuniform LDMA with M omnidirectional micro- phones. The distances between microphones are not uniform, with the distance between the mth microphone and m + 1th ...

work page
[3]

The optimization problem has been solved in Python using JAX and scipy.optimize

NUMERICAL RESULTS Considering the simulation setup as the order of LDMA is N = 2, the number of microphones is M = 5, sound speed is c = 340m/s, f = 1kHz, ω = 2πf = 2π × 1000 ≈ 6, 283, δmin = 0, and δmax = 15cm. The optimization problem has been solved in Python using JAX and scipy.optimize. The nu- merical optimization technique used for this constraint ...

work page
[4]

The optimization problem is solved by considering two variables: filter coefficients and microphone positions

CONCLUSION This paper proposes a novel approach based on differentiable programming for designing constrained adaptive non-uniform LDMAs of any desired order. The optimization problem is solved by considering two variables: filter coefficients and microphone positions. This approach is significantly more efficient in terms of time and effort required for ...

work page
[5]

A. G. Baydin, B. A. Pearlmutter, A. A. Radul, and J. M. Siskind, ”Automatic differentiation in machine learning: A survey,” Journal of Machine Learning Research, vol. 18, pp. 1–43, 2018

work page 2018
[6]

Innes, A

M. Innes, A. Edelman, K. Fischer, C. Rackauckas, E. Saba, V . B. Shah, and W. Tebbutt, ”A differentiable pro- gramming system to bridge machine learning and scien- tific computing,” arXiv e-prints, arXiv:1907, 2019

work page 1907

Show all 17 references

[7]

Blondel and V

M. Blondel and V . Roulet, ”The elements of differen- tiable programming,” arXiv preprint, arXiv:2403.14606, 2024

2024 arXiv
[9]

Masuda and D

N. Masuda and D. Saito, ”Improving semi-supervised differentiable synthesizer sound matching for practical applications,” IEEE/ACM Trans. Audio, Speech, Lang. Process., vol. 31, pp. 863-875, 2023

2023
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Caspe, A

F. Caspe, A. McPherson, and M. Sandler, ”DDX7: Dif- ferentiable FM synthesis of musical instrument sounds,” arXiv preprint, arXiv:2208.06169, 2022

2022 arXiv
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B. Zhi, A. Sharma, D. N. Zotkin, and R. Duraiswami, ”A differentiable image source model for room acous- tics optimization,” in Proc. IEEE Workshop Appl. Sig- nal Process. Audio Acoust. (W ASPAA), pp. 1-5, 2023

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L. Zhao, J. Benesty, and J. Chen, ”Design of robust dif- ferential microphone arrays,” IEEE/ACM Trans. Audio, Speech, Lang. Process., vol. 22, no. 10, pp. 1455-1466, Oct. 2014

2014
[14]

J. Jin, G. Huang, J. Chen, and J. Benesty, ”Design of optimal linear differential microphone arrays based on array geometry optimization,” in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process. (ICASSP), pp. 5741- 5745, 2019

2019
[15]

J. Jin, J. Benesty, G. Huang, and J. Chen, ”On differ- ential beamforming with nonuniform linear microphone arrays,” IEEE/ACM Trans. Audio, Speech, Lang. Pro- cess., vol. 30, pp. 1840-1852, 2022

2022
[16]

J. Jin, G. Huang, X. Wang, J. Chen, J. Benesty, and I. Cohen, ”Steering study of linear differential microphone arrays,” IEEE/ACM Trans. Audio, Speech, Lang. Pro- cess., vol. 29, pp. 158-170, 2021

2021
[17]

X. Luo, J. Jin, G. Huang, J. Chen, and J. Benesty, ”De- sign of steerable linear differential microphone arrays with omnidirectional and bidirectional sensors,” IEEE Signal Process. Lett., vol. 30, pp. 463-467, 2023

2023

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [8]

Masuda and D

N. Masuda and D. Saito, ”Synthesizer sound matching with differentiable DSP,” in Proc. ISMIR, pp. 428-434, 2021

work page 2021

[2] [13]

C. Pan, J. Chen, and J. Benesty, ”Theoretical analysis of differential microphone array beamforming and an improved solution,” IEEE/ACM Trans. Audio, Speech, Lang. Process., vol. 23, no. 11, pp. 2093-2105, Nov. 2015

work page 2015

[3] [1]

INTRODUCTION Derivatives play a vital role in machine learning, especially in optimization processes. Automatic Differentiation (AD) has revolutionized the computation of these derivatives, and its incorporation into machine learning frameworks as differ- entiable programming has opened new avenues for optimiz- ing complex functions with greater efficienc...

work page Pith review arXiv 2024

[4] [2]

SYSTEM MODEL AND PROBLEM DEFINITION 2.1. System Model of the Nonuniform LDMAs We consider a plane wave coming from a far-field sound source with the speed of c = 340 m/s encountering an adaptive nonuniform LDMA with M omnidirectional micro- phones. The distances between microphones are not uniform, with the distance between the mth microphone and m + 1th ...

work page

[5] [3]

The optimization problem has been solved in Python using JAX and scipy.optimize

NUMERICAL RESULTS Considering the simulation setup as the order of LDMA is N = 2, the number of microphones is M = 5, sound speed is c = 340m/s, f = 1kHz, ω = 2πf = 2π × 1000 ≈ 6, 283, δmin = 0, and δmax = 15cm. The optimization problem has been solved in Python using JAX and scipy.optimize. The nu- merical optimization technique used for this constraint ...

work page

[6] [4]

The optimization problem is solved by considering two variables: filter coefficients and microphone positions

CONCLUSION This paper proposes a novel approach based on differentiable programming for designing constrained adaptive non-uniform LDMAs of any desired order. The optimization problem is solved by considering two variables: filter coefficients and microphone positions. This approach is significantly more efficient in terms of time and effort required for ...

work page

[7] [5]

A. G. Baydin, B. A. Pearlmutter, A. A. Radul, and J. M. Siskind, ”Automatic differentiation in machine learning: A survey,” Journal of Machine Learning Research, vol. 18, pp. 1–43, 2018

work page 2018

[8] [6]

Innes, A

M. Innes, A. Edelman, K. Fischer, C. Rackauckas, E. Saba, V . B. Shah, and W. Tebbutt, ”A differentiable pro- gramming system to bridge machine learning and scien- tific computing,” arXiv e-prints, arXiv:1907, 2019

work page 1907