A Novel Geometric Analysis for Two-Dimensional Sound Source Localization

In modern societies, a wide range of devices, materials, and equipment constantly produce sounds that are perceived through pressure variations in the ear. The human brain has an extraordinary capacity to identify and locate these sound sources, a process referred to as sound localization in neuroscience. Inspired by this natural ability, researchers have long sought to replicate it in artificial systems. The task of determining the position of an acoustic source using microphones is widely known as Sound Source Localization (SSL).

With the continuous advancement of technology, national defense requirements have grown increasingly complex. Traditional detection systems such as RADAR and SONAR have long been employed in both military and civilian applications to identify objects within a specific area. These systems operate by transmitting radio or sound waves at various frequencies and wavelengths, then estimating an object’s position and size from the reflected signals. While highly effective, this active emission approach has a critical drawback in military contexts: it exposes the emitter’s location, making it vulnerable to passive detection and precision-guided attacks. In contrast, SSL offers a passive alternative, capable of tracking objects moving at subsonic speeds without revealing the observer’s position. This makes SSL a promising technology for advancing defense systems and enhancing national security. Beyond defense, SSL has also attracted substantial attention due to its broad range of applications, including hearing aids, robotics, navigation, speaker tracking, remote sensing, and security-related systems such as surveillance, gunshot detection, and artillery localization.

Most existing SSL studies have relied on professional grade large microphone arrays (e.g., 4–56 Brüel & Kjær or Eigenmike microphones). These studies were typically conducted in acoustically controlled environments such as anechoic or semi-anechoic chambers [1-5]. While these studies have advanced SSL, their reliance on expensive hardware and ideal conditions limits their applicability in everyday environments. Additionally, there is a lack of research on low-cost and simple setups with only a few nonlinearly placed microphones in real-world environments, characterized by naturally occurring noise and reverberation which cause performance degradation in most existing systems. Despite significant progress in SSL, recent surveys highlight, challenges persist, including artificial intelligence-based models' heavy reliance on training data and the difficulty of achieving reliable and robust localization in realistic, dynamic acoustic environments [6, 7]. Therefore, there is a critical gap between the controlled, professional grade setups dominating the literature and the need for cost-effective, robust SSL methods that function reliably in everyday environments. This study directly addresses this gap by introducing a low-cost, practical SSL approach and validating it in realistic domestic conditions.

Our contribution is twofold: first, we propose a new SSL method that can be implemented with three low-cost MAX9814 microphones and a National Instruments (NI) USB-6216 interface; second, we provide a direct comparison of the proposed method and an Artificial Neural Network (ANN) model using our own data recorded in realistic domestic environments. The economic advantage of our approach is significant. At the component level, the cost of our three microphones is orders of magnitude lower than that of a single microphone module in professional arrays such as Eigenmike or Brüel & Kjær systems. While we used an NI USB-6216 interface in this study for its proven reliability and performance, our localization method is algorithmically independent of this specific hardware. This means that for applications where budget is the primary constraint, the method can work with lower-cost data acquisition solutions, potentially reducing the total system cost further compared to our current setup. Overall, the total cost of our reference setup remains more than an order of magnitude lower than that of professional systems. The use of inexpensive, widely available microphones highlights the feasibility of the method for practical, low-budget implementations. By combining methodological novelty with clear economic advantages, this study demonstrates a practical and cost-effective approach to SSL that can benefit real-world applications requiring both accuracy and affordability.

The structure of this paper is as follows: Section 2 provides a brief literature review of similar studies on SSL. Section 3 is divided into subsections explaining the experimental setup and acoustic characterization, data preparation, the proposed method, and the ANN used for performance comparison. The obtained results are presented, thoroughly compared, and discussed in detail in Section 4. Section 5 addresses the study's limitations and suggests future work. Finally, Section 6 concludes the paper by summarizing the key findings and outlining opportunities for future research.

In this section, we present in subsections the experimental setup and environmental characterization, data preparation and filtering, proposed method for SSL along with the ANN architecture used to benchmark its performance.

3.1 Experimental setup and acoustic characterization

To ensure the validity of the proposed method, both the experimental setup and the acoustic environment were carefully characterized. The experiments were conducted in a real-world indoor environment with echoic and noise-prone conditions, and the acoustic data-hand claps were collected. The setup consisted of three electret microphones, each equipped with a MAX9814 microphone amplifier, positioned at an angle of 120 degrees to each other and placed at an arm length of 50 cm. MAX9814 is a microphone amplifier with 3 adjustment options for gain and AR (attack/release) pins and is built on automatic gain control. The center of the gray platform, shown in Figure 1, was defined as the point (0, 0). The positions of the microphones were determined in meters as (-0.25√3, -0.25), (0, 0.50), (0.25√3, -0.25) respectively. A NI USB-6216 data acquisition card was used for recording. Additionally, a protractor created with MATLAB® and a tape measure were employed to accurately place the sound source at specific angles and distances.

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Figure 1. A schematic view of the experimental setup

The experiments were performed in a residential room with dimensions of approximately 5 m × 4 m × 2.5 m. All recordings used for analysis were carried out under real-world conditions, where the acoustic environment included everyday household noise, outdoor traffic, or continuous fan noise from a computer and a ventilator. To better characterize the influence of these interferences, two reference cases were considered: one set of recordings conducted at night under nearly silent conditions, and another set conducted under typical everyday conditions. As shown in Figure 2, the nighttime recordings are almost free of external disturbances and primarily contain the microphones’ inherent thermal/electronic noise. This reference confirmed that the microphones themselves introduce negligible noise, and highlighted that the variations observed in daytime recordings mainly originate from realistic environmental interferences. These results demonstrate that the experimental data reflect practical everyday conditions, as intended for evaluating SSL performance in real-world scenarios.

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Figure 2. Comparison of recordings captured in a quiet nighttime environment and in a daytime environment

To further analyze the recording environment, noise and signal properties of the microphones, harmonic noise components, and reverberation time were evaluated. The Root Mean Square (RMS) noise levels of the microphones were approximately 0.022 V, while the RMS signal amplitudes during hand-claps ranged between 0.065–0.089 V, yielding signal-to-noise ratios (SNR) between 8.9 dB and 12 dB. The noise floor was estimated at around −33 dB, and the gain mismatch between microphones was measured as 0.56 dB, which is acceptable for source localization tasks. DC offsets were negligible (≈ −0.018 V), and clipping was minimal (≤ 5 counts), confirming reliable acquisition without significant distortion. Low-frequency interferences at 50, 100, and 150 Hz were well below the signal level (all < −60 dB), and the high-frequency noise floor above 10 kHz was measured at about −92 dB. Reverberation time (RT60), estimated via the Schroeder method with T30 analysis, yielded values of 1.288 s, 1.193 s, and 1.595 s across the three microphones, with an average of 1.36 s. These results demonstrate that the experiments were carried out in a moderately reverberant residential environment, representative of real-life acoustic conditions in which SSL systems are expected to operate.

3.2 Data preparation

In the experimental setup, acoustic signals were recorded at various source angles and distances using the NI USB-6216 data acquisition card, which was connected to MATLAB® via the Data Acquisition Toolbox. The sampling rate was set to 48 kHz. Experiments were conducted at room temperature (≈ 20℃) and atmospheric pressure, with 343 m/s assumed as the reference speed of sound in air.

Hand clap sounds were employed as acoustic stimuli in a domestic environment. Using a protractor and a tape measure, the sound source (a human subject) was positioned at multiple angles and distances relative to the origin. At each position, the subject produced a hand clap, and the resulting acoustic data were recorded for further analysis. Data were recorded for scenarios where the sound source was positioned at different angles and distance values between 0 and 360 degrees relative to the origin. As shown in Figure 1, the coordinate system was defined such that Microphone 2 pointed in the +y direction, Microphone 1 in the –x and –y directions, and Microphone 3 in the +x and –y directions. The center of the platform was aligned with the protractor’s origin, which served as the reference for all measurements.

To minimize the effect of environmental noise and irrelevant frequency components, 5th-order band-pass Butterworth filter (100–8000 Hz) was applied to the raw recordings. Low-frequency components (e.g., room hum) and high-frequency components (e.g., electronic noise) were attenuated.

Since the recordings were obtained in a home environment, potential factors such as background noise or slight inaccuracies in positioning could affect data quality. To mitigate this, ten recordings were taken for each source position, and the one with the clearest signal onset and highest signal-to-noise ratio was selected. In total, 1551 recordings were collected at various angles and distances for subsequent SSL analysis.

3.3 Proposed model

In this section, we present the proposed method for SSL. Accurate localization relies on the acoustic signals received by the microphones, and the method estimates the source position by exploiting the time delays between these signals. The following subsections provide details on the time delay estimation and the geometric analysis underlying the approach.

3.3.1 Time delay estimation

SSL commonly relies on Time Delay Estimation (TDE) due to its proven effectiveness in determining the direction of acoustic sources. When a sound is emitted, it arrives at each microphone in an array at slightly different times and with varying waveform characteristics, depending on their spatial positions. These time differences provide crucial information for estimating the source location.

In this study, time delays between microphone signals were computed using the finddelay() function in MATLAB®. This function employs a cross-correlation-based algorithm to determine the delay between two signals. Cross-correlation measures the similarity between two signals by assessing how well they match when one is shifted in time relative to the other. The algorithm computes this similarity by summing the products of the two signals at various time shifts. The cross-correlation function $R_{x y}$ for two discrete time signals x[n] and y[n] is defined by Eq. (1) as [33]:

$R_{x y}[k]=\sum_{n=-\infty}^{\infty} x[n] * y[n+k]$    (1)

where, $k$ is the lag index and the sum is taken over all time indices $n$. The $k$ value, where $R_{x y}[k]$ is the highest, gives the delay between two signals in terms of the number of samples. Time delay estimated in samples was converted into seconds using the sampling period. The conversion of the delay in samples to the delay in seconds is defined as follows with Eq. (2) as:

Delay in Seconds $=m * T_s$    (2)

where, $T_s$ denotes sampling period.

3.3.2 Geometric analysis

In this section, a novel method based on geometric analysis is proposed for two-dimensional sound source localization. The proposed method utilizes distance measurements in meters instead of time-delay estimates in seconds. The time delays calculated in seconds in the previous section were converted into distance values using the assumed speed of sound. This conversion was defined in Eq. (3) as follows

$d_{-} c=$ Delay in Seconds $* c$     (3)

where, $c$ is the speed of sound in air and $d_{-} c$ is distance in meter calculated by using $c$. These distance differences for microphone pairs are referred to as $d_{12}, d_{13}, d_{32}$ in the following sections.

The position of any point can be defined with respect to a given reference. Based on this principle, the proposed method assumes that the location of a stationary sound source can be estimated with geometric techniques using two dimensional vectors. As illustrated in Figure 3, the position of point S relative to the origin O is represented by the position vector $\vec{R}$.

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Figure 3. A position of a point

In polar form, this vector can be expressed as given by Eq. (4).

$\vec{R}=|\operatorname{os}|(\cos \theta \hat{\imath}+\sin \theta \hat{\jmath})$    (4)

where, $\hat{\imath}$ and $\hat{\jmath}$ are unit vectors. Alternatively, in Cartesian coordinates, the position vector is given by Eq. (5) as:

$\vec{R}=x \hat{\imath}+y \hat{\jmath}$    (5)

where, $x$ and $y$ are the distances. Equivalently, the position can be represented in the complex plane as given by Eq. (6).

$R=x+i y$    (6)

where, $i$ denotes the unit imaginary number.

The proposed method models the position vectors of the sound source as forming closed-loop polynomials, analogous to loop closure equations. This formulation enables a novel geometric framework for SSL. As illustrated in Figure 4, the static vectors representing the position of the sound source form vector loops, and the equations that describe the closure of these loops are referred to as loop closure equations. The proposed method for SSL was performed by setting up equations similar to loop closure equations.

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Figure 4. Vector loops of the experimental environment

In Figure 4, the point marked with the purple x represents the origin. Microphone1, Microphone2, Microphone3 and Sound Source was denoted as Mic1, Mic2, Mic3, Source respectively. The microphones positions were given in $\left(x_m, y_n\right)$ coordinates, and their distances from the origin are equal, shown as $r$. The distances from the origin and microphones to the sound source were represented as $R$, $l_1$, $l_2$ and $l_3$. The angles of the distances from the origin and microphones to the source with respect to the horizontal is defined as θ, α, β, γ, respectively. These angle values were measured counterclockwise to be positive.

The coordinate system was defined with the x- and y-axes shown in Figure 1, with point O was selected as the origin. The vector from point O to Mic1 was denoted as $\overrightarrow{O M \iota c 1}$ and similarly, the other vectors were defined in the same manner. According to these assumptions, the vector loops were specified as given by Eqs. (7)-(9) as:

$\overrightarrow{ { OMıc1 }}+\overrightarrow{ { Mvc1Source }}=\overrightarrow{ { OSource }}$    (7)

$\overrightarrow{ { OMıc2 }}+\overrightarrow{{ Mıc2Source }}=\overrightarrow{ { OSource }}$    (8)

$\overrightarrow{ { OMıc3 }}+\overrightarrow{ { Mvc3Source }}=\overrightarrow{ { OSource }}$    (9)

Using the parameters illustrated in Figure 4 and applying Euler’s formula, these vector loops were expressed in parametric form as presented in Eqs. (10)-(12).

$r e^{i\left(\frac{7 \pi}{6}\right)}+l_1 e^{i \alpha}=R e^{i \theta}$    (10)

$r e^{i\left(\frac{\pi}{2}\right)}+l_2 e^{i \beta}=R e^{i \theta}$    (11)

$r e^{i\left(\frac{11 \pi}{6}\right)}+l_3 e^{i \gamma}=R e^{i \theta}$    (12)

where, $r$ is the length of the microphone arms, $l_i$ is defined as the distance of the sound source to the $i^{t h}$ microphone and $R$ is the distance of sound source to the origin point.

To extend this framework, a virtual mirror was assumed along the x-axis. As shown in Figure 5, the black vectors represent the real vector loops from Figure 4, whereas the red vectors illustrate their mirrored counterparts. Each real vector loop in Figure 4 has a corresponding mirrored loop in Figure 5, forming a symmetric representation of the system. The original vector loops and their mirror images have identical magnitudes; however, in the mirror image representation, angles with respect to the horizontal axis are measured clockwise and considered negative.

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Figure 5. Mirror image of vector loops of the experimental environment

Similar to those previously derived from the real vectors, the complex conjugates of the loop closure equations given in Eqs. (10)-(12) were obtained from the mirror image and defined by Eqs. (13)-(15).

$r e^{-i\left(\frac{7 \pi}{6}\right)}+l_1 e^{-i \alpha}=R e^{-i \theta}$    (13)

$r e^{-i\left(\frac{\pi}{2}\right)}+l_2 e^{-i \beta}=R e^{-i \theta}$    (14)

$r e^{-i\left(\frac{11 \pi}{6}\right)}+l_3 e^{-i \gamma}=R e^{-i \theta}$     (15)

By analytically solving the loop closure equations and their conjugates, the distance $R$ from the origin to the source was expressed in terms of different microphone parameters, as shown in Eqs. (16)-(18).

$R=\sqrt{r^2+2 r l_1 \cos \left(\frac{7 \pi}{6}-\alpha\right)+l_1^2}$    (16)

$R=\sqrt{r^2+2 r l_2 \sin (\beta)+l_2^2}$    (17)

$R=\sqrt{r^2+2 r l_3 \cos \left(\frac{11 \pi}{6}-\gamma\right)+l_3^2}$    (18)

In addition to the loop closure equations, the Cosine Theorem was applied in Figure 4 and according to the Cosine Theorem, the distances $l_1$, $l_2$, $l_3$ and $R$ were obtained. The inter microphone distance differences were then written in terms of $l_1$, $l_2$, $l_3$ as in Eqs. (19)-(21).

$d_{12}=l_1-l_2$    (19)

$d_{13}=l_3-l_1$    (20)

$d_{32}=l_3-l_2$    (21)

The distances $l_1$, $l_2$, $l_3$ were also calculated with Euclidean distance, as given in Eqs. (22)-(24), where the sound source position is determined as $(R * \cos (\theta), R * \sin (\theta))$.

$l_1=\sqrt{\left(R \cos (\theta)-x_1\right)^2+\left(R \sin (\theta)-y_1\right)^2}$     (22)

$l_2=\sqrt{\left(R \cos (\theta)-x_2\right)^2+\left(R \sin (\theta)-y_2\right)^2}$    (23)

$l_3=\sqrt{\left(R \cos (\theta)-x_3\right)^2+\left(R \sin (\theta)-y_3\right)^2}$    (24)

where, $R$ is the distance from the origin to the source, $\theta$ is the positive angle between $R$ and the horizontal axis and $x_i$, $y_i$ are the cartesian coordinates of the $i^{\text {th }}$ microphone.

The time delays were also expressed in terms of the newly calculated $l_1$, $l_2$, $l_3$ values as shown in Eqs (25)-(27).

$\begin{gathered}d_{12}=\sqrt{\left(R \cos (\theta)-x_1\right)^2+\left(R \sin (\theta)-y_1\right)^2}-\sqrt{\left(R \cos (\theta)-x_2\right)^2+\left(R \sin (\theta)-y_2\right)^2}\end{gathered}$    (25)

$\begin{gathered}d_{13}=\sqrt{\left(R \cos (\theta)-x_3\right)^2+\left(R \sin (\theta)-y_3\right)^2}-\sqrt{\left(R \cos (\theta)-x_1\right)^2+\left(R \sin (\theta)-y_1\right)^2}\end{gathered}$     (26)

$\begin{gathered}d_{32}=\sqrt{\left(R \cos (\theta)-x_3\right)^2+\left(R \sin (\theta)-y_3\right)^2}-\sqrt{\left(R \cos (\theta)-x_2\right)^2+\left(R \sin (\theta)-y_2\right)^2}\end{gathered}$    (27)

To solve these nonlinear equations, the fsolve() function in MATLAB® was employed. The solution process was performed in multiple stages: subsets of the equations were first solved to obtain preliminary estimates, which were subsequently refined by solving the complete system.

The Levenberg–Marquardt algorithm was selected as the numerical solver within fsolve(). This hybrid optimization technique combines the advantages of gradient descent and Gauss–Newton methods, enabling stable convergence when far from the initial estimate and faster convergence near the solution. The algorithm’s damping parameter ensures robustness by dynamically adjusting the step size. In this study, the maximum number of iterations was limited to 1000, and both function and step tolerances were set to 10^-12. With these settings, the nonlinear systems were solved successfully, yielding the source position estimates.

To validate the proposed method, an ANN was trained using the same dataset, providing a comparative benchmark against the analytical results.

3.4 Artificial Neural Network (ANN) model

In this section, ANN model used for performance comparison was explained. In this study, a feedforward artificial neural network architecture was employed to estimate the direction and distance of a sound source based on time delay measurements between microphones. The dataset comprised a total of 1551 recordings. These samples were grouped in sets of three, from which one instance was randomly selected as the test data, while the remaining two were used for training. This sampling strategy resulted in 1034 training samples and 517 test samples. The ANN was trained exclusively using the training dataset, and its performance was evaluated on the separate test dataset to ensure unbiased assessment. In the neural network model developed for this study, the inputs consist of the true time delays between microphone pairs, expressed in meters, and the Cartesian coordinates of the microphones. As outputs, the network was trained to estimate the radial distance from the origin to the sound source and the positive horizontal angle relative to the x-axis. Based on these predicted values, the two-dimensional Cartesian coordinates (x,y) of the sound source were subsequently computed.

The implemented ANN featured a feedforward structure with three hidden layers. These hidden layers comprised 18, 27, and 10 neurons respectively. The activation functions selected for the hidden layers were the logarithmic sigmoid function for the first layer, the radial basis function for the second layer, and the linear function for the third layer. Several alternative architectures and hyperparameter settings were explored during preliminary tests, and the selected network (with three hidden layers of 18, 27, and 10 neurons) consistently yielded the best performance among the tested options. However, given the virtually infinite range of possible configurations, it cannot be claimed with certainty that this architecture represents the global optimum. Rather, it reflects the most effective structure identified within the practical constraints of this study.

The Levenberg–Marquardt backpropagation algorithm was utilized as the training algorithm due to its proven efficiency in nonlinear regression problems. Prior to training, all input and output features were normalized to enhance training convergence and ensure consistent network behavior. The normalization process was applied uniformly across both training and testing datasets. The ANN was then trained and evaluated exclusively using these normalized data values to ensure compatibility and generalization.

The obtained results using proposed method and ANN are given in the next section.

The present study has several limitations that should be acknowledged. First, the experimental data were collected exclusively in domestic indoor environment with naturally occurring noise and reverberation. While these conditions provide a realistic representation of everyday scenarios, the dataset does not include outdoor environments, industrial spaces, or other complex acoustic contexts. As a result, the generalizability of the findings to broader conditions cannot be fully ensured.

Second, the proposed geometric analysis method has certain constraints. It relies on three microphones arranged in a triangular configuration, which provides the minimum spatial diversity necessary for unique 2D localization. Moreover, in environments with strong reverberation, overlapping sound sources, or rapidly changing acoustic conditions, the robustness of the proposed method may degrade compared to controlled scenarios.

Furthermore, although the proposed method demonstrated real-time feasibility in MATLAB®, with an average processing time of approximately 98 ms for a 3-second input signal (3.3% of the recording duration) on an Intel i7-6700HQ CPU with 16 GB RAM, several practical limitations should be acknowledged. First, the current implementation was tested only on a general-purpose laptop processor, and performance may vary when deployed on resource-constrained embedded platforms such as ARM processors or FPGAs. Second, hardware compatibility and optimization for low-power devices were not investigated in this study. Third, the proposed geometric method was specifically designed for a triangular configuration of three microphones, which ensures the minimum spatial diversity for 2D localization. Extending the method to larger or irregular microphone arrays would require additional adaptations in the algorithm and may increase computational complexity. These considerations highlight the need for further validation and optimization to ensure robust applicability in real-world embedded systems.

Third, the ANN baseline used for comparison also has limitations that influenced its performance. The underperformance of the ANN model in this specific study can be primarily attributed to two factors. First, the scale of the dataset: while the available 1,034 training and 517 test samples were sufficient to validate the proposed geometric method, they are relatively limited for data-driven models like ANNs, which typically require much larger and more diverse datasets to achieve robust generalization. This likely contributed to overfitting on the training set and reduced accuracy on unseen data. Second, although several ANN architectures were explored and the best-performing configuration was selected, the nearly infinite hyperparameter search space (e.g., neuron counts, activation functions, learning rates) makes it infeasible to guarantee a globally optimal solution within the scope of this study. Consequently, the observed performance gap does not necessarily indicate a fundamental weakness of neural networks for SSL, but rather reflects the practical challenges of applying them effectively with limited data and computational resources. Future work will therefore focus on expanding the dataset and conducting a more comprehensive architecture and hyperparameter search to enable a fairer and more definitive comparison.

Future work will focus on addressing these limitations, including optimizing the algorithm for embedded real-time platforms and ensuring scalability for larger microphone arrays. Expanding the dataset to include outdoor and industrial scenarios, as well as more complex noise conditions, will provide further insights into the method’s generalizability. Additionally, extending the evaluation to alternative microphone configurations, including irregular or larger arrays, will help to better assess scalability. Finally, combining the proposed geometric approach with machine learning or adaptive signal processing techniques may improve robustness and accuracy in challenging acoustic environments.