Directional Array Transfer Functions

• Directional Array Transfer Functions are models that mathematically map incident fields to multi-channel sensor outputs, considering spatial, spectral, and array-specific effects.
• They employ complex-valued tensors and spherical harmonic expansions to accurately represent directivity, coupling, and boundary scattering in practical arrays.
• DATFs facilitate advanced applications such as spatial audio, machine learning for sensor arrays, and robust analysis in acoustics and electromagnetics.

A directional array transfer function (DATF) rigorously characterizes the full input–output mapping between arbitrary incident fields and the multi-channel output of a spatially distributed, frequency-dependent, and possibly non-omnidirectional sensor array. Across acoustics, electromagnetics, and signal processing, DATFs represent both the spatial response to direction and frequency as well as internal array coupling/processing, allowing for precise modeling, analysis, and machine learning with real–world (non-ideal) sensors. In practice, DATFs are formulated as complex-valued tensors or operator-valued functions, parameterized by array geometry, element directivity, sensor orientation, mutual coupling, boundary conditions, and often higher-order expansions such as spherical harmonics.

1. Mathematical Formulation and Physical Meaning

The DATF encapsulates the response of a sensor array to an incident field parameterized by direction and frequency, accounting for all array-intrinsic properties (position, coupling, directivity, truncation, boundary effects). For a $P$-channel array, define the directionally-resolved transfer function vector at frequency $f$ as

$\mathbf{h}^f(\theta, \phi) \in \mathbb{C}^P$

such that the array output from a continuous incident amplitude $a^f(\theta, \phi)$ is

$$\mathbf{x}_t^f = \iint a_t^f(\theta,\phi) \cdot \mathbf{h}^f(\theta,\phi) \sin\phi \,d\theta d\phi$$

for every time frame $t$ and frequency $f$ (Heikkinen et al., 30 Jan 2026). This function generalizes to discrete tensors

$$H \in \mathbb{C}^{P \times D \times F_H}, \quad H_{p,d,f} = h_p(f, \theta_d, \phi_d)$$

where $D$ is the number of spatial directions and $F_H$ the number of frequency bins. In classical free-field conditions with baricentric arrays, the function reduces to

$$h_p(f, \mathbf{u}, \mathbf{r}_p) = \exp\left( j2\pi f \frac{\mathbf{u} \cdot \mathbf{r}_p}{c} \right)$$

but practical systems require explicit modeling of frequency-dependent directivity, local boundary scattering, mutual coupling, and non-idealities (Heikkinen et al., 30 Jan 2026, Lee et al., 27 Feb 2025, Goldring et al., 25 Oct 2025, Huang et al., 10 Nov 2025).

2. Modal and Harmonic Representations

For spherical arrays and spatial audio, DATFs are advantageously expanded in the spherical harmonic (SH) domain:

$$T(\mathbf{u}) = \sum_{n=0}^N \sum_{m=-n}^n t_{n,m} Y_n^m(\mathbf{u})$$

with vectorizations as $$\mathbf{t}_N = \int_{S^2} T(\mathbf{u}) \mathbf{y}_N^H(\mathbf{u}) d\mathbf{u}$$ (Politis, 2024). Pointwise products (e.g., element directivity $\times$ phase delay) are mapped via Gaunt coefficients:

$$F_{n_1 m_1, n_2 m_2}^{n m} = \int_{S^2} R_{n_1}^{m_1}(\mathbf{u}) R_{n_2}^{m_2}(\mathbf{u}) R_{n}^{m}(\mathbf{u}) d\mathbf{u}$$

yielding a matrix–tensor multiplication framework for the transfer function. For an array with local directivities $d_q(k, \mathbf{u})$ and phase shifts $\exp(i k \mathbf{u}^T\mathbf{x}_q)$, the SH domain DATF is

$$\hat{\mathbf{h}}_N^{(q)}(k) = [\mathbf{d}_{N'}^{(q)}(k)]^T \mathbf{F}_{N',N''} [\mathbf{J}_{N''}(k d_q) \mathbf{r}_{N''}(\hat{\mathbf{x}}_q)]$$

Stacking over microphones yields the aggregate DATF $\hat{\mathbf{H}}_N(k)$ (Politis, 2024).

3. Mutual Coupling, Truncation, and Decomposition Methods

Physically accurate DATF determination for large electromagnetics or acoustic arrays must account for mutual coupling and truncation effects. The active element pattern (AEP) method constructs transfer matrices that map port excitation to active current distributions, from which the far-field is synthesized:

$$\mathbf{J}^{\text{est}} = [\mathbf{J}_{\rm iso}] [\mathbf{C}]_{\rm 2D} \mathbf{w}$$

$$F(\theta, \phi) = \mathbf{G}(\theta, \phi)[\mathbf{J}_{\rm iso}] [\mathbf{C}]_{\rm 2D} \mathbf{w}$$

where $[\mathbf{C}]_{\rm 2D}$ is a Kronecker-assembled block-diagonal matrix from directional decompositions along $x$ and $y$ axes, enabling efficient computational scaling from $\mathcal{O}(M_B^2 N_x^3 N_y^3)$ to $\mathcal{O}(M_B^2 (N_x^3 + N_y^3))$ (Lee et al., 27 Feb 2025). This enables direct prediction of all active element patterns (hence the full DATF) in arbitrary directions without the expense of full 2-D MoM solutions.

4. Multipath, Reflection, and Directional Modulation

In multipath scenarios or when ground/interface reflections are non-negligible, DATFs must accurately encode both line-of-sight and reflected contributions. For frequency diverse arrays (FDA) with multipath:

$$H_n(f) = \frac{1}{R^{\rm LoS}_n} e^{-j \frac{2\pi f}{c} R_n^{\rm LoS}} + \Gamma \frac{1}{R^{\rm ref}_n} e^{-j\frac{2\pi f}{c} R_n^{\rm ref}}$$

with $R^{\rm LoS}_n, R^{\rm ref}_n$ path lengths, $\Gamma$ reflection coefficient, and $H_n(f)$ assembled for all elements $n$ (Cheng et al., 2019). The array-level DATF thus captures the composite spatio-temporal filtering of arbitrary excitation, steering, and multipath effects, critical for physical-layer security, spatial zero-forcing, and null-shaping.

5. Data-Driven Spatial Processing and Neural Architectures

In spatial audio and modern array processing, DATFs serve as the basis for machine learning pipelines that transcend geometry-only meta-data. In neural Ambisonics encoding, discrete DATT tensors $H_{p,d,f}$ are combined through deep feature encoders, and cross-attention layers merge signal and array metadata into a latent representation agnostic to array-specific biases:

• Audio and DATF encoders generate feature tensors $Z_X, Z_H$.
• Cross-attention fuses these into context-aware filters for spatial audio decoding (Heikkinen et al., 30 Jan 2026).

Quantitative evaluation shows that using full DATF tensors in neural spatial encoding surpasses geometric approaches by $\sim 1$dB SI-SDR and $\sim 4$ units in MS_Err, especially for arrays with strong scattering and directionality.

Similarly, neural directional filtering (NDF) learns to approximate target virtual directivity patterns by mapping array signals to masks that emulate a composite DATF:

$\widehat{Z}(f, t) = \mathcal{M}(f, t) Y_1(f, t)$

where $\mathcal{M}$ is a learned mask approximating the target directionality, evaluated via power pattern estimates and directivity factor metrics (Huang et al., 10 Nov 2025). NDF demonstrates the ability to synthesize frequency-invariant and high-order directivity responses that are robust to aliasing and to SNR variations.

6. Applications in Near-Field, Spatial Audio, and Topological Sensing

DATFs are central in both perceptual and performance-critical systems:

• Binaural Signal Matching (BSM) for wearable arrays utilizes DATFs as the mapping between source positions and array outputs, optimizing for binaural metrics (ILD, ITD) and leveraging near-field extensions (NF-BSM) with field-of-view weighting for perceptual focus. Explicit use of DATF-informed models improves rendering accuracy at close distances and under head movement by accounting for true propagation (phase and amplitude) effects (Goldring et al., 25 Oct 2025).
• In topological microwave amplifier arrays, the non-reciprocal DATF arising from engineered array Hamiltonians ($J e^{\pm i \Delta\phi}$), enforces exponential directional gain and isolation, protected by winding-number invariants and robust against substantial fabrication disorder (2207.13728).

7. Evaluation Metrics and Benchmarking

Performance of DATF-based systems is typically assessed across several metrics, reflecting both spatial fidelity and transfer function accuracy:

Metric	Mathematical Definition	Domain
SI-SDR	$\mathrm{SI}$-$\mathrm{SDR}(s, \hat{s})$	Signal
Magnitude-squared Coherence	$$|\Phi_{b\hat{b}}(f)|^2 / (\Phi_b(f)\Phi_{\hat{b}}(f))$$	Spectral
Mean spectrum error (MS_Err)	$$\frac{1}{FT}\sum_{f,t} ||B_{:,f,t}| - |\hat{B}_{:,f,t}||$$	Spatial
Directivity Factor (DF)	Ratio of reverberant energy before/after	Array/domain
ILD/ITD Error	$$|\mathrm{ILD}_k^{\rm rep} - \mathrm{ILD}_k^{\rm ref}|$$	Binaural

Practical implementations use combinations of these, with MSE $<0.1$dB in main lobes for efficient decomposition methods (Lee et al., 27 Feb 2025), and SI-SDR/ILD/ITD improvements for perceptual renderers and learning-based solutions (Heikkinen et al., 30 Jan 2026, Huang et al., 10 Nov 2025, Goldring et al., 25 Oct 2025).

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Directional array transfer functions thus provide the canonical framework for describing, synthesizing, and inverting the spatial, spectral, and coupling characteristics of real-world sensor arrays, bridging physics-based modeling, computational efficiency, and modern data-driven signal processing.