Frequency-Invariant Directivity Pattern

Updated 17 November 2025

Frequency-Invariant Directivity Pattern is a spatial response function designed to remain constant over a broad frequency band, ensuring consistent beam width and sidelobe behavior.
Various design methodologies, including differential arrays, modal matching, and convex optimization, are employed to achieve near-constant directivity across frequencies.
Empirical studies demonstrate that optimized arrays effectively control spectral coloration and spatial aliasing, delivering stable performance in broadband acoustic and beamforming applications.

A frequency-invariant directivity pattern describes a spatial response function whose angular profile is intentionally engineered to remain constant (or nearly so) as a function of frequency over a wide operational bandwidth. This property is critical in acoustic (microphone and loudspeaker) arrays and phased array applications for ensuring consistent beam width, mainlobe/sidelobe shapes, and null positions—thereby preventing spectral coloration, spatial aliasing, and loss of selectivity in broadband environments. Frequency-invariant directivity patterns are foundational in broadband beamforming, parametric sources, and robust audio sensing, and have spurred the development of multiple algorithmic and physical array-design methodologies.

1. Mathematical Definition and Theoretical Basis

The general formulation of a frequency-invariant directivity pattern involves designing a spatial filter or beampattern, $D(\theta, f)$ , such that

$D(\theta, f) \approx D_0(\theta), \quad \forall f \in [f_{\ell}, f_{h}]$

where $D_0(\theta)$ is the desired angular pattern, $f$ is the frequency, and $\theta$ is the angle of incidence relative to steering direction. Typically $D_0(\theta)$ is specified in forms such as polynomials of $\cos\theta$ (for differential arrays) or via truncated Fourier or circular harmonic expansions: $D_0(\theta) = \sum_{n=0}^{N} \alpha_{N, n} \cos[n(\theta - \theta_s)]$ with coefficients $\alpha_{N,n}$ that define the target directivity order and shape, and $\theta_s$ as the steering angle (Qian et al., 28 Oct 2025).

For linear or planar arrays, the array weights or filters are solved such that the synthesized spatial response matches $D_0(\theta)$ for all $f$ of interest, under constraints on white noise gain, distortionless response, or spatial nulls. In the context of differential beamforming, frequency-invariance is achieved when the system's steering vector dependency on $f$ is sufficiently suppressed—often by leveraging the small-aperture approximation or differential microphone elements (Miotello et al., 17 Aug 2025).

For loudspeakers and parametric arrays, frequency-invariant directivity is typically phrased in terms of maintaining a constant generalized directivity index (GDI) or other Rayleigh quotient metrics across frequency, leading to quadratic-equality-constrained designs (Luo, 2 Jul 2024, Lissek et al., 25 Aug 2025).

2. Array Designs and Algorithms for Frequency Invariance

Several classes of array design strategies have been developed to approach or achieve frequency-invariant directivity:

2.1. Differential and Superdirective Arrays

Differential arrays use spatial finite-difference operations on pressure signals from sub-wavelength spaced elements to approximate spatial derivatives, which are intrinsically frequency-independent (Miotello et al., 17 Aug 2025). For example, in planar arrays of first-order directional elements, the beampattern can be synthesized as a truncated circular harmonic sum with modal weights fitted to the desired target pattern, and the resulting frequency response is flat subject to the array being sufficiently small relative to $\lambda_{min}$ .

Modal-matching frameworks engineer the synthesized pattern via explicit harmonic expansion of both the array's response and the target pattern, solving for array weights that cause modal coefficients to exactly match up to the truncation order $N$ . This is analytically tractable and provably achieves frequency invariance under the maintained assumptions (full row-rank, sufficient modal coverage, compact geometry) (Miotello et al., 17 Aug 2025).

2.3. Geometry Translation and Virtual Array Synthesis

Uniform linear (or planar) arrays can exploit "geometry translation," transforming the frequency dependence of the physical steering vector into a position dilation, such that a single set of weights designed for a fixed "virtual" frequency realizes the same beam across all frequencies. The resulting weights become frequency-independent, yielding an exactly invariant pattern (Son, 2020).

2.4. Joint Optimization of Geometry, Directivity, and Beamforming

State-of-the-art frameworks jointly optimize microphone positions, individual sensor directivity parameters, and filter coefficients to minimize an integrated error

$\overline{\varepsilon}_N[h, x] = \int_{\theta_1}^{\theta_2}\int_{\omega_L}^{\omega_H} \frac{1}{2\pi}\int_0^{2\pi} \left| \mathcal{B}\left[h(\omega), x, \theta \right] - D_0(\theta) \right|^2 d\theta\, d\omega\, d\theta_s$

over the design region (Qian et al., 28 Oct 2025). The optimal array geometry and sensor directivity are searched using a genetic algorithm, and filters are computed using Jacobi–Anger series expansion truncated at a chosen order.

2.5. Frequency-Regularized Quadratic Programming

For arrays (notably loudspeakers) with heterogeneous element responses, frequency-invariant directivity index constraints are enforced through quadratic-equality-constrained QPs. Regularization terms account for unusable passbands and maintain physical feasibility, with solutions obtained analytically via secular equations or projected-ascent algorithms (Luo, 2 Jul 2024).

2.6. Null Constraint and Convex Optimization

Recent methods apply explicit null constraints (enforcing zeros in the spatial response at chosen directions and distortionless constraints at the steering angle) formulated as convex QCQPs. This approach ensures the main-lobe width, null positions, and sidelobe levels remain flat across broad frequency bands and is practically solved using interior-point methods (Zhang et al., 25 Aug 2025).

2.7. Physical Source Superposition

The stacking of monopole and dipole acoustic sources, as in dual corona discharge transducers (CDTs), provides a mechanical means of synthesizing invariant cardioid-type patterns. The monopole and dipole contributions are independently controlled to yield the desired weighted sum, yielding directivity patterns

$D(\theta, f) = M(f) + D(f)\cos\theta$

normalized such that the ratio $D(f)/M(f)$ is constant and maps to any desired cardioid-like pattern over a wide bandwidth (Lissek et al., 25 Aug 2025).

2.8. Deep Learning Approaches

Neural directional filtering (NDF) trains deep recurrent networks to produce a complex mask applied to the reference microphone, with the loss based on the time-domain signal corresponding to the desired pattern. The network implicitly learns to "hallucinate" spatial cues that enforce the target directivity at all frequencies, even above the theoretical aliasing limit, exploiting cross-frequency correlations (Huang et al., 10 Nov 2025).

3. Performance Metrics, Physical Constraints, and Error Measures

Frequency-invariant directivity is rarely achieved exactly; practical realizations are quantitatively assessed by metrics such as:

Approximation error $\varepsilon_N$ : $L_2$ -norm error between synthesized and ideal beampatterns integrated over angles and frequency (Qian et al., 28 Oct 2025).
Directivity Factor (DF): Ratio of mainlobe to average pattern power—ideally remains constant over the frequency band (Zhang et al., 25 Aug 2025, Huang et al., 10 Nov 2025).
White Noise Gain (WNG): Ratio of white-noise suppression; serves as a robustness constraint or objective (Qian et al., 28 Oct 2025).
Null depth and width: Depth and angular sharpness of imposed nulls, which must not degrade at high frequency (e.g., via aliasing or truncation errors).
Beampattern invariance plots: Overlaid (dB) spatial responses at multiple frequencies, demonstrating pattern identity (Miotello et al., 17 Aug 2025, Huang et al., 10 Nov 2025).

Physical limitations stem from aperture size (which sets angular resolution and spatial aliasing limit), the spacing between elements (imposing lower/upper frequency bounds), sensor directivity model accuracy, and manufacturing tolerances. Geometric randomness (e.g., hyperuniform disorder) can suppress grating lobes and maintain pattern invariance where conventional periodic arrays develop frequency-dependent artifacts (Tang et al., 2023).

4. Comparative Analysis of Representative Methods

Class/Technique	Design Variables/Process	Frequency-Invariance Properties
Differential modal matching	Modal coefficients, array/sec. geometry	Exact (up to modal truncation)
Null-constraint QCQP	Beamformer coefficients, null locations	Flat pattern, robust to errors
Geometry translation	Reference pattern, coordinate transform	Exact by design (if model holds)
Joint GA-based optimization	Mic positions, element directivity, filter	Near-invariant, globally optimal
Secular-equation QEQP	Loudspeaker weights, penalty terms	Constant GDI over band
Neural directional filtering	End-to-end DNN mask, time-loss	Approximate (handles aliasing)
Hyperuniform array design	Emitter positions (statistical)	Sidelobe/grating lobe suppression
Monopole/dipole superposition	Voltage drive amplitude and phase	Flat for sub-aperture frequencies

Methods that rely on modal matching or null constraints tend to achieve the highest degree of frequency invariance with theoretical guarantees, provided array geometry and filter truncation parameters are satisfied. Neural approaches provide approximate invariance and superior performance in highly constrained hardware (compact apertures), leveraging learned spectral and spatio-temporal priors.

5. Experimental Verification and Practical Guidelines

Empirical results across a wide set of works demonstrate:

In optimized superarrays, directivity factor and white noise gain remain flat over 200 Hz–8 kHz, with approximation error up to 50% lower than benchmark designs (Qian et al., 28 Oct 2025).
Null-constraint beamforming yields stable mainlobe width and null depth over 200 Hz–5 kHz with MSE maintained below −40 dB, outperforming Jacobi–Anger truncation methods, and maintaining pattern shape in real arrays up to at least 3 kHz (Zhang et al., 25 Aug 2025).
Planar differential arrays with $M=9$ elements can sustain invariant high-order patterns up to 4 kHz even under geometry randomization (Miotello et al., 17 Aug 2025).
Hyperuniform arrays suppress grating lobes and retain mainlobe width over ultrasonics bands; duplicability allows for large-scale arrays without position optimization overhead (Tang et al., 2023).
Neural directional filtering with compact ( $\sim$ 3–9 cm) arrays can synthesize 6th-order patterns invariantly up to and beyond the spatial aliasing frequency, outperforming linear parameteric and least-squares beamformers in both SDR and pattern fidelity (Huang et al., 10 Nov 2025).
Dual CDT prototypes experimentally realize cardioid-type directivity flat within ±1 dB from 125 Hz up to 1 kHz, with mild degradation at higher frequencies attributable to piston diffraction in finite apertures (Lissek et al., 25 Aug 2025).

Key guidelines for implementation include:

Ensure $M \geq 2N+1$ for $N$ th-order modal matching.
Regularize matrix inversions (e.g., Tikhonov) for stability.
Discretize angle and frequency integrations at sufficient resolution.
For neural approaches, condition the network on steering direction and train with full-band, realistic data.
Physical designs (CDTs, HUD arrays) require careful calibration and validation to ensure mechanical and statistical properties are maintained over the operational band.

6. Outstanding Challenges and Future Directions

Continuing research seeks to extend frequency-invariant directivity principles to:

Arbitrary irregular, wide-aperture arrays beyond the small-aperture regime.
Dynamic, programmable hardware (MEMS, reconfigurable arrays) enabling variable pattern orders and steering in real time.
Robust, low-latency implementations for time-domain processing, enabling real-time applications such as noise/reverberation suppression in nonstationary environments.
Further integration of learned and model-based methods, bridging DNN approaches with physical priors to exploit invariance even under adverse conditions or for non-ideal sensor transfer functions.

A longstanding challenge is enforcing frequency invariance in the presence of significant multipath, environmental mismatches, or under severe spatial aliasing—areas where data-driven or hybrid model+learning approaches show promise (Huang et al., 10 Nov 2025).

Frequency-invariant directivity has critical implications in broadband audio capture, communication, spatial audio rendering, sonar/radar, and ultrasonic imaging. It prevents spectral coloration and improves intelligibility in speech applications, suppresses unintentional lobes in parametric arrays, and underpins robust sound field control in complex environments.

Recent advances leverage frequency-invariant directivity principles for:

Active acoustic metamaterials with programmable angular responses (Lissek et al., 25 Aug 2025)
Steerable spatial filters and parametric loudspeaker arrays without grating lobe artifacts (Tang et al., 2023)
Constant directivity horns and loudspeaker arrays for consistent sound field shaping (Luo, 2 Jul 2024).

Collectively, these methods grant new degrees of freedom in array design, enabling both high-order angular selectivity and broadband operation previously limited by hardware or algorithmic constraints.