Parametric Array Loudspeakers (PALs) Overview

• PALs are ultrasonic transducer systems that use the parametric acoustic effect to generate highly directional audio beams for focused sound applications.
• Advanced designs like HUD arrays and SPPALs achieve superior beamforming by minimizing grating lobes and optimizing array configurations.
• Recent integration of deep-learning methods with traditional modulation techniques effectively reduces nonlinear distortion, enhancing audio quality and system versatility.

Parametric Array Loudspeakers (PALs) are ultrasonic transducer systems that exploit the intrinsic quadratic nonlinearity of air to synthesize highly directional, narrow-audio beams from modulated ultrasonic carriers. Their ability to produce tight acoustic directivity at low audio frequencies, unattainable with conventional electrodynamic loudspeakers (EDLs) at comparable aperture sizes, underpins a wide range of applications, including targeted audio delivery, personal audio zones, ultrasonic communications, and underwater neutrino detection.

1. Nonlinear Acoustic Principle and Fundamental Modeling

PALs synthesize audible sound via the parametric acoustic effect, governed by the Westervelt equation:

$$\frac{\partial^2 p}{\partial t^2} - c_0^2 \nabla^2 p = -\frac{\beta}{\rho_0 c_0^2} \frac{\partial^2}{\partial t^2} (p^2)$$

where $p$ is acoustic pressure, $c_0$ is ambient sound speed, $\rho_0$ is density, and $\beta$ characterizes air’s nonlinearity. When two high-amplitude ultrasonic waves at frequencies $f_1, f_2$ co-propagate, the nonlinear interaction in air produces a difference-frequency (audio) pressure field. In the far field, this is approximately:

$$P_\text{diff}(r) \propto P_1(r) P_2^*(r) e^{i(k_1 - k_2)\cdot r}$$

The difference-frequency field ($f_a = |f_2 - f_1|$) possesses a wavelength much larger than the ultrasonic primaries, yet retains their directivity—allowing the generation of audio beams with $-3$ dB widths as narrow as a few degrees even at $f_a \lesssim 1$ kHz (Tang et al., 2023).

For amplitude-modulated PALs, the modulated ultrasonic carrier is represented as:

$p_{e}(t) = A_c[1 + m x(t)] \cos(\omega_c t)$

Self-demodulation in air yields an audio signal proportional to the second time derivative of the squared envelope, producing both the intended linear audio and nonlinear (harmonic/intermodulation) distortion components (Li et al., 2024).

2. PAL Architectures and Array Design

Conventional Arrays

Periodic arrays of piezoelectric or Langevin-type ultrasonic transducers remain standard, with array directivity governed by the array factor:

$$A(\theta, \phi) = \frac{1}{N} \sum_{j=1}^N e^{i k \cdot r_j}$$

Spatial aliasing imposes the Nyquist criterion: no grating lobes occur if element spacing $a \leq \lambda / 2$ (where $\lambda$ is the ultrasonic wavelength). Exceeding this introduces strong off-axis grating lobes, which are inherited by the audible secondary field due to the spatial Fourier properties of the parametric effect (Tang et al., 2023).

Hyperuniform Disordered (HUD) Arrays

HUD arrays deploy elements in a pattern that is random at short range but highly correlated at long range, characterized by a structure factor $S(k)=0$ for $|k|\leq k_c$. These suppress grating lobes independent of element spacing, while preserving narrow main lobes and deep sidelobe rejection ($\leq -18$ dB for all tested frequencies), even for $\lambda \ll 2a$. HUD arrays are trivially scalable via periodic tiling, allowing large array designs without recomputation (Tang et al., 2023).

Single-Transducer Plate Designs

The Stepped Plate PAL (SPPAL) employs a single Langevin transducer mechanically coupled to a flexural stepped plate, designed such that its velocity field approximates that of a rigid piston at the ultrasonic carrier frequency. By multi-objective optimization (“dual-resonance tuning”), plate geometry and transducer parameters are engineered for efficient generation of narrow audio beams down to 500 Hz. Spherical wave expansion and equivalence ratio analyses provide accurate predictions of on-axis sound pressure and beam patterns (Kim et al., 29 Apr 2025).

3. Signal Modulation, Nonlinearity, and Distortion

Modulation and Demodulation

PALs typically employ double-sideband amplitude modulation (DSB-AM) or lower-sideband AM (LSB-AM), encoding audio onto the ultrasonic carrier. The parametric audio field is generated by nonlinear demodulation, described by the virtual source model:

$$q(r_v, \omega_a) = -i \beta \omega_a \rho_0^2 c_0^4 p_1^*(r_v, \omega_1) p_2(r_v, \omega_2)$$

The radiated audio pressure results from the convolution of these “virtual sources” with the Green’s function of the secondary wave (Zhuang et al., 24 Apr 2025, Zhuang et al., 2024).

Nonlinear Distortion

The demodulation process inherently introduces even- and odd-order nonlinearities—manifest as total harmonic distortion (THD) and intermodulation distortion (IMD)—which challenge high-fidelity audio reproduction. While classical Volterra-inverse filtering reduces distortion to the low to mid-teens percent, higher-order nonlinearities prevent full correction (Li et al., 2024).

Recent application of deep-learning–based feedforward WaveNet models for identification and inverse compensation has achieved substantial reduction in THD/IMD, to averages below 5% and 3% respectively, surpassing 2nd- and 3rd-order Volterra benchmarks (Li et al., 2024).

4. Multi-Channel, Beamforming, and Sound Zone Applications

Multi-Channel and Sound Zone Control (SZC)

In multi-channel PALs (MCPALs), each array element or “virtual channel” is driven with independent phase/amplitude modulation for spatial control. The resulting nonlinear audio field is a bi-quadratic form of the drive vectors, precluding simple superposition but enabling advanced zone control (Zhuang et al., 2024, Zhuang et al., 30 Jul 2025).

Multi-carrier PALs now enable SZC with a single transducer: multiple audio signals are mapped to separate ultrasonic carriers, all launched simultaneously. Through structured nonlinear demodulation and spatial selectivity, existing acoustic contrast control (ACC) algorithms can be applied unmodified, yielding virtual multi-channel outputs in air (Zhuang et al., 24 Apr 2025). Simulations report contrast improvements of up to 15 dB using 4 carriers over single-carrier configurations. Compared to EDLs, PAL arrays offer greater contrast and robustness at audio frequencies $\geq4$ kHz, even under substantial coupling uncertainty (Zhuang et al., 2024).

Beamforming and Modeling

PAL beamforming relies on amplitude/phase steering at the ultrasonic carrier level. The “k-space” fast simulation methodology leverages 2D/3D FFTs to evaluate Westervelt-based models without paraxial or far-field restrictions, enabling field predictions for MCPALs with arbitrary array geometries and drive laws. This approach achieves $>10^4$ speedup over direct integration with no loss of accuracy (Zhuang et al., 30 Jul 2025).

5. Measurement, Characterization, and Practical Implementation

Microphone nonlinearity, resulting in spurious detection of high-amplitude ultrasonic components, poses challenges for reliable PAL characterization. Half-wavelength resonator filters fabricated by stereolithography (SLA) offer $60$ dB transmission loss at target ultrasonic frequencies (e.g., 40 or 60 kHz), suppressing artifacts regardless of angle or distance. The approach outperforms previous phononic or Helmholtz techniques in both attenuation and bandwidth, and should be integrated into ISO-compliant PAL test protocols for reproducibility (Kim et al., 16 Apr 2025).

Standard implementation practice for arrays includes PWM-driven FPGA control (sub-$\mu$s phase delays) and phased-drive beam steering up to $30^\circ$ off-axis, with validation in semi-anechoic facilities (Tang et al., 2023). For SPPALs, transducer fabrication uses full-wavelength, double-stack Langevin designs with optimized preloading and horn geometry to ensure robust dual-resonance behavior and prevent combination resonance at unwanted frequencies (Kim et al., 29 Apr 2025).

6. Experimental Systems and Special Applications

Highly compact PALs employing three-element FFR rings at 400 kHz are used as acoustic calibrators for neutrino telescopes, producing parametric “pancake” beams with $1^\circ$ width and $0.1$ Pa bipolar pulses exceeding $5\sigma$ detection thresholds at $1$ km in water (Adrián-Martínez et al., 2012). Envelope design (e.g., Gaussian windowed tone bursts) steers the temporal waveform, while array configuration shapes the beam for in situ calibration tasks.

7. Design Trade-offs, Limitations, and Outlook

Summarized key guidelines and open challenges:

• Array Layouts: HUD arrays offer grating-lobe–free, scalable solutions; periodic arrays require spacings $\leq \lambda/2$ to suppress aliases (Tang et al., 2023).
• Single-Transducer PALs: Stepped-plate and MCPL configurations dramatically reduce system complexity while retaining spatial selectivity (Zhuang et al., 24 Apr 2025, Kim et al., 29 Apr 2025).
• Distortion Mitigation: Deep-learning compensation provides superior nonlinearity suppression versus traditional inverse filters (Li et al., 2024).
• Measurement Standards: Half-wavelength resonator filters should become standard for PAL measurement to ensure artifact-free characterization (Kim et al., 16 Apr 2025).
• Engineering Constraints: Ultrasonic transducer bandwidth, stable high-voltage drive, and environmental adaptation (e.g., for underwater or reverberant spaces) remain practical challenges (Zhuang et al., 24 Apr 2025, Kim et al., 29 Apr 2025).
• Applications: Personal audio zones, targeted medical sonics, underwater neutrino calibration, and immersive spatial audio are current application domains.

A significant ongoing direction is the integration of model-based and data-driven (deep learning) methods for PAL system identification, control, and inverse design, as well as the extension of PAL arrays into large-scale, arbitrarily steerable and dynamically reconfigurable platforms.