Higher-Order Non-Linear Beamforming

• Higher-order non-linear beamforming is a set of methods that utilize non-linear combinations of delayed sensor signals to harness higher-order statistics for improved noise suppression and signal enhancement.
• These techniques extend traditional delay-and-sum methods by employing polynomial correlations and neural network architectures, leading to sharper resolution and better artifact rejection.
• Efficient implementations via closed-form polynomial expansions and GPU parallelism enable real-time applications, yielding significant gains in SNR, dynamic range, and imaging contrast.

Higher-order non-linear beamforming refers to spatial signal processing techniques that incorporate products or other non-linear combinations of delayed sensor signals across orders greater than two, in order to exploit higher-order statistical dependencies for improved noise/artifact suppression and signal enhancement. These methods surpass traditional linear approaches—such as delay-and-sum—as well as basic second-order non-linear methods like delay-multiply-and-sum (DMAS), by systematically harnessing higher-order interactions among sensor channels in array signal processing tasks. This class includes both polynomial-correlation beamformers and neural architectures designed to realize, approximate, or extend such non-linear filtering.

1. Mathematical Principles and Non-Linear Extensions

Linear beamforming methods, typified by delay-and-sum (DAS), assume perfect coherence across delayed channels, summing the delayed observations $s_i(t-\tau_i)$ as $S_\mathrm{DAS}(t) = \sum_{i=1}^N s_i(t-\tau_i)$. While computationally efficient ($O(N)$ per beam/pixel), this approach suffers from broad main lobes, high sidelobes, and low contrast/resolution in noisy or reverberant environments due to its inability to discriminate signal from incoherent noise or clutter (Mulani et al., 2022, Jansen et al., 12 Nov 2025).

Second-order DMAS improves upon these limitations by introducing pairwise signal products, evaluating $$S_\mathrm{DMAS} = \sum_{i<j} \operatorname{sign}(s_i s_j) \sqrt{|s_i s_j|}$$. This pairwise multiplication acts as a simple coherence detector, reinforcing true source directions while attenuating incoherent/noise-like signals, but incurs a quadratic $O(N^2)$ complexity.

The general $k$-th order DMAS beamformer extends this by summing over all $k$-tuples of delayed signals:

$$S_k = \sum_{1 \le i_1 < \dots < i_k \le N} \operatorname{sign}\bigg(\prod_{\ell=1}^k s_{i_\ell}\bigg) \bigg|\prod_{\ell=1}^k s_{i_\ell}\bigg|^{1/k}$$

with $k=3,4,5$ yielding progressively stronger suppression of off-axis incoherent artifacts while further distinguishing coherent signal arrivals (Mulani et al., 2022, Jansen et al., 12 Nov 2025).

In the domain of multichannel speech enhancement, higher-order non-linear beamformers also emerge from an MMSE-optimal Bayesian perspective when the noise is modeled as non-Gaussian, e.g., a Gaussian mixture, resulting in estimators that are fundamentally non-linear and jointly exploit spatial and spectral statistics (Tesch et al., 2021). Neural approximators such as TaylorBeamformer employ high-order nonlinear transformations, recursively derived from Taylor expansion terms, where each higher-order component serves as a data-driven residual canceller complementing the 0th-order spatial filter (Li et al., 2022).

2. Efficient Implementation via Closed-Form and Neural Methods

Direct computation of $k$-th order DMAS is combinatorially expensive ($O(N^k)$ for $N$ sensors and order $k$). Closed-form polynomial expansions, derived using Newton–Girard identities, enable efficient $O(N)$ computation for all practical $k$ (typically $k \leq 5$), reducing higher-order sums to a small number of sums and products of vector powers. For example, third-order DMAS is implemented as:

$$S_3 = \tfrac{1}{6}\left[(\sum_i r_i)^3 + 2\sum_i s_i - 3(\sum_i r_i)(\sum_i r_i^2)\right]$$

with $r_i = s_i^{1/3}$ (Mulani et al., 2022, Jansen et al., 12 Nov 2025).

Real-time deployment is achieved by parallelizing these vector calculations across array pixels or time-frequency bins on GPU architectures, with memory traffic and root/sign operations managed to sustain high throughput (e.g., 23 frames per second for images with $2048 \times 2048$ pixels on commodity GPUs for $k=3$) (Mulani et al., 2022). Embedded GPU platforms support real-time in-air acoustic imaging with similar techniques, using a per-pixel CUDA thread model (Jansen et al., 12 Nov 2025).

End-to-end neural architectures, such as TaylorBeamformer, replace explicit higher-order analytic terms with learnable neural modules for each derivative order. These networks (e.g., stacks of S-TCN blocks) are trained with loss functions balancing spatial and spectral reconstruction, and achieve competitive inference cost ($\sim$8 Giga MAC/s on 6 mics, 7.25M parameters for $Q=5$) (Li et al., 2022).

3. Quantitative Performance in Imaging and Speech Applications

Systematic evaluations have demonstrated that increasing the correlation order $k$ yields monotonic improvements in contrast, SNR, and artifact suppression, up to an empirically optimal value (usually $k=5$). In photoacoustic imaging, progressing from DAS to DMAS-5 led to FWHM reductions (from $\sim$3.2 mm to $\sim$1.6 mm) and SNR improvements ($+25$ dB versus DAS, $+17$ dB versus DMAS) (Mulani et al., 2022). In in-air acoustic imaging, dynamic range increased from $\sim$30 dB (DAS) to $\sim$80 dB (DMAS-5), with SNR and contrast rising accordingly (Jansen et al., 12 Nov 2025).

In multichannel speech enhancement, analytic higher-order non-linear filters outperformed classical and two-stage linear beamformers, particularly in heavy-tailed (kurtotic) or multi-interferer environments. The non-linear joint MMSE filter, or its neural approximation, delivered SI-SDR gains up to $\sim$4.5 dB and perceptual speech quality (POLQA) advantages over linear approaches in both simulated and real environments (Tesch et al., 2021). Neural higher-order methods (TaylorBeamformer, $Q=5$) outperformed frame-wise oracle MVDR baselines by $+0.4$ PESQ, $+9$ ESTOI points, and $+2.3$ dB SI-SDR in causal 6-microphone speech enhancement settings (Li et al., 2022).

Method/Order	FWHM (mm)	SNR Gain (dB, vs. DAS)	Dynamic Range (dB)
DAS (Linear, $k=1$)	$\sim$3.2	0	$\sim$30
DMAS ($k=2$)	$\sim$1.9	$+8$–$9$	$\sim$50
DMAS-5 ($k=5$)	$\sim$1.6	$+25$	$\sim$80

4. Comparison with Classical Linear and Hybrid Approaches

Classical linear approaches, such as MVDR or multichannel Wiener filtering, are optimal only under Gaussian noise due to the sufficiency of second-order statistics. Two-step cascades (linear spatial filter plus postfilter) are suboptimal for non-Gaussian fields because they cannot fully exploit higher-order spatial or spectral dependencies.

Higher-order non-linear methods, whether analytic or learned, can (a) suppress more than $D-1$ directional interferers for $D$ array elements, and (b) adapt to non-stationary or heavy-tailed noise via higher-order moment exploitation. This effect is particularly pronounced in heavy-tailed (super-Gaussian) or mixture-based noise environments (Tesch et al., 2021). Linear cascades lose spatial detail by collapsing mixture components, whereas non-linear spatial filtering leverages individual component covariances, realizing higher spatial selectivity.

Neural architectures inspired by Taylor expansion (e.g., TaylorBeamformer) generalize this principle, where each additional order corresponds to a data-driven non-linear correction that further reduces residual noise or reverberation, with performance saturating after $Q=5$ (Li et al., 2022).

5. Practical Implementation Strategies

For efficient deployment of higher-order non-linear beamformers:

• Use closed-form polynomial expansions for analytic DMAS to reduce computation from $O(N^k)$ to $O(N)$ per beam/pixel (Mulani et al., 2022, Jansen et al., 12 Nov 2025).
• Exploit GPU parallelism by allocating one computation thread per output pixel/angle, with delayed signal lookups and vectorized operations.
• Coherence Factor (CF) weighting can further clean up side lobes and residual artifacts, being especially useful in reverberant scenes (Jansen et al., 12 Nov 2025).
• For neural methods, stack modular networks corresponding to higher-order residual terms and supervise both spatial and spectral outputs for optimal training convergence (Li et al., 2022).
• In both signal processing and neural contexts, orders beyond $5$ may yield diminishing returns or signal saturation/distortion, so system parameters are typically tuned for $k, Q \leq 5$.

6. Application Domains and Limitations

Higher-order non-linear beamforming is applicable to:

• Photoacoustic and ultrasonic imaging, where image quality and artifact rejection are critical, and coherent signal peaks require maximal reinforcement (Mulani et al., 2022).
• In-air acoustic imaging and real-time sonar/ultrasound applications, with practical deployment on embedded GPU processors for industrial, autonomous robotic, and medical imaging scenarios (Jansen et al., 12 Nov 2025).
• Multichannel speech enhancement, especially in environments characterized by non-Gaussian, diffuse or distributed interferers, or where outlier robustness is required (Tesch et al., 2021, Li et al., 2022).

Key limitations include increased memory bandwidth and per-pixel compute (root/sign calculation), slight sensitivity to calibration errors, and signal peak saturation for orders above five (Mulani et al., 2022). Analytic construction in high-dimensional arrays may become intractable; neural approximators mitigate this cost at the expense of requiring extensive labeled data and careful model selection.

7. Extensions and Future Research Directions

For large arrays or time-varying environments:

• Fit more flexible non-Gaussian mixture models or non-parametric noise models for Bayesian filters (Tesch et al., 2021).
• Use attention or recurrent architectures to capture time-varying spatial statistics.
• Generalize analytic higher-order beamforming to arbitrary heavy-tailed (e.g., α-stable) noise through learned nonlinearities (Tesch et al., 2021).
• Combine higher-order analytic and neural approaches: hand-crafted closed-forms as initialization or regularization for trainable systems (Li et al., 2022).

A plausible implication is that as embedded computing capabilities expand, advanced non-linear beamformers—both analytic and learned—will become increasingly prevalent in real-time, resource-constrained deployment for robust imaging, sensing, and speech enhancement.