PAM Beamforming Framework

• PAM Beamforming Framework is a spatial signal processing approach that uses pulse amplitude modulation to enable efficient, high-resolution beam synthesis with reduced hardware complexity.
• Key technical advances include row–column multiplexing, co-prime sampling, and convolutional time-domain formulations that boost imaging speed and suppress noise.
• The framework is applicable in photonic, acoustic, and RF arrays, enhancing passive acoustic mapping and ultrasound therapy with significant computational gains.

Pulse Amplitude Modulated (PAM) Beamforming Frameworks encompass a class of spatial signal processing techniques in which transducer or radiator arrays are driven and/or controlled via pulse amplitude modulation, enabling efficient electronic and photonic beam synthesis, high spatial resolution, and favorable trade-offs in hardware complexity. The PAM paradigm is prominent in both acoustic and optical fields, particularly where real-time computational or device constraints force fundamental design innovations. Key technical advances in PAM beamforming include row-column multiplexing to reduce driver count in photonic arrays, data-adaptive and deep learning driven apodization for passive acoustic mapping, and convolutional time-domain formulations for efficient inverse problem solution.

1. PAM Row–Column Drive in Co-Prime Photonic Beamforming Arrays

The co-prime photonic transceiver architecture utilizes PAM multiplexing in conjunction with co-prime sampling to achieve beamwidth and sidelobe performance equivalent to conventional dense arrays with a drastically reduced number of electrical drivers (Khachaturian et al., 2021). Each N×N array (transmitter and receiver) is equipped with phase modulators at every element, traditionally requiring O(N²) individual drivers. PAM row-column methodology organizes phase modulators into N rows and N columns, each addressed by a separate amplitude (power) driver. The instantaneous amplitude at location (i,j) becomes the product of the row and column drive signals, $A_i·B_j$. This enables programmable independent control over each modulator via $w_{ij} = e^{j(\alpha_i+\beta_j)}$, while the total hardware complexity drops to O(N) drivers and the number of resolvable spots remains O(N²).

Co-prime sampling in this context involves configuring the aperture spacings as $d_{Tx}=P d_x$, $d_{Rx}=Q d_x$ with $P, Q$ coprime. Grating lobes in the transmitted and received arrays only overlap at the main lobe, ensuring full field-of-view (FOV) coverage without side-lobe ambiguities.

2. Mathematical Description and Analytical Expressions

For a generic 2D array of M×M radiators, the PAM-driven beam synthesis is described by the array factor: $$AF_{Tx}(θ_x, θ_y) = \sum_{m=0}^{M-1}\sum_{n=0}^{M-1} w_{mn} e^{j k [m d_{Tx} \sinθ_x + n d_{Tx} \sinθ_y]}$$ with $w_{mn}$ realized as products of row and column drive phases. The synthesized heterodyne beam pattern in the co-prime architecture is given by: $$P_{TRx}(θ;\,θ_{Tx},θ_{Rx}) = P_{Tx}(θ - θ_{Tx}) \cdot P_{Rx}(θ - θ_{Rx}) \cdot \rho(θ)$$ The minimum grating-lobe angle per aperture is $θ_{GL}= \sin^{-1}(\lambda/d_{Tx})$, and full-FOV is achieved when the base spacing $d_x = \lambda/2$.

3. Complexity and Implementation Trade-offs

Conventional uniformly spaced arrays of $N^2$ radiators require $N^2$ phase drivers, scaling linearly with desired spatial resolution. PAM row-column driving reduces the driver count to $2N +$dummy elements for an $N\times N$ grid, preserving beam synthesis flexibility over $N^2$ spatial spots.

Table: Comparison of Array Complexity

Architecture	Radiators	Electrical Drivers	Sidelobe Level (SLL)
Conventional ($d=\lambda/2$ grid)	$N^2$	$N^2$	–11.3 dB/typical
Co-prime PAM (row-column)	$N^2$	$2N$ (+dummy)	–11.3 dB (measured)

A silicon photonic implementation with $8\times8=64$ Tx and Rx elements and co-prime spacings ($P=3, Q=4$) achieves a beamwidth of 0.6°, SLL of –11.3 dB, and 1026 resolvable spots using only 34 PAM electrical drivers (Khachaturian et al., 2021).

4. Design Guidelines and Applicability

For large photonic or RF arrays, PAM row-column drive is optimal when:

• Co-prime spacings $(P, Q)$ are chosen to minimize grating lobes, balancing sidelobe rejection and spot count.
• Aperture spacings are realized in routing layers with $d_x = \lambda/2$.
• The modulation bandwidth of PAM drivers is well below the multiplexing rate, ensuring quasi-static phase control.
• The principle extends to RF arrays by analog multipliers or switched row-column driver architectures.

Choosing larger $P, Q$ values increases spatial diversity but requires more radiators. Row-column PAM allows arbitrary separable weight synthesis $w_{ij} = r_i c_j$ for full-aperture control.

5. PAM Beamforming in Signal Processing Frameworks

In the context of passive acoustic mapping (PAM) for ultrasound therapy, deep and data-adaptive beamforming methods augment or supplant classical time exposure acoustics (TEA) via learned mappings from raw radio-frequency (RF) signals to spatial energy maps. The deep beamformer architecture leverages a generative adversarial network to produce PAM images with reduced energy spread and improved signal-to-noise ratio (ISNR), matching traditional beamformer quality at three orders of magnitude lower computational cost (Zeng et al., 2024).

Switchable deep PAM beamformers accept transducer information as a one-hot mask, enabling flexible reconstruction across different array geometries and frequency regimes. Data-adaptive PAM beamformers utilize robust Capon beamforming, eigenspace projection, and dual apodization cross-correlation for interference suppression and optimal spatial focus.

6. Extensions: Convolutional Time-Domain PAM and Inversion

Efficient PAM beamforming frameworks also recast the forward model as a convolutional mapping from spatiotemporal source distributions to measured RF signals, allowing inversion via regularized optimization (ℓ₁ and regularization by denoising terms) and solution by ADMM (Gelvez-Barrera et al., 12 Jan 2026). This enables sub-100µs temporal tracking and superior cavitation localization compared to delay-and-sum or frequency-domain techniques.

Key steps include:

• Physical forward model: $y_j(t) = (A s)_j(t) + w_j(t)$, with Green’s function $h_j(r, t)$.
• Discrete convolutional reformulation: $$Y = \mathcal{E}( \sum_{j=1}^{N_z} K_{:,j,:} * X_{:,N_z-j+1,:}) + W$$.
• Regularized least squares inversion: $$\hat{x} = \arg\min_x \frac{1}{2} \|A x - y\|_2^2 + \lambda\|x\|_1 + \mu R_2(x)$$, solved via ADMM with auxiliary denoising and shrinkage steps.

This approach is most effective when sensor pitch matches pixel width, and computational gains are leveraged via FFT-based convolutional operators.

7. Limitations and Contemporary Research Directions

PAM beamforming frameworks with row-column multiplexing and time-domain convolutional inversion are subject to device and modeling constraints, such as array geometry, modulation bandwidth, spatial element uniformity, and accurate propagation delay modeling. For convex or 2D arrays, retraining or extension of the switch index is required. In deep learning-based PAM, bubble interactions and nonlinear cavitation physics are typically ignored and remain open challenges. Extensions include style-transfer approaches for array generalization and joint source-speed-of-sound inversion for aberration correction (Zeng et al., 2024, Gelvez-Barrera et al., 12 Jan 2026).

PAM beamforming, via pulse amplitude modulation in either hardware or algorithmic implementation, offers a unifying framework for high-resolution, efficient, and scalable spatial imaging across photonic, acoustic, and clinical domains.