Compressed Beamformer: Sparse Reconstruction

• Compressed beamformer is a family of sparsity-driven methods that recasts traditional beamforming as a lower-dimensional inverse problem.
• It uses techniques like ℓ1-regularized optimization and atomic norm minimization to recover sparse source maps from reduced measurements.
• Applications in acoustics, ultrasound, radar, and communications benefit from improved resolution and reduced data requirements through compressed beamforming.

Searching arXiv for relevant papers on compressed beamforming and compressed beamformers. Searching arXiv for compressed beamforming across acoustics, ultrasound, radar, and mmWave beamforming. A compressed beamformer is a beamforming formulation in which sensing, beam selection, or image formation is recast as a lower-dimensional or sparse inverse problem, so that array data are processed from fewer measurements, fewer channels, fewer Fourier coefficients, or a reduced representation of the scene. Across acoustics, ultrasound, radar, and wireless communications, the unifying premise is that the underlying source field, channel, angular support, or beamspace representation is sparse, compressible, or otherwise low dimensional, permitting recovery or detection from compressed observations with resolution that often exceeds conventional beamforming (Zhong et al., 2013, Gerstoft et al., 2015, Wagner et al., 2012, Pezeshki et al., 2022). The term therefore denotes a family of methods rather than a single algorithm: sparse reconstruction beamformers for direction-of-arrival estimation, covariance-based compressed array imagers, Fourier-domain and $k$-space ultrasound beamformers, compressed-domain radar detectors, random-projection adaptive beamformers, and beamspace or tensor-compressed beamforming architectures for communication systems (Xenaki et al., 2015, Tohidi et al., 2020, Mittal et al., 8 Jul 2025, Zheng et al., 2024).

1. Conceptual scope and defining formulations

In array acoustics and direction-of-arrival estimation, a compressed beamformer is typically a sparsity-driven reconstruction method. Rather than evaluating a conventional beampattern over a scan grid, the measurements are modeled as a linear superposition of steering vectors, and the unknown source map is recovered with $\ell_1$-regularized optimization. For narrowband sensing on a discretized angular grid, the canonical model is

$\mathbf{y}=\mathbf{A}\mathbf{x}+\mathbf{n},$

with $\mathbf{x}$ sparse over candidate directions, making the beamformer effectively a sparse inverse solver rather than a classical spatial filter (Gerstoft et al., 2015). A closely related aeroacoustic formulation writes the sensor outputs as

$Y=G_{m_I}S+N,$

where the source vector is assumed spatially sparse and mutually incoherent, and compressive sensing is applied directly to the source map rather than to the array surface field, which is not assumed sparse in a generic basis (Zhong et al., 2013).

In continuous-angle formulations, the compressed beamformer is defined without a discrete grid. The source field is modeled as a sparse measure,

$x(t)=\sum_{i=1}^{K} x_i\,\delta(t-t_i),$

and recovered by atomic norm minimization or an equivalent semidefinite program. This replaces grid-based sparse coding with continuous support estimation, specifically to avoid basis mismatch when true directions do not coincide with predefined angular bins (Xenaki et al., 2015).

In ultrasound, the expression “compressed beamforming” has a more literal acquisition meaning. The beamformed scanline or image is reconstructed from sub-Nyquist samples, partial Fourier coefficients, or a reduced set of synthesized receive plane waves. Early Xampling-based work defined compressed beamforming as estimating the Fourier coefficients of the beamformed signal directly from low-rate multichannel measurements, exploiting a finite rate of innovation model for the beamformed line (Wagner et al., 2011, Wagner et al., 2012). Later work generalized this to frequency-domain beamforming, where the beam is formed from a subset of Fourier coefficients rather than from oversampled time-domain delays, enabling reductions of up to $1/28$ over standard beamforming rates on in vivo cardiac data (Chernyakova et al., 2013). More recent far-field compressive ultrasound beamforming, termed KK beamforming, compresses receive data into virtual receive plane waves and performs reconstruction entirely in $k$-space, with coarray-based control over spatial-frequency sampling (Khetan et al., 23 Mar 2026).

In radar and communications, compressed beamformer can denote either compressed-domain beamforming itself or beamforming enabled by compressed channel inference. In colocated MIMO radar, a Capon beamformer is applied after a first compression stage, followed by a second compression of the sparse beamformer output before GLRT detection, yielding a “compress → beamform → compress again → detect” architecture (Tohidi et al., 2020). In mmWave systems, compressed sensing is used to estimate sparse angular channel structure from low-dimensional analog beam measurements, after which beams are selected or designed using the inferred raw channel rather than a fixed codebook (Pezeshki et al., 2022, Yang et al., 2022).

2. Sparse inverse formulations in acoustic and array beamforming

A principal lineage of compressed beamforming treats beamforming as sparse source localization. For a uniform linear array under narrowband plane-wave propagation, the steering vector at direction $\theta_i$ is

$\mathbf{a}(\theta_i)=\frac{1}{\sqrt{M} \left[1,\ e^{j\frac{2\pi d}{\lambda}\sin\theta_i},\ \ldots,\ e^{j\frac{2\pi d}{\lambda}(M-1)\sin\theta_i}\right]^T ,$

and the inverse problem is constructed over a fine angular grid (Gerstoft et al., 2015). The single-snapshot compressed beamformer solves

$\ell_1$0

or its constrained variant

$\ell_1$1

This formulation is explicitly derived as a maximum a posteriori estimate under complex Gaussian noise and a Laplacian prior on the source amplitudes (Gerstoft et al., 2015).

For multiple snapshots, the same paper extends the model to

$\ell_1$2

with shared support across snapshots. The corresponding multiple-measurement-vector compressed beamformer imposes row sparsity via

$\ell_1$3

followed by unbiased least-squares amplitude recovery on the detected active set (Gerstoft et al., 2015). An important stated property is that these CS beamformers operate directly on the raw snapshots and do not require estimating or inverting the sample covariance matrix, in contrast to MVDR and MUSIC; this is why they can work with a single snapshot and remain effective for coherent arrivals (Gerstoft et al., 2015).

The aeroacoustic formulation of compressive sensing beamforming in noisy measurements introduces two specific algorithms, CSB-I and CSB-II. CSB-I solves the direct sparse reconstruction problem

$\ell_1$4

and reports the beamforming output as

$\ell_1$5

CSB-II instead uses second-order statistics: $\ell_1$6 vectorizes the cross-spectrum matrix, and solves

$\ell_1$7

The distinction is substantive: CSB-I is described as computationally lightweight but highly sensitive to measurement noise, while CSB-II is explicitly designed to be more robust by working on covariance structure rather than raw pressure snapshots (Zhong et al., 2013).

The simulation evidence in that work gives a concrete contrast. In a free-space monopole case at $\ell_1$8 kHz with 10 randomly chosen microphones and white noise, CSB-I performs excellently at $\ell_1$9, showing more than $\mathbf{y}=\mathbf{A}\mathbf{x}+\mathbf{n},$0 dB dynamic range, but breaks down at $\mathbf{y}=\mathbf{A}\mathbf{x}+\mathbf{n},$1 dB with false sources scattered across the image. CSB-II still identifies the mainlobe at $\mathbf{y}=\mathbf{A}\mathbf{x}+\mathbf{n},$2 dB and preserves good sidelobe rejection, though its dynamic range drops to about $\mathbf{y}=\mathbf{A}\mathbf{x}+\mathbf{n},$3 dB. Against conventional beamforming,

$\mathbf{y}=\mathbf{A}\mathbf{x}+\mathbf{n},$4

the compressive methods are reported to have narrower mainlobes and better resolution (Zhong et al., 2013). In a practical aeroacoustic test with 56 microphones, CSB-II produced images with narrower mainlobes and lower sidelobe levels than conventional beamforming at $\mathbf{y}=\mathbf{A}\mathbf{x}+\mathbf{n},$5 kHz and $\mathbf{y}=\mathbf{A}\mathbf{x}+\mathbf{n},$6 kHz, while its reported computational cost was about 450 s on a modest laptop for 56 sensors and 1600 imaging gridpoints (Zhong et al., 2013).

3. Grid-based, grid-free, and off-grid compressed beamformers

A recurring issue in compressed beamforming is basis mismatch. Grid-based CS formulations assume that sources lie exactly on a chosen angular grid. When they do not, energy spreads across neighboring atoms and the reconstruction becomes biased or blurred. “Grid-free compressive beamforming” addresses this by using a continuous representation of the source field and solving a convex dual problem with semidefinite programming (Xenaki et al., 2015).

The continuous measurement model is written as

$\mathbf{y}=\mathbf{A}\mathbf{x}+\mathbf{n},$7

or, with noise,

$\mathbf{y}=\mathbf{A}\mathbf{x}+\mathbf{n},$8

where $\mathbf{y}=\mathbf{A}\mathbf{x}+\mathbf{n},$9 is the sparse measure over continuous angle parameter $\mathbf{x}$0 (Xenaki et al., 2015). The continuous analog of the $\mathbf{x}$1 norm is the atomic norm,

$\mathbf{x}$2

leading to the noiseless problem

$\mathbf{x}$3

and the noisy problem

$\mathbf{x}$4

The dual constraint

$\mathbf{x}$5

is reformulated as a semidefinite program using bounded trigonometric polynomial theory (Xenaki et al., 2015).

A notable feature of the grid-free method is that support recovery is extracted from the dual polynomial: true source locations correspond to points where the dual polynomial magnitude reaches one. The paper gives a root-finding construction via

$\mathbf{x}$6

with DOAs recovered from unit-circle roots of $\mathbf{x}$7 (Xenaki et al., 2015). It also states a maximum number of resolvable sources,

$\mathbf{x}$8

and, for complex amplitudes, a separation condition

$\mathbf{x}$9

This formulation is reported to work with non-uniform arrays, single-snapshot data, and noisy experimental towed-array measurements (Xenaki et al., 2015).

The distinction between grid-based and grid-free compressed beamforming is not merely numerical. The former discretizes the inverse problem and uses sparse regression; the latter promotes sparsity directly in a continuous measure space. This suggests that “compressed beamformer” is best understood as an inverse-modeling class whose specific instantiation depends on whether discretization, covariance statistics, or continuous convex geometry is taken as primary.

4. Ultrasound compressed beamforming: Xampling, frequency domain, and $Y=G_{m_I}S+N,$0-space

In ultrasound, compressed beamforming emerged from the observation that standard delay-and-sum beamforming imposes sampling rates far above the Nyquist rate of the received bandpass signal because fine time resolution is needed to implement dynamic delays digitally. The basic receive beamformer is

$Y=G_{m_I}S+N,$1

with

$Y=G_{m_I}S+N,$2

and

$Y=G_{m_I}S+N,$3

for the dynamic focusing law (Wagner et al., 2012, Chernyakova et al., 2013).

The 2011–2012 compressed beamforming work defines the beamformed signal as approximately finite rate of innovation: $Y=G_{m_I}S+N,$4 and derives Fourier coefficients of the beamformed scanline from low-rate measurements across the array (Wagner et al., 2011, Wagner et al., 2012). The resulting CS recovery model is

$Y=G_{m_I}S+N,$5

with sparse $Y=G_{m_I}S+N,$6 representing reflector amplitudes on a quantized delay grid, and OMP used for recovery (Wagner et al., 2011). The motivation is explicit: low-rate parameter extraction on each channel separately is unstable in the presence of noise and speckle, whereas beamforming the sub-Nyquist data first enhances SNR (Wagner et al., 2012).

On cardiac ultrasound data, that framework reported nearly eight-fold sample-rate reduction relative to standard techniques (Wagner et al., 2011, Wagner et al., 2012). In one experiment with a 64-channel phased array and 120 beams over a $Y=G_{m_I}S+N,$7 sector, the standard reference used 1662 real samples per element after conventional downsampling, while the compressed beamforming schemes used 100 complex coefficients per scanline in one version and an average of 116 complex samples per receiving element in the approximate formulation (Wagner et al., 2012).

“Fourier Domain Beamforming: The Path to Compressed Ultrasound Imaging” generalizes this idea by moving beamforming itself into the frequency domain (Chernyakova et al., 2013). Beam Fourier-series coefficients are written as weighted sums of nearby Fourier coefficients from each channel: $Y=G_{m_I}S+N,$8 with a rapidly decaying geometry-dependent kernel. The paper reports that about 20 significant coefficients of $Y=G_{m_I}S+N,$9 capture on average more than 95% of the energy (Chernyakova et al., 2013). By computing only the beam’s significant frequency bins, the method avoids oversampling required by time-domain interpolation and reports 416 real-valued samples per line versus 3360 for standard beamforming in one cardiac experiment, about $x(t)=\sum_{i=1}^{K} x_i\,\delta(t-t_i),$0 reduction, and up to $x(t)=\sum_{i=1}^{K} x_i\,\delta(t-t_i),$1 of standard beamforming rates when combined with compressed sensing on partial beam bandwidth (Chernyakova et al., 2013).

A second ultrasound branch merges Fourier-domain beamforming with convolutional beamforming and sparse arrays. “Compressed Fourier-Domain Convolutional Beamforming for Wireless Ultrasound Imaging” defines CFCOBA as a combination of Fourier-domain delay implementation, COBA’s virtual-aperture sum co-array, and compressed sensing recovery from partial Fourier coefficients (Mamistvalov et al., 2020). The beamformed signal is shown to admit an FRI model,

$x(t)=\sum_{i=1}^{K} x_i\,\delta(t-t_i),$2

leading to the sparse recovery problem

$x(t)=\sum_{i=1}^{K} x_i\,\delta(t-t_i),$3

Using in vivo data, the method reported up to $x(t)=\sum_{i=1}^{K} x_i\,\delta(t-t_i),$4 reduction in sampling rate and up to $x(t)=\sum_{i=1}^{K} x_i\,\delta(t-t_i),$5 total data reduction versus DAS (Mamistvalov et al., 2020).

The most recent ultrasound instance in the provided material is far-field compressive ultrasound beamforming, or KK beamforming (Khetan et al., 23 Mar 2026). Here the receive RF data $x(t)=\sum_{i=1}^{K} x_i\,\delta(t-t_i),$6 are compressed into virtual receive plane waves by

$x(t)=\sum_{i=1}^{K} x_i\,\delta(t-t_i),$7

reducing the data matrix from $x(t)=\sum_{i=1}^{K} x_i\,\delta(t-t_i),$8 to $x(t)=\sum_{i=1}^{K} x_i\,\delta(t-t_i),$9 and giving compression factor

$1/28$0

Image formation is rewritten in $1/28$1-space via pairwise differences of transmit and receive plane-wave directions,

$1/28$2

and the beamforming itself becomes

$1/28$3

The paper introduces dense low-frequency, shifted vernier, and confocal or hybrid sampling strategies to manage the resolution-versus-contrast tradeoff. It reports compression factors of about $1/28$4, $1/28$5, $1/28$6, and $1/28$7 depending on the receive-sampling strategy, with image quality comparable to DAS in phantom and in-vivo data, and faster runtimes such as 15.8 ms versus 53.9 ms in one setting (Khetan et al., 23 Mar 2026).

A separate ultrasound branch replaces handcrafted beamforming altogether with a learned compressive beamformer. “Adaptive and Compressive Beamforming Using Deep Learning for Medical Ultrasound” defines DeepBF as an encoder-decoder CNN that maps full or sub-sampled delayed RF cubes directly to IQ output (Khan et al., 2019). Compressive beamforming here means reconstructing high-quality images from 32, 24, 16, 8, and 4 Rx channels instead of 64, corresponding to approximately $1/28$8, $1/28$9, $k$0, $k$1, and $k$2 compression, with the same network handling random and fixed masks (Khan et al., 2019). The reported inference time is about 4.8 ms per depth plane, and at $k$3 subsampling on in vivo data DeepBF reaches approximately CR 11.80 dB, CNR 1.38, GCNR 0.65, PSNR 23.55 dB, and SSIM 0.87, outperforming DAS on those metrics (Khan et al., 2019).

5. Compressed-domain and reduced-dimension beamformers in radar and communications

In radar, the compressed beamformer may be an intermediate stage in a detection pipeline rather than a final image reconstruction. The colocated MIMO radar formulation in (Tohidi et al., 2020) begins with Nyquist data modeled as

$k$4

applies a first compression

$k$5

then designs a Capon beamformer in the compressed domain by solving

$k$6

After beamforming,

$k$7

a second compression is applied,

$k$8

and detection proceeds via a GLRT over angle cells. The method explicitly exploits the fact that the beamformer output becomes sparse after clutter suppression, making the second compression worthwhile. In some settings, it reports an 8-fold reduction of sample complexity relative to conventional CS-MIMO radar (Tohidi et al., 2020).

In mmWave hybrid and analog beamforming, compressed beamforming frequently means that the beam management problem itself is enabled by compressed sensing channel recovery. The underlying channel at delay tap $k$9 is observed only through low-dimensional beamformed measurements

$\theta_i$0

which may be as low-dimensional as $\theta_i$1 while the raw channel is $\theta_i$2 (Pezeshki et al., 2022). The sparse angular-domain channel model

$\theta_i$3

with dictionary

$\theta_i$4

yields the compressed sensing system

$\theta_i$5

OMP can then identify the strongest angular tuple and steer a custom beam toward it, while DLISTA replaces classical sparse recovery with an unrolled learned model based on ISTA and a LASSO-type objective (Pezeshki et al., 2022). The paper reports comparable spectral efficiency for OMP and DLISTA, while DLISTA reduces dictionary size from about 524k atoms to 2000 atoms in the reported setting (Pezeshki et al., 2022).

A closely related hybrid-beamforming channel estimation paper compares three CS frameworks: 1-D CS, two-stage CS using row-group sparsity, and 2-D CS using a 2-D dictionary (Yang et al., 2022). The 1-D vectorized model,

$\theta_i$6

is computationally burdensome because the dictionary grows as $\theta_i$7. Two-stage CS first estimates AoAs through SOMP on a row-sparse matrix, then estimates AoDs on the reduced problem; 2-D CS keeps the angular matrix structure and solves over atoms

$\theta_i$8

The paper concludes that 2-D OMP matches 1-D OMP in performance while having lower complexity and memory, whereas two-stage SOMP has somewhat lower performance but significantly lower complexity (Yang et al., 2022).

Beamspace massive MIMO introduces yet another meaning of compressed beamformer: a reduced-dimensional beamspace representation produced after a beamspace transformation and a discrete beam combination network (Jiang et al., 2017). There the receive chain is

$\theta_i$9

with $\mathbf{a}(\theta_i)=\frac{1}{\sqrt{M} \left[1,\ e^{j\frac{2\pi d}{\lambda}\sin\theta_i},\ \ldots,\ e^{j\frac{2\pi d}{\lambda}(M-1)\sin\theta_i}\right]^T ,$0 constrained to low-resolution phase shifters,

$\mathbf{a}(\theta_i)=\frac{1}{\sqrt{M} \left[1,\ e^{j\frac{2\pi d}{\lambda}\sin\theta_i},\ \ldots,\ e^{j\frac{2\pi d}{\lambda}(M-1)\sin\theta_i}\right]^T ,$1

The design objective is the spatial compression efficiency

$\mathbf{a}(\theta_i)=\frac{1}{\sqrt{M} \left[1,\ e^{j\frac{2\pi d}{\lambda}\sin\theta_i},\ \ldots,\ e^{j\frac{2\pi d}{\lambda}(M-1)\sin\theta_i}\right]^T ,$2

optimized via branch-and-bound or a sequential greedy beam combination scheme. The reported result is up to $\mathbf{a}(\theta_i)=\frac{1}{\sqrt{M} \left[1,\ e^{j\frac{2\pi d}{\lambda}\sin\theta_i},\ \ldots,\ e^{j\frac{2\pi d}{\lambda}(M-1)\sin\theta_i}\right]^T ,$3 RF-chain reduction with a one-bit digital phase-shifter network (Jiang et al., 2017).

6. Compression of beamforming objects: random projections, codebooks, and weight tensors

Not all compressed beamformers compress the scene; some compress the beamformer input or the beamformer object itself. “Beamforming with Random Projections: Upper and Lower Bounds” proposes preprocessing microphone-array data with multiple random projections

$\mathbf{a}(\theta_i)=\frac{1}{\sqrt{M} \left[1,\ e^{j\frac{2\pi d}{\lambda}\sin\theta_i},\ \ldots,\ e^{j\frac{2\pi d}{\lambda}(M-1)\sin\theta_i}\right]^T ,$4

followed by compressed-domain MVDR beamforming

$\mathbf{a}(\theta_i)=\frac{1}{\sqrt{M} \left[1,\ e^{j\frac{2\pi d}{\lambda}\sin\theta_i},\ \ldots,\ e^{j\frac{2\pi d}{\lambda}(M-1)\sin\theta_i}\right]^T ,$5

The compressed-domain MVDR solution is

$\mathbf{a}(\theta_i)=\frac{1}{\sqrt{M} \left[1,\ e^{j\frac{2\pi d}{\lambda}\sin\theta_i},\ \ldots,\ e^{j\frac{2\pi d}{\lambda}(M-1)\sin\theta_i}\right]^T ,$6

Instead of trusting any single projection, the method uses several compressed beamformers and applies a hard time-frequency selection rule

$\mathbf{a}(\theta_i)=\frac{1}{\sqrt{M} \left[1,\ e^{j\frac{2\pi d}{\lambda}\sin\theta_i},\ \ldots,\ e^{j\frac{2\pi d}{\lambda}(M-1)\sin\theta_i}\right]^T ,$7

Under a computational budget $\mathbf{a}(\theta_i)=\frac{1}{\sqrt{M} \left[1,\ e^{j\frac{2\pi d}{\lambda}\sin\theta_i},\ \ldots,\ e^{j\frac{2\pi d}{\lambda}(M-1)\sin\theta_i}\right]^T ,$8, the paper argues that many small projections can outperform a single larger one; it reports higher SNR and SINR gain than full sensor-space MVDR over a wide range of projection dimensions in simulated distributed microphone arrays (Mittal et al., 8 Jul 2025). Theoretical upper and lower bounds are derived using an RIP-type assumption on the projection matrix (Mittal et al., 8 Jul 2025).

Compression can also target feedback or fronthaul representations of beamforming matrices. “Beamforming Matrix Quantization with Variable Feedback Rate” compresses a unitary beamforming matrix using Givens Rotation factorization,

$\mathbf{a}(\theta_i)=\frac{1}{\sqrt{M} \left[1,\ e^{j\frac{2\pi d}{\lambda}\sin\theta_i},\ \ldots,\ e^{j\frac{2\pi d}{\lambda}(M-1)\sin\theta_i}\right]^T ,$9

but adapts the number of quantization bits for phase parameters based on the values of the rotation angles. At average 8 bits feedback for a $\ell_1$00 beamforming vector, the paper reports MSE 0.110 and MAD 0.312 for the traditional fixed-rate scheme versus MSE 0.092 and MAD 0.282 for the proposed variable-rate scheme (0806.3329). For a $\ell_1$01 unitary beamforming matrix, it reports average effective feedback rate about 12.71 bits when adaptive allocation and Huffman coding are used (0806.3329).

For hybrid mmWave channel estimation, deterministic beamformer and pilot codebook design is framed as coherence minimization of the equivalent sensing matrix

$\ell_1$02

The mutual coherence objective is

$\ell_1$03

and the paper designs DFT-based hybrid beamformers and pilots, along with a greedy precoder column ordering, to reduce coherence and improve OMP-based channel recovery (Sung et al., 2019). Although this is not a compressed beamformer in the acoustic-imaging sense, it is a compressed beamforming strategy in the broader communications sense: the beamformers are designed to produce a sensing matrix favorable to sparse beam-space inference.

A more infrastructure-oriented example is tensor compression of ZF beamforming weights for massive MU-MIMO fronthaul (Zheng et al., 2024). The weights are arranged in a third-order tensor

$\ell_1$04

and compressed by sparse Tucker decomposition,

$\ell_1$05

followed by complex Givens decomposition and run-length encoding of the factor matrices (Zheng et al., 2024). The method specifically targets beamforming weights sent over eCPRI from BBU to RRU. For Dataset 1 at around 15% compression ratio, it reports RL $\ell_1$06 for TD, RL $\ell_1$07 for STD, and RL $\ell_1$08 for STD+FC; for Dataset 2 at around 15% compression ratio, RL is $\ell_1$09, $\ell_1$10, and $\ell_1$11, respectively (Zheng et al., 2024).

7. Performance themes, tradeoffs, and recurring misconceptions

Several themes recur across these diverse meanings of compressed beamformer. First, sparsity assumptions are domain specific. In aeroacoustics, the source map is assumed spatially sparse even though the measured array field is not sparse in a typical basis (Zhong et al., 2013). In mmWave communications, sparsity is in angular-domain channel structure, often after discretization into AoA/AoD dictionaries (Pezeshki et al., 2022, Yang et al., 2022). In ultrasound, sparse or compressible structure may apply either to strong reflectors in the beamformed scanline or to spatial-frequency sampling of the image transfer function (Chernyakova et al., 2013, Khetan et al., 23 Mar 2026). This suggests that “compressed beamformer” does not imply a universal sparsifying transform; it denotes exploitation of whichever latent structure is physically justified in the application.

Second, covariance use is not consistent across subfields. Some compressed beamformers deliberately avoid covariance inversion, as in single- and multiple-snapshot sparse DOA estimation (Gerstoft et al., 2015). Others become robust precisely by moving to second-order statistics, as in CSB-II for noisy aeroacoustic measurements (Zhong et al., 2013). In compressed-domain radar, Capon beamforming remains central because clutter suppression is needed before the second compression stage (Tohidi et al., 2020). A plausible implication is that compressed beamforming should not be conflated with covariance-free beamforming; the choice depends on whether covariance is a burden or a resource.

Third, compression does not automatically imply robustness. The CSB-I versus CSB-II comparison is explicit: direct sparse inversion of noisy measurements fails quickly, whereas covariance-based sparse power recovery remains usable at $\ell_1$12 dB and fails only below about $\ell_1$13 dB in the simple example (Zhong et al., 2013). Similarly, early ultrasound compressed beamforming based on exact sparsity preserved strong reflectors but lost speckle, which later $\ell_1$14-based compressible models were designed to retain (Chernyakova et al., 2013). This contradicts the common misconception that all compressed beamformers simply trade data rate for little else; in the cited literature, the main issue is often whether the structural prior matches the physics of noise, clutter, or weak scattering.

Fourth, higher resolution often comes from global inverse optimization rather than from narrower physical beams. Sparse DOA beamformers produce narrower effective mainlobes and better sidelobe suppression because they solve for a globally sparse explanation of the measurements (Gerstoft et al., 2015, Zhong et al., 2013). Grid-free CS achieves super-resolution by continuous sparse recovery rather than by denser gridding (Xenaki et al., 2015). KK beamforming manages resolution and contrast through $\ell_1$15-space sampling density and support, not through conventional receive-aperture delay laws (Khetan et al., 23 Mar 2026).

Finally, the literature uses “compressed beamformer” for at least four distinct objects: a sparse reconstruction algorithm, a reduced-dimension adaptive beamformer, a compressed-domain detection pipeline, or a compressed representation of beamforming weights or channel observations. The terminology is therefore context dependent. A compressed beamformer in ultrasound may reconstruct an image from sub-Nyquist or far-field-compressed receive data (Wagner et al., 2012, Khetan et al., 23 Mar 2026), whereas in mmWave it may denote CS-enabled beam selection from underdetermined analog measurements (Pezeshki et al., 2022), and in fronthaul systems it may refer to transport compression of the beamforming tensor itself (Zheng et al., 2024). The common denominator is dimensionality reduction guided by beam physics and inverse modeling, not a single standardized architecture.