DFT Beamspace Processing Overview

• DFT beamspace processing is a technique that applies the discrete Fourier transform to convert array data into orthogonal spatial beams for focused energy and efficient dimension reduction.
• It enables applications in automotive MIMO radar, mmWave/THz communications, and wideband beamforming by facilitating sectorization and reducing computational overhead.
• The approach leverages FFT-based low-complexity computation and hardware-efficient designs to overcome array size, power, and cost constraints while maintaining high-resolution performance.

DFT beamspace processing refers to the use of the discrete Fourier transform (DFT) as a spatial or slow-time transformation to project antenna-domain or pulse-domain data into an orthogonal domain of beams. This enables spatial energy focusing, angular sectorization, or explicit dimension reduction for high-resolution sensing, communications, and radar. DFT beamspace techniques have become fundamental across automotive MIMO radar, mmWave/THz communications, wideband spectral processing, and array hardware design: they offer a mathematically principled, physically meaningful, and computationally tractable alternative to element-space or tapped-delay-line processing, especially in systems where array size, regulatory power constraints, or hardware cost are dominant engineering bottlenecks.

1. Principles of DFT Beamspace Mapping

DFT beamspace processing applies a unitary DFT matrix $F_N$ across the antenna, pulse, or snapshot dimension(s) of a sensor array. For a uniform linear array (ULA) of $N$ elements (or a sequence of $N$ pulses in MIMO radar), the DFT matrix

$$[F_N]_{k,n} = \frac{1}{\sqrt{N}} e^{-j \frac{2\pi}{N} k n}$$

maps the sampled measurements to $N$ orthogonal beam directions ("DFT bins"), each approximating a spatial frequency (or DOA) given by $\sin\theta_k = 2k/N - 1$. The DFT beamspace is thus an angular spectrum: each output corresponds to the signal amplitude incident from a specific direction. Unlike physical phased array steering or analog phase shifting, DFT beamspace operates in digital or baseband. For multidimensional arrays (planar, volumetric), separable Kronecker DFTs yield 2D or 3D beam covering.

In slow-time (pulse) domains—such as in Doppler division multiple access (DDMA) MIMO radar—the DFT across the pulse dimension yields virtual beams in Doppler (or "slow-time beamspace"), which can then be mapped back to physical transmit/receive beams using code modulation schemes (Xu et al., 2021).

The DFT is widely adopted because it (i) produces orthogonal, physically meaningful beams, (ii) enables energy focusing and sectorization, (iii) admits FFT-based low-complexity computation, and (iv) provides analytic tractability for signal processing and hardware analysis (Taner et al., 2021).

2. Applications and Signal Models

MIMO Radar—Slow-Time Beamspace for Transmit Focusing

Transmit beamspace (TB) DDMA automotive MIMO radar demonstrates DFT beamspace applied across the slow-time (pulse) dimension. Here, a bank of phase codes with intentionally designed "empty" Doppler spectrum are assigned to transmit elements. The DFT is used to form a set of transmit beams in slow-time, each focusing energy into an angular sector of interest. This achieves sectoral SNR gain ($\approx 10\log_{10}(M)$ dB for $M$ transmitters), mitigates the trade-off between antenna active-time and total frame time, and resolves Doppler ambiguities via an empty-guard band and sequential test function (Xu et al., 2021). Only those beamspace indices corresponding to the target sector are processed, reducing computational and detection search overheads.

Massive MIMO Communications and Hybrid Arrays

In mmWave and THz communications, DFT beamspace transforms high-dimensional antenna data into a sparse set of beams aligned with the dominant propagation directions. Under the standard model $$\mathbf{h} = \sum_\ell \alpha_\ell \mathbf{p}(\omega_\ell)$$ (with steering vectors $\mathbf{p}(\omega)$), the DFT resolves each plane wave into a localized beamspace bin. For hardware-limited hybrid MIMO, a subset of DFT beams are selected and RF chains are mapped to them, enabling hybrid digital/analog processing with near-optimal performance and greatly reduced hardware cost (Taner et al., 2021, Şenyuva, 30 Nov 2025).

Wideband Arrays and Beamforming

In wideband scenarios or under frequency-selective propagation, the DFT approach decomposes the spatio-temporal problem into parallel narrowband beamforming sub-problems ("beamspace snapshot" model). Tapped-delay-line (TDL) implementations otherwise scale quadratically with array size and bandwidth. DFT beamspace reduces both covariance computation and weight optimization to per-frequency-bin problems, yielding orders-of-magnitude complexity reduction without SINR loss (Hamza et al., 2019, Hamza et al., 2020, Noroozi et al., 15 Aug 2025). Block-Toeplitz completion and group-sparse optimization are used to maintain performance under sparse sensor selection or missing-lag regimes.

Near-Field Beam Training and High-Dimensional Arrays

In massive-antenna (ELAA/XL-MIMO) near-field systems, the DFT codebook—originally designed for far-field planar-wave steering—still produces beam patterns whose angular support (center) tracks user direction and whose spread (width) encodes user range (Wang et al., 27 Mar 2025, Wu et al., 2023, Wang et al., 25 Jun 2025, Heo et al., 26 Feb 2025). Closed-form, O(1)-complexity algorithms infer range from measured DFT beam spreads; hybrid model-driven and neural methods further improve alignment within reduced support sets (Heo et al., 26 Feb 2025). Sparse DFT codebooks and subarray partitioning enable periodic, scalable, and low-overhead beam training, with training overhead scaling as $O(\sqrt{N})$ rather than $O(N^2)$ (Zhou et al., 2024).

3. Algorithms, Optimization, and Performance

Beam Selection, Sectorization, and Covariance Guidance

Optimal performance with constrained RF budgets or computational limits is achieved by selecting a small number of DFT beams—either by sectorization or by covariance-guided optimization over contiguous beam blocks. Structured signal-plus-noise covariance estimation and Toeplitz-PSD projection yield denoised signal subspaces for DFT beam selection. Maximizing a well-conditioned "capture" score (combining trace and condition number metrics) ensures robust DOA estimation within a tight beam budget (Şenyuva, 30 Nov 2025).

Sparse Array Design and Group-Sparsity

To minimize sensor count while preserving SINR, weighted mixed-norm (e.g., $\ell_{1,\infty}^2$) and group-sparsity penalties are introduced within the DFT beamspace optimization. Semidefinite relaxation (SDR) and iterative reweighting techniques identify physically implementable sparse subarrays and assign frequency-dependent beamformers across all bins (Hamza et al., 2020, Hamza et al., 2019). This exploits DFT bin orthogonality and narrowband decoupling for scalable array optimization.

Robust Beamforming and Reconstructions

Low-complexity robust beamformers reconstruct the spatial power spectrum and correlation sequence via diagonal averaging and DFT/IDFT, then carve out noise-plus-interference angular sectors to reconstruct interference covariances. The difference yields an interference-suppressed desired-signal covariance, to which standard MVDR (or Capon) weights can be applied (Mohammadzadeh et al., 2021). These data-driven methods require little prior information and maintain robust SINR performance under angular mismatch, non-Toeplitz error, and scattering impairments.

Hardware-Efficient and Approximate DFTs

Current-mode analog CMOS and digital ASIC/FPGA designs implement DFT beamspace transformation using approximate integer-matrix DFTs, sparse-add-only factorizations, or multiplierless transforms (Ariyarathna et al., 2015, Madanayake et al., 2022, Madanayake et al., 2022). Such designs exploit the beamspace orthogonality and energy localization of the DFT, enabling $>4$ GHz instantaneous bandwidth and power/area reductions up to 55% vs. multiplier-rich FFTs. For massive planar arrays, 2D (Kronecker) DFTs are standard; high spatial-frequency (evanescent) codewords are identified as non-propagating and pruned without loss in beamforming or throughput (Yang et al., 2024).

4. Analytical and Practical Trade-Offs

Energy Concentration and Dimension Reduction

DFT beamspace leverages energy localization—most plane wave or localized sources map to a small subset of DFT bins/beams. Selecting only the relevant bins achieves computational scaling from $O(N^3)$ (full MVDR) to $O(N\log N)$ (FFT-based beamspace), with no loss in detection or nulling for typical array sizes and sectoral coverage (Noroozi et al., 15 Aug 2025). Empirical results demonstrate near-equality in detection probability, false-alarm rate, and interference suppression for well-chosen window sizes or beam budgets.

Angular Resolution and Range-Angle Ambiguity

By choosing DFT bins that most tightly cover the field-of-interest, the angular search space is drastically reduced, leading to improved angle estimation accuracy and narrower beamwidths. In transmit beamspace MIMO radar, the selection of k-indices within specific angle sectors focuses transmit power, improves SNR, and enhances DOA estimation precision (Xu et al., 2021).

For near-field beam training, the width of the DFT beam lobe yields a direct, invertible relation to user range; algorithms achieve $\O(1)$ complexity in the range domain (distance estimation is achieved by a one-shot algebraic formula, not a grid search) (Wang et al., 27 Mar 2025, Wang et al., 25 Jun 2025). Energy spread sets the minimal subspace needed for DNN-based fine alignment, enabling hybrid model-driven and learning approaches to scale efficiently (Heo et al., 26 Feb 2025).

Codebook Redundancy and Evanescent Beams

In codebook-based systems (3GPP NR), the Kronecker-product DFT bases for planar arrays theoretically span beyond the physically radiating region; high spatial-frequency beams are evanescent (non-radiating) and are pruned by checking if $$(l'/(\alpha_1 N_1 O_1))^2 + (m'/(\alpha_2 N_2 O_2))^2 > 1$$. Simulations confirm that throughput is unchanged upon pruning, while codebook size, signaling, and computational overhead are cut by 15–25% (Yang et al., 2024).

5. Hardware, Architectural, and Implementation Aspects

Multiplier-Free and Mixed-Signal Architectures

Practical array implementations exploit the DFT’s factorability, allowing both low-SWaP digital beamformer cores (32$\times$32 ADFT for 1024 beams (Madanayake et al., 2022, Madanayake et al., 2022)) and current-mode analog approximations (8-point, $N$-point at >4 GHz (Ariyarathna et al., 2015)). Integer-aperture designs (entries in $\{0,\pm1,\pm2\}$ only) provide direct current-mirror or adder-friendly mapping, supporting high bandwidth and spatial fidelity, with small mean-squared error in pattern compared to exact DFT.

Hybrid Analog-Digital Beamspace, Toeplitz Reconstruction, and OFDM/IDFT

Hybrid analog-digital arrays select contiguous DFT beams for analog RF front-ends, then reconstruct covariance or subspace structure via digital Toeplitz projection and non-negative least squares, preserving array aperture for fine-grained ESPRIT parameter estimation (Şenyuva, 30 Nov 2025). In wideband or OFDM scenarios, spatial IDFTs are used to restore squint-free steering, eliminating inter-carrier interference and maintaining constant EVM across all subcarriers (Beshary et al., 2023).

Group-Sparsity, Phased Networks, and Power/RF-Chain Reduction

Coupling DFT beam selection with low-resolution digital phase-shifter networks achieves spatial compression efficiencies up to 25–40%, reducing the number of digital/RF chains below what beam-selection alone allows, without significant SINR or throughput loss (Jiang et al., 2017). Branch-and-bound and greedy algorithms design optimal or near-optimal phase networks atop the DFT domain.

6. Limitations, Assumptions, and Opportunities

DFT beamspace processing presumes regular array geometries (ULA/URA) and critical spacing; off-grid or non-uniform arrays may require generalized Gohberg-Semencul operators or eigenbeam approaches. In the near-field, the DFT codebook’s range-insensitivity may require hybrid or off-grid refinement, as in hybrid model/deep learning or polar-domain sweeps (Heo et al., 26 Feb 2025, Zhou et al., 2024). Multipath, coherent local scattering, and strong NLoS may degrade the tight DFT energy localization; robust covariance estimation and Toeplitz completion techniques partly compensate.

Future research targets the integration of DFT beamspace with hardware-augmented learning, real-time covariance-fitting and beam-selection algorithms, scalable squint correction in wideband arrays, and ever more physically accurate sector pruning and spatial codebook optimization. The DFT’s role as the backbone of beamspace processing remains fundamental, enabling scalable, high-performance, and hardware-efficient implementations across modern radar, communication, and sensing systems.

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References:

• "Transmit Beamspace DDMA Based Automotive MIMO Radar" (Xu et al., 2021)
• "Low-Complexity Near-Field Beam Training with DFT Codebook based on Beam Pattern Analysis" (Wang et al., 27 Mar 2025)
• "Sparse Array DFT Beamformers for Wideband Sources" (Hamza et al., 2019)
• "Optimality of the Discrete Fourier Transform for Beamspace Massive MU-MIMO Communication" (Taner et al., 2021)
• "Multi-beam 4 GHz Microwave Apertures Using Current-Mode DFT Approximation on 65 nm CMOS" (Ariyarathna et al., 2015)
• "Scaling Wideband Massive MIMO Radar via Beamspace Dimension Reduction" (Noroozi et al., 15 Aug 2025)
• "Study of Robust Adaptive Beamforming Based on Low-Complexity DFT Spatial Sampling" (Mohammadzadeh et al., 2021)
• "Joint Linear Precoding and DFT Beamforming Design for Massive MIMO Satellite Communication" (Ha et al., 2022)
• "Fast Radix-32 Approximate DFTs for 1024-Beam Digital RF Beamforming" (Madanayake et al., 2022)
• "Beamspace Multidimensional ESPRIT Approaches for Simultaneous Localization and Communications" (Jiang et al., 2021)
• "Revealing the evanescent components in Kronecker-product based codebooks: insights and implications" (Yang et al., 2024)
• "Optimal Discrete Spatial Compression for Beamspace Massive MIMO Signals" (Jiang et al., 2017)
• "DFT-based Near-field Beam Alignment: Model-based and Data-Driven Hybrid Approach" (Heo et al., 26 Feb 2025)
• "Spatial IDFT for Squint-Free Massive Arrays" (Beshary et al., 2023)
• "Near-field Beam Training with Sparse DFT Codebook" (Zhou et al., 2024)