Sparse Array DFT Beamformers

• Sparse Array DFT beamformers are spatial signal processing techniques that use a subset of antennas and DFT transforms to achieve efficient beam steering and spatial filtering in mmWave/THz systems.
• They employ sparsification methods such as ℓ4-norm maximization and penalized optimization to reduce hardware complexity and baseband processing in wideband applications.
• By integrating learned unitary beamspace transforms and hybrid analog-digital architectures, these beamformers deliver near-optimal SINR performance in massive MIMO and sensing systems.

Sparse Array DFT beamformers are a class of spatial signal processing techniques in which a sparse subset of antenna elements is combined with spatial or temporal DFT transforms to achieve efficient beam steering, spatial filtering, or spatial multiplexing, particularly in scenarios characterized by hardware constraints, wideband signals, or sparsity in the physical propagation environment. These beamformers have found prominence in massive MU-MIMO communications, wideband source separation, near-field massive array beam training, and hybrid analog-digital architectures in mmWave and terahertz (THz) systems.

1. Physical and Channel Modeling in Sparse Array DFT Beamforming

Sparse Array DFT beamformers operate within environments where the propagation channel is inherently sparse, often due to the high directivity and multipath sparsity at mmWave/THz frequencies. The canonical model features a narrowband (or wideband) channel with $L$ discrete paths and an underlying uniform linear array (ULA) of $N$ sensors:

$$\mathbf{H} = \sum_{\ell=1}^L g_\ell \mathbf{a}_\text{RX}(\theta_\ell) \mathbf{a}_\text{TX}^H(\theta_\ell)$$

where $g_\ell$ are complex path gains and $\mathbf{a}_{(\cdot)}(\theta)$ denote steering vectors parameterized by angle of arrival/departure. For DFT beamforming, the spatial DFT matrix $F \in \mathbb{C}^{N \times N}$ acts upon ULA data, diagonalizing plane-wave responses and yielding output beams associated with discrete spatial directions. In sparse scenarios ($L \ll N$), DFT beamspace transformation renders the channel or received signal sparse in the beam domain, enabling reduced baseband complexity and facilitating spatial filtering (Taner et al., 2021).

In the wideband context, the array output is processed as parallel narrowband signals via the time-frequency DFT, with each frequency bin featuring its own spatial response and steering vector. This parallelization allows concurrent design of spatial and temporal beamforming weights, supporting selective sensor activation for array sparsification (Hamza et al., 2019, Hamza et al., 2020).

2. DFT Beamspace Transform and Sparsification

The spatial DFT matrix $F$ for a ULA is defined by:

$$F_{n, k} = \frac{1}{\sqrt{N}} \exp\left( -j \frac{2\pi}{N} n k \right)$$

where the $k$th column is the steering vector towards angle $\phi_k$ with $\sin\phi_k = 2k/N - 1$. Application of the DFT transform ($F^H$) to antenna data produces $N$ orthogonal beams, each sampling the angular spectrum at regular intervals.

In massive MU-MIMO systems utilizing beamspace processing, the transformed data is dominated by $L$ beams carrying significant energy, corresponding to the $L$ dominant paths (Taner et al., 2021). The DFT's efficacy in sparsifying signal representations motivates its use in hardware-constrained systems with limited RF chains or for baseband complexity reduction.

To measure sparsification, the $\ell_4$-norm objective is employed:

$$\| F^H \mathbf{H} \|_4^4 = \sum_{i,k} \left| (F^H \mathbf{H})_{i,k} \right|^4$$

Maximizing the $\ell_4$-norm promotes localization of signal energy into a minimal number of coefficients/beams, thus optimizing sparsity in beamspace (Taner et al., 2021).

3. Sparse Array Selection and Penalized Optimization (Wideband and Sensor Selection)

When hardware or power limitations require sensor count reduction, sparse array DFT beamformers select a subset of $P \ll N$ sensors while optimizing spatial beamforming. For the wideband case, the array design problem is structured as a quadratic-constrained quadratic program (QCQP):

$$\min_{\{\mathbf{w}_l\}} \sum_l \mathbf{w}_l^H \mathbf{R}_{i,l} \mathbf{w}_l \quad \text{s.t.} \quad \sum_l \mathbf{w}_l^H \mathbf{R}_{s,l} \mathbf{w}_l = 1$$

To induce sensor sparsity, weighted mixed $\ell_{1-\infty}$-norm penalization is applied on the beamformer weights:

$$\lambda \sum_{k=1}^N \|\mathbf{w}_k\|_\infty^2 = \lambda \sum_{k=1}^N \max_l |w_{k,l}|^2$$

This formulation encourages most sensors to have zero weights across all frequencies, yielding $P$ active sensors (Hamza et al., 2019, Hamza et al., 2020).

Semidefinite relaxation (SDR) lifts the problem to a convex domain using $W_l = \mathbf{w}_l \mathbf{w}_l^H \succeq 0$, drops the rank constraint, and introduces auxiliary reweighting matrices to iteratively drive the solution toward exact sparsity. Principal eigenvector extraction of each $W_l$ recovers the optimal weights once the sparsity pattern stabilizes.

Successive convex approximation (SCA) approaches further reduce computational complexity by linearizing the objective around prior iterates, iteratively updating sparsity-inducing weights $u_k$ and optimizing a convex subproblem per iteration (Hamza et al., 2020). Pseudocode for both SDR and SCA approaches reflects iterative updating, sensor counting, and thresholding for exact $P$-sensor selection.

Matrix completion via block Toeplitz constraints addresses missing autocorrelation lags for partial aperture arrays. Low-rank PSD Toeplitz completion and indefinite Toeplitz completion techniques reconstruct the full aperture covariance for reliable beamformer design (Hamza et al., 2020).

4. Algorithms for Unitary Beamspace Transform Learning and Covariance-Guided Beam Selection

In scenarios where non-ideal channel conditions or hardware impairments distort DFT sparsity, learned unitary beamspace transforms can outperform fixed DFT matrices. Two principal algorithms are:

• Matching–Stretching–Projection (MSP):
  • Compute gradient $G_t$ of $\mathbb{E}_\Omega \|A_t y(\Omega)\|_4^4$
  • SVD of gradient: $G_t = U\Sigma V^H$
  • Update: $A_{t+1} = U V^H$
• Coordinate Ascent (CA):
  • Parametrize $A \in U(N)$ by Givens rotations and phase rotations
  • Loop over index pairs $(i, k)$, optimize rotation angles to maximize objective
  • Sequentially update unitary $A$

Both algorithms optimize the $\ell_4$-norm objective for maximal sparsification over the unitary group (Taner et al., 2021).

In hybrid analog-digital mmWave MIMO, covariance-guided DFT beam selection reconstructs a denoised full-aperture covariance from the subarray, scores candidate blocks of contiguous DFT beams by their covariance-capture ability (with penalties for poor conditioning), and selects the optimal subset according to explicit beam-budget constraints. Beamspace Unitary ESPRIT then operates on the selected sparse beams for direction-of-arrival estimation (Şenyuva, 30 Nov 2025).

5. Near-Field Sparse DFT Codebook Construction and Training Protocols

In XL-ULA systems operating in the Fresnel (near-field) region, classical far-field DFT codebooks lose angular selectivity due to spherical wavefront propagation. The sparse DFT codebook method activates every $U$th antenna (yielding $\approx Q$ active elements), collapses the DFT codewords into a lower-dimensional codebook, and exploits angular periodicity in the near-field beam pattern:

$$\mathbf{w}_n = \frac{1}{\sqrt{Q}} [1, e^{-j\pi U \psi_n}, \dots, e^{-j\pi (Q-1) U \psi_n}]^T$$

A three-phase training protocol is deployed:

1. Sparse-DFT angular sweep: Sweep $Q$ beams to identify candidate angular sector via periodicity.
2. Subarray ambiguity elimination: Use a central $M$-element subarray to discriminate among $U$ ambiguous angles.
3. Polar-domain range search: Fix angle and sweep $V$ range codewords for fine user localization.

This protocol reduces training overhead from $O(NV)$ to $O(\sqrt{N} + V)$ and preserves high-SNR rate performance (Zhou et al., 2024).

6. Hardware Realizations and Computational Complexity

Sparse array DFT beamformers map naturally onto hardware-efficient architectures. The Vandermonde (DFT) matrix admits a radix-2 sparse factorization analogous to the FFT:

$$V_N(\tau) = P_N^T \cdot \begin{pmatrix} V_{N/2}(\tau) & 0 \ 0 & V_{N/2}(\tau) \end{pmatrix} \cdot \text{(twiddle delays + butterflies)}$$

This recursive decomposition reduces circuit complexity for true-time-delay (TTD) wideband beamforming from $O(N^2)$ to $O(N \log N)$ delay–amplifier blocks, with stable error bounds even in analog implementations (Perera et al., 2022). Sparse signal-flow graphs with regular fan-out and low nonzero count per stage enable efficient analog beamformer ICs essential for mmWave and THz front-end design.

In contrast, delay-line approaches for wideband sparse beamforming incur $O(N^6 L^6)$ complexity per SDP iteration, while the DFT domain approach reduces this by a factor of $L^3$ (Hamza et al., 2019). SDR–DFT complexity is $O(N^{4.5}\log(1/\epsilon))$, and SCA–DFT achieves $O(N^3\log(1/\epsilon))$, suitable for modestly large apertures (Hamza et al., 2020).

7. Performance, Design Guidelines, and Practical Impact

Sparse-array DFT beamformers, in both simulation and practice, achieve near-optimal SINR and spatial selectivity with drastically reduced hardware, computational, and training overhead:

• In wideband interference scenarios, DFT-SDR and DFT-SCA methods achieve within $0.3$ dB of exhaustive search SINR, reliably finding optimal $P$-sensor solutions (Hamza et al., 2019, Hamza et al., 2020).
• For massive MIMO LoS channels, the DFT beamspace is provably locally optimal in the $\ell_4$ sparsity sense; learned transforms yield negligible gain in such ideal settings (Taner et al., 2021).
• In non-ideal environments (NLOS, hardware impairments, failures), learning a tailored beamspace transform can yield up to $4\times$ $\ell_4$-norm improvement and $\sim$5 dB SNR BER improvement (Taner et al., 2021).
• Covariance-guided DFT beam selection followed by sparse beamspace ESPRIT achieves near-Cramér–Rao bound accuracy for direction-of-arrival estimation, reducing outlier rate and runtime under tight beam and RF chain budgets (Şenyuva, 30 Nov 2025).
• Near-field sparse DFT codebooks provide $98.67\%$ reduction in training beams over exhaustive search while retaining high-SNR rate (Zhou et al., 2024).

Key practical guidelines:

• Evaluate DFT beamspace sparsity ($\ell_4$-norm) on representative channel data; use DFT if improvement from learning is modest ($<10\%$).
• For wideband and hardware-constrained systems, prioritize group-sparsity penalization in optimization, and employ matrix completion techniques where partial aperture data is available.
• For near-field XL arrays, leverage sparse DFT codebooks and multi-phase training for overhead reduction and ambiguity resolution.
• FFT-like sparse-factorization of delay Vandermonde matrices should be used for chip-efficient wideband multi-beam analog beamforming.

These methods collectively enable scalable, high-performance spatial processing across massive MIMO, wideband, and near-field array modalities, addressing stringent design constraints in next-generation wireless and sensing systems.