Tiled Windowed-Beamspace MVDR

• The tiled windowed-beamspace MVDR framework decomposes full-array processing into per-tile 2D FFT and localized windowing, significantly reducing computational complexity and memory usage.
• It aggregates windowed outputs from each tile into a global beamspace MVDR process that retains full-aperture performance while lowering training demands.
• The approach balances resolution and interference suppression with adjustable tile sizes and window parameters, making it ideal for wideband massive MIMO radar applications.

The tiled windowed-beamspace MVDR (Minimum Variance Distortionless Response) framework provides a scalable method for digital beamforming in wideband massive MIMO radar arrays. By leveraging energy concentration in beamspace, it decomposes an otherwise computationally intractable full-array MVDR problem into a distributed pipeline—partitioning the array into tiles, projecting each subarray’s data via 2D spatial FFTs, and processing only a compact, windowed subset of beamspace coefficients. These windowed, per-tile outputs are aggregated and subjected to a reduced-dimension global MVDR process, supporting coherent full-aperture adaptive beamforming with diminished computational, memory, and training demands. The methodology balances system scalability with detection and interference rejection accuracy, enabling deployment on dense arrays where conventional MVDR is infeasible (Noroozi et al., 6 Dec 2025, Noroozi et al., 15 Aug 2025).

1. Array Partitioning and Data Model

Consider a two-dimensional uniform planar array (UPA) with $T_z \times T_x$ tiles, each consisting of $N_z\times N_x$ elements. This yields $T = T_z T_x$ tiles, $N = N_z N_x$ elements per tile, and a total aperture of $TN$ antenna elements. The element steering vector for spatial frequency $\Omega \in \mathbb{R}$ is

$$a_N(\Omega) = [1,\,e^{j\Omega},\,e^{j2\Omega},\,\ldots,\,e^{j(N-1)\Omega}]^\top \in \mathbb{C}^N,$$

and for a source with azimuth $\varphi_k$ and elevation $\theta_k$,

$$\Omega^{ref}_k = \pi\begin{bmatrix} \cos\theta_k\sin\varphi_k\ \sin\theta_k \end{bmatrix},$$

scaling as a function of frequency $f$ via

$\Omega_k(f) = (f/f_d)\,\Omega^{ref}_k.$

The per-tile steering vector at frequency $f$ for tile $t = 1,\ldots, T$ is

$a_k^{(t)}(f) = [\Psi_k(f)]_t \cdot \psi_k(f),$

where $\psi_k(f)$ is the intra-tile steering ($N \times 1$), and $\Psi_k(f)$ controls phase progression across tiles ($T \times 1$) (Noroozi et al., 6 Dec 2025).

The received signal at tile $t$ and snapshot $n$ is

$$y^{(t)}[n] = \sum_{k=1}^K \alpha_k^{(t)}\,a_k^{(t)}\,p_k[n-\tau_k] + I^{(t)}[n] + n^{(t)}[n],$$

with $\alpha_k^{(t)}$ complex gain, $p_k[\cdot]$ the transmit pulse, $I^{(t)}$ interference/clutter, and $n^{(t)}$ white noise.

2. Spatial FFT and Tilewise Beamspace Projection

Each tile applies a 2D spatial DFT. Let $D_{N_x}$ and $D_{N_z}$ denote normalized $N_x$– and $N_z$–point DFT matrices. Define the 2D DFT for each tile as

$V_{\rm tile} = F_x \otimes F_z,$

with $F_x = D_{N_x}^\top, F_z = D_{N_z}$. For a tile's element vector $y^{(t)}$, the beamspace projection is $b^{(t)} = V_{\rm tile}\,y^{(t)} \in \mathbb{C}^N$. Only a localized window of beamspace bins is needed for most signals due to angular energy concentration (Noroozi et al., 6 Dec 2025, Noroozi et al., 15 Aug 2025).

3. Angle-of-Arrival Windowing and Global Concatenation

For each target $k$, a window in beamspace is defined by binary selectors $S_{W_x}^{(k)} \in \{0,1\}^{W_x\times N_x}$ and $S_{W_z}^{(k)} \in \{0,1\}^{W_z\times N_z}$. The composite window operator is

$$S_k = (S_{W_x}^{(k)})^\top \otimes S_{W_z}^{(k)} \in \{0,1\}^{W\times N}, \quad W = W_x W_z.$$

The windowed tile output is

$$\tilde y_k^{(t)}[n] = S_k V_{\rm tile} y^{(t)}[n] = B_k y^{(t)}[n], \; B_k = S_k V_{\rm tile},$$

and tile outputs are concatenated:

$$\widetilde y_k[n] = \{\,\tilde y_k^{(1)}[n],\,\ldots,\,\tilde y_k^{(T)}[n]\,\} \in \mathbb{C}^{T W}.$$

Global projection from full TN to $TW$ dimensions is

$\widetilde y_k[n] = (I_T \otimes B_k) y[n].$

This drastically reduces observation dimensionality while retaining dominant spatial features (Noroozi et al., 6 Dec 2025).

4. Centralized Beamspace MVDR Beamforming

The global reduced-dimensional covariance for target $k$ is

$$R_{bs,k} = \mathbb{E}[\,\widetilde y_k[n]\,\widetilde y_k[n]^H\,] \in \mathbb{C}^{TW \times TW},$$

with empirical estimator

$$\widehat R_{bs,k} = \frac{1}{n_t} \sum_{n=1}^{n_t} \widetilde y_k[n]\,\widetilde y_k[n]^H.$$

The global beamspace steering vector is

$$\widetilde a_k = \{\,\widetilde a_k^{(1)},\,\ldots,\,\widetilde a_k^{(T)}\,\},\;\widetilde a_k^{(t)} = B_k a_k^{(t)}.$$

MVDR weights in beamspace solve

$$\min_c\; c^H \widehat R_{bs,k} c \; \text{ s.t. } c^H \widetilde a_k = 1,$$

with closed-form solution

$$\widetilde c_k = \frac{\widehat R_{bs,k}^{-1} \widetilde a_k}{\widetilde a_k^H \widehat R_{bs,k}^{-1} \widetilde a_k}.$$

To inspect the synthesized array pattern or for postprocessing, these weights can be lifted to the full aperture:

$$\hat c_k = (I_T \otimes B_k^H) \,\widetilde c_k \in \mathbb{C}^{T N} [2512.06536].$$

5. Complexity, Scalability, and Trade-offs

Full-array MVDR requires $O((TN)^3)$ matrix inversion and $O((TN)^2 n_t)$ covariance estimation per subband. The tiled windowed-beamspace approach yields:

• Per-tile 2D DFT: $T \times O(N\log N)$
• Covariance estimation: $O((T W)^2 n_t)$
• Matrix inversion: $O((T W)^3)$ (with $T W \ll T N$)
• Memory: Only $(T W)\times (T W)$ covariance storage
• Training: $O(T W)$ snapshots sufficient for stable estimation

System parameters govern scalability and performance:

• Tile size $(N_z, N_x)$: Determines local FFT granularity and maximal beamspace resolution.
• Window width $(W_z, W_x)$:
  • Larger window: Higher resolution, better interferer nulling, greater complexity
  • Smaller window: Lower cost, risk of mainlobe distortion or insufficient degrees of freedom

Because signal energy is concentrated in a few beamspace bins, the global beamspace dimension $TW$ grows far slower than antenna count, supporting scalability to large apertures (Noroozi et al., 6 Dec 2025, Noroozi et al., 15 Aug 2025).

6. Empirical Performance and Implementation Architecture

Numerical evaluations on a $16\times64$ array (partitioned into $4\times2$ tiles of $4\times32$ elements) across multiple interference scenarios show:

• Observation dimension $TW = 32$ (e.g., $8$ tiles $\times$ $4$ beams per tile) matched full-aperture MVDR performance ($TN = 1024$) for detection and interference suppression.
• In severe clutter scenarios, tiling produced deeper nulls and narrower mainlobes, increasing detection SINR and reducing missed detections.
• Window sizes as low as $2\times4$ beams suffice for low interference; $4\times8$ for harsh conditions (Noroozi et al., 6 Dec 2025, Noroozi et al., 15 Aug 2025).

Implementation proceeds as:

1. Acquire element-space IQ snapshots;
2. Channelize in frequency (fast-time FFT to $L$ subbands);
3. For each subband:
   • Apply spatial FFT per tile ($V_{\rm tile}$),
   • Window and stack outputs,
   • Estimate global covariance,
   • Compute MVDR weights;
4. Optionally lift weights to the full array.

Block-level and pseudo-code implementations are specified in the source texts.

7. Extensions, Limitations, and Practical Considerations

The framework reveals parameter trade-offs crucial to practical system design. Proper window sizing is required for energy containment and nulling capacity. The reduced training requirement and memory footprint address dominant bottlenecks in conventional MVDR. While tiling and windowing limit degrees of freedom per tile, energy concentration in beamspace ensures that full-aperture performance is attainable with a fraction of the original dimensionality, except in extreme interference regimes where window growth may be required. Applications span wideband radar and potentially wideband MIMO communications, wherever coherent spatial processing at scale is required (Noroozi et al., 6 Dec 2025, Noroozi et al., 15 Aug 2025).