Reduced Dimension Beamspace LMMSE

• The paper introduces a LMMSE receiver architecture that exploits spatial DFT-based beamspace transformation and user-specific windowing to capture dominant channel paths.
• It reduces computational complexity from O(M³) to O(UW³) and minimizes pilot overhead by operating on a smaller number of DFT bins per user.
• Performance analysis demonstrates near-optimal spectral efficiency under channel sparsity and angular separability, validated via deterministic-equivalent frameworks and simulation benchmarks.

Per-subcarrier reduced dimension beamspace linear minimum mean-squared error (LMMSE) processing refers to a class of receiver architectures for multiuser (MU) massive MIMO-OFDM systems that combine spatial Fourier transforms (DFT/FFT) with user-specific dimensionality reduction and LMMSE filtering on each OFDM subcarrier. These methods leverage channel sparsity and user angular separation to reduce computational complexity and training overhead, while retaining near-optimal performance. Two principal frameworks have been developed: the statistical two-stage beamformer design using deterministic equivalents (Asgharimoghaddam et al., 2019), and geometric windowed-beamspace reduction strategies validated by information-theoretic benchmarks (Cebeci et al., 6 Dec 2025).

1. System Model and Beamspace Transformation

Consider a MU-MIMO-OFDM uplink with $M$ BS antennas, $U$ single-antenna users, and $K$ OFDM subcarriers. For each subcarrier $\ell$, the baseband input-output relationship is

$$\mathbf{y}_\ell = \mathbf{H}_\ell \mathbf{x}_\ell + \mathbf{n}_\ell,$$

where $\mathbf{y}_\ell \in \mathbb{C}^{M}$ is the received signal, $\mathbf{x}_\ell \in \mathbb{C}^{U}$ is the vector of user symbols (assume uncorrelated symbols, $$\mathbb{E}[\mathbf{x}_\ell \mathbf{x}_\ell^H] = \sigma_x^2 \mathbf{I}_U$$), $\mathbf{H}_\ell \in \mathbb{C}^{M \times U}$ is the frequency-domain channel, and $\mathbf{n}_\ell$ is $\mathcal{C}\mathcal{N}(0, \sigma_n^2 \mathbf{I}_M)$ noise.

A spatial DFT (unitary FFT) matrix $\mathbf{F} \in \mathbb{C}^{M \times M}$ with entries $$[\mathbf{F}]_{m,n} = M^{-1/2}\exp\left(-j2\pi(m-1)(n-1)/M\right)$$ transforms antenna space to the beamspace domain. The beamspace observation on subcarrier $\ell$ is

$$\mathbf{z}_\ell = \mathbf{F}^H \mathbf{y}_\ell \in \mathbb{C}^{M},$$

where energy from each user is concentrated in a small number of angular (DFT) bins.

2. Dimensionality Reduction via Per-User Beamspace Windows

Exploiting the angular sparsity and spatial separation of users, for each user $k$, a contiguous window $\mathcal{W}_k$ of $W$ DFT bins is selected to contain the dominant energy for that user. Define user-specific selection matrices $\mathbf{S}_k \in \{0,1\}^{M\times W}$, formed from the canonical basis vectors indexed by $\mathcal{W}_k$. The reduced-dimension received vector for user $k$ is:

$$\overline{\mathbf{y}}_{\ell,k} = \mathbf{S}_k^T \mathbf{z}_\ell = \mathbf{S}_k^T \mathbf{F}^H \mathbf{y}_\ell \in \mathbb{C}^W.$$

The corresponding reduced-dimension channel (beamspace projection) is $$\overline{\mathbf{h}}_{\ell,k} = \mathbf{S}_k^T \mathbf{F}^H \mathbf{h}_{k,\ell}$$, where $\mathbf{h}_{k,\ell}$ is the column of $\mathbf{H}_\ell$ for user $k$.

In the two-stage beamforming setting (Asgharimoghaddam et al., 2019), the DFT basis is partitioned into $S$ angular sectors each containing $D = N/S$ beams. The projector for sector $s$ is $\mathbf{P}_s = \mathbf{U}_{(s)}\mathbf{U}_{(s)}^H$, derived from columns $(s-1)D+1$ through $sD$ of the DFT matrix $\mathbf{U}$.

3. Reduced-Dimension LMMSE Filter Derivation

The per-subcarrier, reduced-dimension LMMSE filter for user $k$ on subcarrier $\ell$ is computed as follows. The projected observation is

$$\overline{\mathbf{y}}_{\ell} = \overline{\mathbf{H}}_{\ell} \mathbf{x}_\ell + \overline{\mathbf{n}}_\ell,$$

where $\overline{\mathbf{H}}_\ell$ is the stack of all users' $W$-bin windows (block-diagonal or concatenated by user), and $$\overline{\mathbf{n}}_\ell \sim \mathcal{CN}(0, \sigma_n^2\mathbf{I}_{UW})$$.

The LMMSE estimator for $\mathbf{x}_\ell$ is

$$\widehat{\mathbf{x}}_\ell = \mathbf{G}_\ell \overline{\mathbf{y}}_\ell, \qquad \mathbf{G}_\ell = \sigma_x^2 \overline{\mathbf{H}}_\ell^H \left( \overline{\mathbf{H}}_\ell \sigma_x^2 \overline{\mathbf{H}}_\ell^H + \sigma_n^2 \mathbf{I}_{UW} \right)^{-1}.$$

Per-user, for user $k$,

$$\mathbf{g}_{\ell,k} = \sigma_x^2 \left(\overline{\mathbf{H}}_\ell \sigma_x^2 \overline{\mathbf{H}}_\ell^H + \sigma_n^2 \mathbf{I}_W\right)^{-1} \overline{\mathbf{h}}_{\ell,k}.$$

This filter suppresses interference from other users whose leakage into user $k$'s window is typically low-rank due to angular concentration (Cebeci et al., 6 Dec 2025).

In the deterministic-equivalent framework, the outer beamformer is designed by computing sector-wise projections $\bar a_{k,s}^{(f)}$ using fixed point equations for the channel statistics, followed by thresholding or water-filling sector selection, and then constructing the reduced-dimension effective channel for the LMMSE inner stage (Asgharimoghaddam et al., 2019).

4. Beamspace Window Selection and Dimensioning Strategies

Selection of the beamspace window (or sector) size per user is critical to balancing complexity and performance:

• Fixed Thresholding: Sectors/windows with amplitude projections above a fraction $\delta$ of the peak per user per subcarrier are retained (Asgharimoghaddam et al., 2019).
• Water-Filling / SINR Targeting: Find the minimal set of sectors/windows whose weighted contribution satisfies a target SINR (rate-constrained allocation). The optimal allocation is given by

$$D_{k,s}^{(f)} = \begin{cases} 0, & \bar a_{k,s}^{(f)} \leq 1/\lambda^*\ D, & \bar a_{k,s}^{(f)} \geq 1/\lambda^*\ \text{intermediate}, & \text{otherwise} \end{cases}$$

where $\lambda^*$ is determined by the SINR constraint (Asgharimoghaddam et al., 2019).

Empirically, a window size $W=4$–$5$ captures over $90\%$ of a dominant path’s energy for any $M$ (Cebeci et al., 6 Dec 2025). Guard intervals of several DFT bins between users ensure residual interference in the window is low-rank, enabling near-optimal suppression.

5. SINR and Information-Theoretic Performance

The post-LMMSE SINR for user $k$ on subcarrier $\ell$ with reduced-dimension beamspace processing is

$$\mathrm{SINR}_{\ell,k} = \frac{\sigma_x^2}{\sigma_n^2} \overline{\mathbf{h}}_{\ell,k}^H \left(\overline{\mathbf{H}}_\ell \sigma_x^2 \overline{\mathbf{H}}_\ell^H + \sigma_n^2\mathbf{I}_W \right)^{-1} \overline{\mathbf{h}}_{\ell,k}.$$

A lower bound on the achievable sum-rate per subcarrier is

$$R_\mathrm{red}(\ell) = \sum_{k=1}^U \log_2(1 + \mathrm{SINR}_{\ell,k}),$$

and spectral efficiency is $$C_\mathrm{red} = \frac{1}{K} \sum_{\ell=0}^{K-1} R_\mathrm{red}(\ell)$$. Performance benchmarks are given by the unconstrained (full-rank MMSE) and full-dimension LMMSE, with simulations showing that for moderate system loading ($U \approx M/2$) and sparse channels, reduced-dimension beamspace LMMSE with $W=5$ yields $$C_\mathrm{red} \approx C_\mathrm{fullLMMSE} \approx C_\mathrm{unconstrained}$$ over an SNR range of $10$–$30$ dB (Cebeci et al., 6 Dec 2025).

6. Complexity and Practical Implementation

Full-dimension LMMSE requires per-subcarrier, per-receiver inversion of $M \times M$ matrices ($O(M^3)$ flops). Beamspace reduction enables inversion of only $W \times W$ matrices per user ($O(U W^3)$), with $W \ll M$. In training/pilot overhead, only $W$ pilots per user per subcarrier are needed to estimate the reduced channel, versus $M$ in full dimension (Cebeci et al., 6 Dec 2025). In the two-stage framework, only a $K \times K$ inversion is required for the deterministic-equivalent update, amortizable over channel statistic updates (Asgharimoghaddam et al., 2019).

For practical MU-MIMO-OFDM with $M \gg 1$ and large $K$, this approach provides a scaling reduction of computational and training cost from $O(M^3)$ and $O(M)$, respectively, to $O(U W^3)$ and $O(W)$ per subcarrier, under the conditions of beamspace sparsity and suitable user scheduling.

7. Open Questions and Future Directions

Several open challenges remain:

• Adaptive selection of window/sector size $W$ per user to optimize the tradeoff between performance and cost.
• Joint scheduling of multiple user paths to minimize window overlap and subsequent interference.
• Extension of this framework to sub-6 GHz systems, where angular sparsity is typically weaker.
• Hybrid analog-digital implementations leveraging beamspace insights.
• Robustness in scenarios with dense secondary multipath, where unmodeled paths can increase interference leakage (Cebeci et al., 6 Dec 2025).

A plausible implication is that scheduling based on users' lowest-frequency spatial separation offers robust performance across wideband channels. Zeropadding the FFT grid can marginally improve noise-limited energy concentration, but may degrade interference suppression; the unpadded DFT is generally preferred in interference-limited regimes (Cebeci et al., 6 Dec 2025).

Table: Summary of Beamspace LMMSE Complexity

Approach	Matrix Inversion per Subcarrier	Pilot Overhead per User
Full-dimension LMMSE	$O(M^3)$	$O(M)$
Reduced beamspace LMMSE	$O(U W^3)$	$O(W)$

The per-subcarrier reduced dimension beamspace LMMSE paradigm thus offers computationally efficient, information-theoretically validated multiuser decoding for next-generation massive MIMO-OFDM systems, especially in regimes characterized by channel sparsity and user angular separability (Asgharimoghaddam et al., 2019, Cebeci et al., 6 Dec 2025).