Beamspace Data Detection Algorithms

• Beamspace data detection algorithms exploit the sparse DFT-transformed mmWave channel structure to reduce baseband processing and improve detection efficiency.
• CSPADE and its threshold-based variants use adaptive filtering and skip logic to achieve significant power savings (up to 66%) with minimal SNR and BER degradation.
• Hardware implementations in both adder-tree and MAC architectures demonstrate near-optimal performance and scalable, power-efficient solutions in massive-MIMO systems.

Beamspace data detection algorithms exploit the sparsity inherent in mmWave and massive MIMO wireless channels when represented in the beamspace (DFT) domain. These techniques, notably the Complex Sparsity-Adaptive Equalizer (CSPADE) and its threshold-based derivatives, are designed to reduce baseband processing power and complexity while maintaining nearly optimal bit-error-rate (BER) and signal-to-noise-ratio (SNR) performance. CSPADE and related frameworks utilize data-dependent skip logic and fractional-norm–adaptive filtering to selectively mute low-importance operations, achieving substantial silicon power savings in hardware implementations without significant loss to detection reliability (Mirfarshbafan et al., 13 Nov 2025, Mirfarshbafan et al., 2024, Abrar, 2017).

1. Beamspace Channel Model and Data Detection Paradigm

The antenna-domain uplink baseband model for a massive-MIMO system with $B$ base station antennas and $U$ user equipment (UEs) is

$$\bar{\mathbf{y}} = \bar{\mathbf{H}}\mathbf{s} + \bar{\mathbf{n}}$$

where $\bar{\mathbf{H}}$ describes channel propagation, $\mathbf{s}$ is the vector of transmitted symbols, and $\bar{\mathbf{n}}$ is Gaussian noise. By applying a unitary DFT matrix $\mathbf{F}$ to $\bar{\mathbf{y}}$, the received vector transforms into the beamspace domain as $\mathbf{y} = \mathbf{F}\bar{\mathbf{y}}$, with an analogous transformation for $\bar{\mathbf{H}}$ and $\bar{\mathbf{n}}$.

This DFT-based transformation leverages the physical propagation structure of mmWave channels, which are well-modeled by a sum of planar waves, concentrating meaningful channel coefficients into a sparse subset of beams. Consequently, both the channel matrix $\mathbf{H} = \mathbf{F}\bar{\mathbf{H}}$ and the processed received vector $\mathbf{y}$ become approximately sparse, underpinning the operational advantage of beamspace detection (Mirfarshbafan et al., 2024, Mirfarshbafan et al., 13 Nov 2025).

Linear minimum mean-square error (LMMSE) equalization in the beamspace domain is given by

$\mathbf{W} = (\mathbf{H}^\Herm \mathbf{H} + \rho\mathbf{I})^{-1}\mathbf{H}^\Herm, \quad \hat{\mathbf{s}} = \mathbf{W}\mathbf{y}$

where $\rho = N_0 / E_s$ controls the regularization according to SNR. The sparsity of $\mathbf{H}$ and, consequently, $\mathbf{W}$, makes selective data detection algorithms in beamspace tractable and advantageous.

2. Fractional-Norm-Based Blind Equalization and Sparse Channel Estimation

CSPADE originated in blind sparse-channel equalization domains, using a fractional-norm penalty alongside the constant modulus (CM) cost function. The objective function is

$J(\mathbf{w}) = E[(|\mathbf{w}^\Herm\mathbf{x}(n)|^2 - R)^2] + \lambda \|\mathbf{w}\|_p^p$

where $\mathbf{x}(n)$ is the received vector, $R$ is the target modulus, and $\|\mathbf{w}\|_p^p = \sum_{i=1}^N |w_i|^p$ for $0 < p < 1$. The sparsity parameter $p$ controls the degree of zero-attraction (as $p\to 0^+$, the penalty approximates the $\ell_0$-norm), and $\lambda$ balances the trade-off between sparsity and tracking error (Abrar, 2017).

The iterative update is

$$\mathbf{w}_{k+1} = \mathbf{w}_k - \mu [\mathbf{g}_k - \gamma_k \mathbf{b}_k]$$

with $\mathbf{g}_k$ denoting the CM gradient, $\mathbf{b}_k$ the fractional-norm gradient, and $\gamma_k$ a data-adaptive scalar determined by the Hermitian angle between $\mathbf{b}_k$ and $\mathbf{g}_k$. The geodesic interpretation ensures the step stays on the constant-$\ell_p$ norm manifold, yielding mean-square stability and reliable convergence (Abrar, 2017).

3. Threshold-Based Sparsity-Adaptive Equalization (SPADE and CSPADE)

Recent CSPADE algorithms for beamspace MIMO data detection define two real thresholds: $T_h$ for filter coefficients and $T_x$ for input samples. The multiplication $W_{u,b} y_b$ is skipped (muted) if both operands are below their respective thresholds. Formally,

$$\delta_{u,b} = \mathbf{1}\left\{ |W_{u,b}| \ge T_h \vee |y_b| \ge T_x \right\}, \quad \hat{s}_u = \sum_{b=1}^B \delta_{u,b} W_{u,b} y_b$$

Row-wise scaling is incorporated so that all equalizer rows use a uniform threshold by normalizing the dynamic range. Parameters are chosen offline to optimize the BER-vs-power trade-off (Mirfarshbafan et al., 2024, Mirfarshbafan et al., 13 Nov 2025).

This skip logic leads to substantial reductions in executed operations, quantified by the multiplier activity rate $\alpha$ (fraction of multiplies performed). In typical LoS mmWave scenarios ($B=64, U=8, 16$-QAM), CSPADE achieves $\alpha \approx 0.21$ with less than 0.5 dB SNR loss at BER $10^{-3}$ (Mirfarshbafan et al., 13 Nov 2025).

4. Hardware Architecture and VLSI Implementations

CSPADE is realized in both adder-tree (fully parallel) and multiply-accumulate (MAC, sequential) VLSI architectures. The key component is the mute-capable complex multiplier (MCM), which implements skip logic by clock-gating input registers based on thresholds applied to the $\ell_{\tilde\infty}$-norm (maximum absolute real or imaginary part).

• Adder-tree (AT) architecture: $U \times B$ parallel MCMs per matrix-vector multiply, summed in a pipelined tree structure for high throughput.
• MAC-based architecture: $U$ parallel CMAC units operating sequentially over $B$ cycles, where both the multiplier and the adder can be muted, leading to maximal power savings.

Measured metrics in a 22 nm FDSOI process (GlobalFoundries 22FDX) are as follows:

Design	Area	Clock	Throughput	Power	Energy eff.	Power Saving
AT-ALMMSE	0.40 mm²	1 GHz	32 Gb/s	500 mW	15.6 pJ/b	–
AT-CSPADE (BS)	0.50 mm²	1 GHz	32 Gb/s	229 mW	7.2 pJ/b	54 %
MAC-ALMMSE (8 MACs)	0.006 mm²	1 GHz	0.5 Gb/s	18 mW	36 pJ/b	–
MAC-CSPADE (8 MACs)	0.008 mm²	1 GHz	0.5 Gb/s	6.2 mW	12.4 pJ/b	66 %

These hardware results confirm that CSPADE can provide up to 54% power savings in fully-parallel architectures and up to 66% in sequential designs while incurring less than 0.6 dB SNR loss at typical operating points (Mirfarshbafan et al., 13 Nov 2025).

5. Performance Trade-offs and Power Savings

Threshold selection is critical. Offline Monte Carlo sweeps identify Pareto-optimal $(T_h, T_x)$ pairs. The main trade-off is between the BER/SNR loss and dynamic power savings via multiplier muting. For realistic LoS/non-LoS channels, SPADE and CSPADE achieve BER degradation below 0.7 dB at reasonable power savings, and these numbers are confirmed across multiple implementations (Mirfarshbafan et al., 2024).

The error induced by skipping (inner-product error) is

$e_u = \sum_{b=1}^B (1-\delta_{u,b}) W_{u,b} y_b$

which increases the post-equalization error variance. The resulting SNR loss can be bounded and characterized analytically (Mirfarshbafan et al., 2024). Empirical results demonstrate robust performance even at aggressive skipping thresholds.

6. Extensions: Explicit Fractional-Norm Regularization and Blind Equalization

Second-stage explicit regularization (e.g., $\ell_{1/2}$) may be used to further prune small filter taps after the main update, via closed-form minimization

$$\min_{h_i} \; |h_i - w_{i,k}|^2 + \lambda_R |h_i|^{1/2}$$

with non-trivial analytical solutions controlling the bias and zero-attraction for near-zero taps (Abrar, 2017).

The principles of CSPADE, especially Hermitian-angle geodesic steps and adaptive regularization, can be applied to both blind sparse ISI equalization and to coherent beamspace MIMO detection.

7. Future Opportunities and Scalability Considerations

Beamspace data detection algorithms such as CSPADE remain effective as system dimensionality grows. Sparsity increases with array size in mmWave, enhancing the operational efficacy of skip logic. The scalability of hardware is linear with respect to $B$ and $U$, but threshold broadcast logic does not increase in complexity. Additional support for higher-order modulations is possible with higher word-length multipliers and modest adjustments to thresholds. Channel dynamics and user mobility may benefit from per-coherence-block or adaptive threshold selection. These findings suggest beamspace adaptive equalization will remain central to energy-efficient massive-MIMO systems as array and bandwidth scaling continues (Mirfarshbafan et al., 2024, Mirfarshbafan et al., 13 Nov 2025).