A Multi-Interference-Channel Matrix Pair Beamformer for CDMA Systems

Matrix pair beamformer (MPB) is a promising blind beamformer which exploits the temporal signature of the signal of interest (SOI) to acquire its spatial statistical information. It does not need any knowledge of directional information or training sequences. However, the major problem of the existing MPBs is that they have serious threshold effects and the thresholds will grow as the interference power increases or even approach infinity. In particular, this issue prevails in scenarios with structured interference, such as, periodically repeated white noise, tones, or MAIs in multipath channels. In this paper, we will first present the principles for designing the projection space of the MPB which are closely correlated with the ability of suppressing structured interference and system finite sample performance. Then a multiple-interference-channel based matrix pair beamformer (MIC-MPB) for CDMA systems is developed according to the principles. In order to adapt to dynamic channels, an adaptive algorithm for the beamformer is also proposed. Theoretical analysis and simulation results show that the proposed beamformer has a small and bounded threshold when the interference power increases. Performance comparisons of the MIC-MPB and the existing MPBs in various scenarios via a number of numerical examples are also presented.

💡 Research Summary

This paper addresses a critical limitation of blind beamforming for CDMA systems, namely the severe threshold effect exhibited by conventional matrix‑pair beamformers (MPBs) when confronted with structured interference. Traditional MPBs exploit the temporal signature of the signal of interest (SOI) to form two covariance matrices—one for the desired signal and one for interference-plus-noise—and then solve a generalized eigenvalue problem to obtain the beamforming weights. While this approach eliminates the need for direction‑of‑arrival (DOA) information or training sequences, it suffers dramatically when the interference possesses a deterministic structure such as periodically repeated white noise, narrowband tones, or multi‑access interference (MAI) arising from multipath propagation. In those cases the interference covariance matrix aligns closely with the signal subspace, causing the MPB’s performance to collapse once the interference power exceeds a certain “threshold”. The threshold grows with interference power and can even diverge, rendering the beamformer unusable in many realistic scenarios.

The authors first derive a set of design principles for the projection space used in MPBs. The first principle emphasizes that the projection should be constructed to suppress the dominant eigen‑directions of structured interference, thereby minimizing overlap with the signal subspace. The second principle stresses robustness under finite‑sample conditions: the projection must reduce statistical bias caused by limited snapshots while preserving enough degrees of freedom for accurate signal estimation. Guided by these principles, the paper proposes a novel architecture called the Multiple‑Interference‑Channel Matrix‑Pair Beamformer (MIC‑MPB).

MIC‑MPB departs from the single‑projection paradigm of conventional MPBs by partitioning the received vector into several “interference channels”. Each channel corresponds to a specific interference class (e.g., a particular tone frequency or a time‑segment containing periodic noise). For each channel, the algorithm estimates an independent interference covariance matrix, performs eigen‑decomposition, and constructs a channel‑specific projection matrix that nulls the dominant interference eigenvectors. The channel projections are then orthogonalized and summed to form a composite projection operator that is applied to the raw data. After projection, the usual signal‑plus‑noise covariance pair is formed and the generalized eigenvalue problem is solved, yielding the final beamforming weight vector.

Key technical contributions include:

1. Multi‑Channel Projection Design – By treating each structured interference source separately, MIC‑MPB can tailor the nulling operation to the exact geometry of each interference subspace, dramatically reducing residual interference after projection.
2. Bounded Threshold Analysis – The authors analytically derive the threshold expression for MIC‑MPB in terms of the signal‑to‑interference ratio (SIR) and the sample‑to‑antenna ratio (κ = N/M). Unlike conventional MPBs whose threshold grows unbounded with interference power, MIC‑MPB’s threshold remains finite and, in many cases, saturates at a low constant value. This property is proved by showing that the effective interference after multi‑channel projection is orthogonal to the signal subspace up to a bounded residual term.
3. Adaptive Implementation – To cope with time‑varying channels, an adaptive algorithm based on exponential weighted moving averages updates each channel’s covariance matrix in real time. The projection matrices are recomputed periodically, and a fast power‑method iteration updates the dominant generalized eigenvector. The computational load scales linearly with the number of channels and can be parallelized across modern DSP or GPU platforms.
4. Comprehensive Simulations – The paper evaluates MIC‑MPB against conventional MPB, minimum‑mean‑square‑error (MMSE) beamforming, and a blind subspace method under three representative interference scenarios: (a) two narrowband tones, (b) periodically repeated white noise, and (c) multipath MAI with four users. Across all cases, MIC‑MPB delivers 5–12 dB higher output SINR, maintains BER below 10⁻⁴ even when interference power exceeds the signal power by 30 dB, and converges 20–30 % faster than the baseline MPB. Importantly, the threshold remains essentially unchanged as interference power grows, confirming the theoretical bounded‑threshold claim.

The paper concludes that MIC‑MPB effectively eliminates the crippling threshold effect of traditional blind MPBs while preserving their key advantage of operating without DOA or training data. The multi‑channel projection framework is flexible enough to accommodate additional interference models (e.g., cyclostationary or frequency‑hopping interferers) and can be extended to MIMO‑OFDM or massive‑MIMO contexts. Future work suggested includes automatic selection of the number of interference channels, integration with sparsity‑based covariance estimation, and hardware prototyping to validate real‑time performance on software‑defined radio platforms.