[Paper Review] Subspace Tracking Algorithms for Millimeter Wave MIMO Channel Estimation with Hybrid Beamforming

Stefano Buzzi, Carmen D’Andrea|arXiv (Cornell University)|Apr 22, 2017

Millimeter-Wave Propagation and Modeling4 citations

TL;DR

This paper proposes subspace tracking algorithms—PASTd and OOJA—adapted for mmWave MIMO channel estimation in hybrid beamforming systems. By exploiting the clustered mmWave channel model, the algorithms estimate dominant left and right singular vectors at the receiver and transmitter, respectively, achieving superior performance over AML and SE-ARN with short training sequences, even under hybrid analog/digital beamforming constraints.

ABSTRACT

This paper proposes the use of subspace tracking algorithms for performing MIMO channel estimation at millimeter wave (mmWave) frequencies. Using a subspace approach, we develop a protocol enabling the estimation of the right (resp. left) singular vectors at the transmitter (resp. receiver) side; then, we adapt the projection approximation subspace tracking with deflation (PASTd) and the orthogonal Oja (OOJA) algorithms to our framework and obtain two channel estimation algorithms. The hybrid analog/digital nature of the beamformer is also explicitly taken into account at the algorithm design stage. Numerical results show that the proposed estimation algorithms are effective, and that they perform better than two relevant competing alternatives available in the open literature.

Motivation & Objective

• Address the challenge of accurate and efficient MIMO channel estimation in mmWave systems with high path loss and limited spectrum.
• Overcome the limitations of conventional pilot-based training in mmWave MIMO by leveraging low-rank structure of the channel matrix via subspace tracking.
• Design subspace tracking algorithms that are compatible with hybrid analog/digital beamforming architectures to reduce hardware complexity.
• Extend existing algorithms like AML and SE-ARN to multi-antenna mobile stations and compare performance under realistic mmWave conditions.
• Demonstrate the feasibility of pilot-less differential modulation using the estimated channel subspace for low-latency communication.

Proposed method

• Adapt the Projection Approximation Subspace Tracking with Deflation (PASTd) algorithm to estimate the right singular vectors at the transmitter and left singular vectors at the receiver using training signals.
• Apply the Orthogonal Oja (OOJA) algorithm to track the dominant subspace of the channel matrix in a recursive, low-complexity manner.
• Incorporate the hybrid beamforming structure explicitly into the algorithm design, modeling the RF chains as analog beamformers with beam-steering response vectors.
• Use the initial 10 training symbols to compute the sample covariance matrix and perform SVD to initialize PASTd and OOJA algorithms.
• Generalize the Approximate Maximum Likelihood (AML) algorithm to multi-antenna mobile stations and compare it with the proposed schemes.
• Implement a differential 16-PSK modulation scheme using the estimated subspace for M=1 multiplexing order, enabling pilot-less transmission.

Experimental results

Research questions

• RQ1 Can subspace tracking algorithms effectively estimate the dominant singular vectors of mmWave MIMO channels under hybrid beamforming constraints?
• RQ2 How do PASTd and OOJA perform compared to AML and SE-ARN in terms of estimation accuracy and spectral efficiency under realistic mmWave conditions?
• RQ3 What is the impact of training sequence length on the performance of subspace tracking algorithms in mmWave MIMO systems?
• RQ4 Can the estimated subspace be used to enable pilot-less differential modulation with acceptable symbol error rate?
• RQ5 How does the performance of hybrid beamforming implementations compare to fully digital implementations in terms of subspace estimation accuracy?

Key findings

• The proposed PASTd and OOJA-based algorithms outperform AML and SE-ARN in terms of subspace tracking accuracy, as measured by the coherence metrics η_U and η_V.
• At 10 dB SNR, the CDF of η_U and η_V shows that the proposed algorithms achieve higher probability of achieving high subspace alignment than competing methods.
• Even with a short training sequence of 30 symbols (10 for initialization), the algorithms achieve good spectral efficiency, with the M=1 case showing near-optimal performance.
• For M=1, the symbol error probability (SER) of the differential 16-PSK scheme is below 10⁻² at 10 dB SNR when using 50 training symbols, confirming feasibility of pilot-less operation.
• The hybrid beamforming implementations achieve performance close to fully digital (FD) versions, especially when the number of RF chains is sufficiently large.
• The performance gain of the proposed algorithms is most evident in low-SNR and short-training scenarios, highlighting their robustness and efficiency.