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Channel Acquisition for Massive MIMO-OFDM with Adjustable Phase Shift Pilots

Li You, Xiqi Gao, A. Lee Swindlehurst, Wen Zhong

arXiv:1511.03812v1cs.IT

TL;DR

The paper addresses pilot overhead and channel acquisition in wideband massive MIMO-OFDM. It introduces APSPs together with a physically motivated sparse channel model and phase-shift scheduling based on angle-delay separation. The proposed approach shows substantial achievable-spectral-efficiency gains over conventional PSOP-based acquisition in typical mobility scenarios.

  • Problem

    Wideband massive MIMO-OFDM requires effective channel acquisition, while conventional phase-shift orthogonal pilots impose pilot overhead.

  • Method

    The paper develops APSPs, relates space-frequency correlations to angle-delay power spectra, and derives scheduling for channel estimation and prediction.

  • Results

    Significant achievable-spectral-efficiency gains over conventional PSOP-based channel acquisition are demonstrated in several typical mobility scenarios.

  • Takeaways & Limitations

    Non-overlapping user channel power distributions in the angle-delay domain provide a common optimal condition for APSP-based channel estimation and prediction.

Abstract

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We propose adjustable phase shift pilots (APSPs) for channel acquisition in wideband massive multiple-input multiple-output (MIMO) systems employing orthogonal frequency division multiplexing (OFDM) to reduce the pilot overhead. Based on a physically motivated channel model, we first establish a relationship between channel space-frequency correlations and the channel power angle-delay spectrum in the massive antenna array regime, which reveals the channel sparsity in massive MIMO-OFDM. With this channel model, we then investigate channel acquisition, including channel estimation and channel prediction, for massive MIMO-OFDM with APSPs. We show that channel acquisition performance in terms of sum mean square error can be minimized if the user terminals' channel power distributions in the angle-delay domain can be made non-overlapping with proper phase shift scheduling. A simplified pilot phase shift scheduling algorithm is developed based on this optimal channel acquisition condition. The performance of APSPs is investigated for both one symbol and multiple symbol data models. Simulations demonstrate that the proposed APSP approach can provide substantial performance gains in terms of achievable spectral efficiency over the conventional phase shift orthogonal pilot approach in typical mobility scenarios.

I. INTRODUCTION

The paper motivates adjustable phase shift pilots for wideband massive MIMO-OFDM by linking channel sparsity in the angle-delay domain to reduced pilot overhead and improved channel acquisition. Its APSP design schedules phase shifts so users’ channel power distributions can be separated, enabling lower-overhead estimation and prediction while improving spectral efficiency.

  • Motivation: Massive MIMO-OFDM is important for wideband transmission, but its performance depends strongly on channel acquisition and pilot design.Massive MIMO is presented as a candidate technology for high-rate systems, while OFDM supports high-rate wideband transmission.
  • APSP design: APSPs use adjustable frequency-domain phase shifts of one pilot sequence, providing more pilot options and reducing pilot overhead relative to fixed-shift PSOPs.The difference from PSOPs is that APSP phase shifts are adjustable rather than fixed and separated by the maximum channel delay.
  • Channel model: The channel model links space-frequency covariance eigenstructure to angle-delay power spectra, revealing sparse and approximately decorrelatable massive MIMO-OFDM channels.With sufficiently many antennas, eigenvectors tend to be common across users while eigenvalues depend on their angle-delay spectra.
  • Channel acquisition: Proper phase-shift scheduling minimizes both channel-estimation and channel-prediction sum MSE when users’ angle-delay power distributions are non-overlapping.The same non-overlap condition applies to estimation and prediction, and motivates a simplified scheduling algorithm.
  • Results: The proposed APSP channel acquisition approach achieves significant achievable-spectral-efficiency gains over conventional PSOP acquisition, especially in high-mobility scenarios.The evaluation covers typical propagation and mobility scenarios and includes both one-symbol and multiple-symbol operation.

III. CHANNEL ACQUISITION WITH APSPS OVER ONE SYMBOL

The paper applies the sparse massive MIMO-OFDM channel model to APSP-based channel acquisition over one OFDM symbol, covering both channel estimation and prediction.

  • A. APSPs over One Symbol: APSP-based channel acquisition over one OFDM symbol is developed for massive MIMO-OFDM using the preceding sparse channel model.The approach includes both channel estimation and channel prediction; the multiple-symbol case is treated separately.

A. APSPs over One Symbol

APSPs use one basic pilot sequence with adjustable frequency-domain phase shifts for different user terminals. Their phase-shift differences control pilot cross-correlations and can increase pilot availability while reducing storage requirements.

  • Pilot transmission: During the uplink pilot segment, synchronized user terminals transmit scheduled pilots simultaneously, and the base station receives their space-frequency pilot signals.The received signal includes each terminal’s channel, frequency-domain pilot, and additive white Gaussian noise.
  • Pilot construction: An APSP is a phase-shifted version of a shared basic pilot matrix, with the phase shift assigned per user terminal in the frequency domain.The basic matrix is unitary, and all terminals use shifted versions of it.
  • Pilot construction: The proposed APSP preserves the basic pilot’s peak-to-average power ratio, allowing existing low-PAPR sequence designs to be incorporated.This retains implementation flexibility while changing the pilot phase structure.
  • Implementation: Only the basic pilot matrix and phase-shift indices need storage, rather than complete pilot matrices for every terminal.This significantly reduces the required pilot-storage space.
  • Pilot scheduling: APSP cross-correlations depend only on phase-shift differences, and adjustable shifts can even be shared across terminals.Unlike conventional PSOPs, APSPs need not enforce fixed shifts separated by the maximum channel delay, increasing available pilots and reducing pilot overhead.

B. Channel Estimation with APSPs

The channel-estimation procedure transforms received pilot observations into the angle-delay domain, where channel elements are approximately uncorrelated and estimation becomes element-wise. Proper phase-shift scheduling can eliminate pilot interference when shifted user-terminal power distributions do not overlap.

  • Estimation framework: Direct MMSE estimation in the space-frequency domain is difficult because it requires large channel covariance information and matrix inversion.The sparse massive-MIMO-OFDM model motivates estimating the angle-delay channel response first, then transforming it back.
  • Estimation framework: After decorrelation and power normalization, the base station obtains an angle-delay channel observation with normalized AWGN whose variance is determined by the pilot-segment SNR.The normalized noise has independent complex Gaussian entries, and ρ_tr = σ_xtr/σ_ztr.
  • Pilot interference: Pilot interference from another terminal appears as a cyclically shifted, column-truncated version of its extended angle-delay channel response.The shift is determined by the difference between the two terminals’ pilot phase shifts.
  • MMSE estimation: The angle-delay channel estimate and its mean square error can be computed element-wise under the MMSE criterion because the transformed channel elements are statistically uncorrelated.The sum channel-estimation MSE is defined across all user terminals.
  • Optimal scheduling: The sum channel-estimation MSE is minimized when phase shifts make the user terminals’ equivalent angle-delay power distributions non-overlapping.Frequency-domain phase shifts create cyclic delay-domain shifts, so proper scheduling can eliminate pilot interference.
  • Scope and sparsity: APSP performance depends on channel sparsity: with common support spanning s columns, at most floor(N_c/s) terminals can avoid pilot interference, while greater sparsity improves performance.In practical channels, the non-overlap condition may not hold exactly, causing degradation.

C. Channel Prediction with APSPs

Channel prediction uses pilot observations together with the channel temporal correlation function to estimate data-segment channels, addressing the degradation of directly reusing pilot-segment estimates in high mobility. As with estimation, suitable phase-shift scheduling can remove pilot interference from the prediction bound.

  • Motivation: Directly using pilot-segment channel estimates during the data segment can be inappropriate in high-mobility scenarios.The motivation is the temporal evolution of the channel between pilot and data symbols.
  • Motivation: When the channel temporal correlation becomes sufficiently small, direct channel estimation can have error exceeding the total channel power, motivating channel prediction.The cited condition is 1 − 2ϱ_k(Δℓ) > 0, equivalently ϱ_k(Δℓ) < 1/2.
  • Prediction method: The predictor combines received pilot signals with the channel temporal correlation function to estimate the angle-delay channel during the data segment.Under MMSE estimation, the prediction can be obtained element-wise using the massive-MIMO-OFDM channel structure.
  • Prediction method: Optimal data-segment channel estimates can be obtained through prediction from pilot-segment estimates, reducing the complexity of channel prediction.The predicted estimate is related directly to the initial channel estimate obtained during the pilot segment.
  • Optimal scheduling: Pilot interference affects channel prediction similarly to channel estimation, but proper phase-shift scheduling can eliminate its effect.The result applies within the considered TDD frame structures.
  • Optimal scheduling: The sum channel-prediction MSE has a lower bound achieved when shifted user-terminal angle-delay power distributions satisfy the stated non-overlap condition.This extends the interference-elimination principle from channel estimation to prediction.

D. Frame Structure

The paper considers two TDD frame structures: one places the uplink pilot segment before uplink and downlink data, while the other places it between those data segments. APSPs reduce pilot duration but can increase the delay to tail-end data symbols.

  • Frame structures: Type-A frames begin with the uplink pilot segment, followed by uplink and downlink data segments.This is the first of the two TDD frame structures considered.
  • Frame structures: Type-B frames place the uplink pilot segment between the uplink data segment and the downlink data segment.The pilot location changes which data symbols are nearest to the pilot segment.
  • Frame structures: With APSPs, the delay from tail-end data symbols to the pilot segment is longer than with PSOPs because the pilot segment is reduced.This affects the temporal separation relevant to channel acquisition.

E. Pilot Phase Shift Scheduling

The paper extends APSP channel acquisition to multiple OFDM symbols and develops scheduling based on minimizing overlap between users’ angle-delay channel distributions. Multiple-symbol APSPs divide phase shifts into groups, while proper scheduling can achieve optimal estimation and prediction performance when pilot interference is eliminated.

  • Scheduling algorithm: The overlap-based scheduling problem is combinatorial, so exhaustive search gives optimal solutions, while the proposed thresholded algorithm provides a practical alternative.The scheduling objective is to keep the overlap function between users’ angle-delay channel power matrices below a preset threshold.
  • Multiple-symbol APSPs: Multiple-symbol APSPs extend the available pilot phase shifts to QN_c−1 and divide them into Q groups based on phase-shift residues modulo Q.Pilot interference affects only users whose phase shifts belong to the same group.
  • Channel acquisition: The receiver obtains element-wise MMSE channel estimates and delayed channel predictions from the multiple-symbol pilot observations, with corresponding sum MSE expressions.The pilot segment is assumed short enough that channels remain constant across its Q consecutive OFDM symbols.
  • Optimal condition: Both channel-estimation and channel-prediction sum-MSE lower bounds are achieved when users’ phase-shifted angle-delay channel power distributions satisfy the stated non-overlap condition.The same optimal scheduling condition applies to channel estimation and prediction.
  • Relation to single-symbol APSPs: For Q=1, the multiple-symbol proposition reduces to the single-symbol results, linking the extended formulation to the earlier APSP analysis.Pilot interference is confined to users using phase shifts within the same group.
  • Scheduling algorithm: When the optimal condition is unattainable, scheduling can divide users into Q groups and apply the single-group algorithm within each group.A preset overlap threshold balances channel acquisition performance against algorithm complexity.

V. NUMERICAL RESULTS

Simulations evaluate APSP-CA against PSOP-CA under typical propagation and mobility settings, showing reduced pilot overhead with comparable MSE and higher spectral efficiency.

  • Simulation setup: The simulations use a 128-antenna ULA, K = 42 user terminals, 20-tap channels, and suburban, urban macro, and urban micro mobility scenarios.The channel model assumes exponential power delay profiles, Laplacian power angle spectra, and synchronized transmissions.
  • Pilot configurations: PSOP-CA requires Q = 3 pilot symbols, whereas APSP-CA uses Q = 1 or 2 with Algorithm 1 for phase-shift scheduling.The simulations use APSP pilot segments shorter than the conventional PSOP segment.
  • Pilot-segment performance: 66.7% and 33.3% pilot-overhead reductions are achieved with APSPs using Q = 1 and Q = 2, respectively, while pilot-segment MSE-CE approaches PSOP performance.The comparison uses conventional PSOPs with Q = 3 as the benchmark.
  • Data-segment acquisition: With Q = 1, APSP channel-prediction MSE approaches PSOP performance despite 66.7% lower pilot overhead, and prediction outperforms estimation across evaluated scenarios.Both channel acquisition MSE measures grow almost linearly with acquisition delay.
  • Spectral efficiency: APSP-CA provides substantial achievable-spectral-efficiency gains over PSOP-CA, especially under high mobility and high SNR conditions.The comparison includes APSP type-A and type-B frame structures and uses a 500 µs frame with MMSE receivers and precoders.

VI. CONCLUSION

The paper concludes that APSP-based channel acquisition reduces pilot overhead while supporting estimation and prediction, with spectral-efficiency gains over conventional PSOP-based acquisition.

  • Contribution: The proposed APSP approach targets reduced pilot overhead in massive MIMO-OFDM channel acquisition.The paper addresses channel estimation and prediction using adjustable phase-shift pilots.
  • Method: APSP-CA uses an optimal phase-shift scheduling condition and a simplified scheduling algorithm for both channel estimation and prediction.The approach is investigated with one-symbol and multiple-symbol pilot models.
  • Result: APSP-CA achieves significant achievable-spectral-efficiency gains over conventional PSOP-CA in the reported evaluations.The supplied conclusion passages state the comparison outcome without specifying a numerical gain.

APPENDIX A DERIVATION OF (6)

The appendix derives a channel-statistical expression using vectorization, Kronecker-product identities, and bounded angle-delay power spectra.

  • Index mapping: The derivation introduces floor and modulo index variables to map vectorized matrix elements to their corresponding row and column positions.For Ω_k ∈ R^(M×N_g), the dth vectorized element is mapped to indices (m_d, n_d).
  • Channel expansion: The derivation substitutes channel expressions and delta-function relations into integrals over angle and Doppler variables.The displayed terms include steering-vector products, temporal phase factors, and delta functions over angle, delay, and Doppler.
  • Proof condition: The proof concludes that the required limits exist and are finite because the channel power angle-delay spectrum is bounded.The boundedness of S_AD,k(θ, τ) supplies the final integrability condition.

APPENDIX C PROOF OF PROPOSITION 2

The proof of Proposition 2 reduces the target relation to established channel expressions and Kronecker-product identities.

  • Proof strategy: The appendix begins the proof by stating that establishing the target equation is sufficient for proving Proposition 2.The subsequent steps rely on vectorization and Kronecker-product identities.
  • Justification: The derivation invokes deterministic transform matrices, an earlier channel equation, and Proposition 1 to justify successive equalities.These ingredients complete the proof according to the appendix.

APPENDIX D PROOF OF PROPOSITION 3

The proof manipulates permutation, truncation, and zero-product conditions before substituting the resulting relation into the MSE-CE expression. It then concludes that the minimum in (30) is achieved.

  • The proof begins from a non-negativity condition involving k′ and derives a subsequent result.
  • A stated condition ensures that corresponding elements remain zero after the same column permutation and truncation.
  • Using the definition in (26) and the permutation-matrix property, the proof reformulates the condition in (60).
  • The reformulated relation is substituted into the MSE-CE expression ǫCE.
  • The proof concludes that the minimum in (30) is achieved.
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