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Spatially Common Sparsity Based Adaptive Channel Estimation and Feedback for FDD Massive MIMO

Zhen Gao, Linglong Dai, Zhaocheng Wang, Sheng Chen

arXiv:1604.03695v3cs.IT

TL;DR

FDD massive MIMO needs downlink CSI, but conventional orthogonal-pilot acquisition can impose excessive overhead. The paper proposes non-orthogonal pilots, CS-based adaptive acquisition with DSAMP, and temporal-correlation-based closed-loop tracking, supported by GMMV analysis and a CRLB. Simulations show reliable CSI acquisition with adaptively reduced overhead and performance approaching the bound.

  • Problem

    FDD massive MIMO requires downlink CSI, while conventional orthogonal-pilot estimation can impose overhead that grows with the number of BS antennas.

  • Method

    The scheme uses non-orthogonal pilots, CS-based adaptive CSI acquisition with DSAMP, and closed-loop pilot adaptation exploiting spatially common sparsity and temporal correlation.

  • Results

    For P = 64, GMMV reaches detection probability one with G = 11 versus G = 17 for MMV, reducing required overhead by approximately 35%.

  • Takeaways & Limitations

    The scheme reliably acquires massive-MIMO CSI with adaptively determined overhead and approaches the performance bound.

Abstract

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This paper proposes a spatially common sparsity based adaptive channel estimation and feedback scheme for frequency division duplex based massive multi-input multi-output (MIMO) systems, which adapts training overhead and pilot design to reliably estimate and feed back the downlink channel state information (CSI) with significantly reduced overhead. Specifically, a non-orthogonal downlink pilot design is first proposed, which is very different from standard orthogonal pilots. By exploiting the spatially common sparsity of massive MIMO channels, a compressive sensing (CS) based adaptive CSI acquisition scheme is proposed, where the consumed time slot overhead only adaptively depends on the sparsity level of the channels. Additionally, a distributed sparsity adaptive matching pursuit algorithm is proposed to jointly estimate the channels of multiple subcarriers. Furthermore, by exploiting the temporal channel correlation, a closed-loop channel tracking scheme is provided, which adaptively designs the non-orthogonal pilot according to the previous channel estimation to achieve an enhanced CSI acquisition. Finally, we generalize the results of the multiple-measurement-vectors case in CS and derive the Cramer-Rao lower bound of the proposed scheme, which enlightens us to design the non-orthogonal pilot signals for the improved performance. Simulation results demonstrate that the proposed scheme outperforms its counterparts, and it is capable of approaching the performance bound.

I. INTRODUCTION

FDD massive MIMO requires accurate downlink CSI, but conventional orthogonal-pilot estimation incurs overhead that grows with the number of BS antennas. The paper exploits spatially common sparsity and temporal correlation to adapt pilots and training overhead for lower-overhead CSI acquisition and tracking.

  • Motivation: Accurate downlink CSI is needed for massive-MIMO beamforming and resource allocation, yet FDD lacks channel reciprocity and makes downlink estimation necessary.TDD can exploit uplink CSI, but calibration errors may reduce its downlink accuracy; FDD remains prevalent in current wireless networks.
  • Motivation: Orthogonal pilots become unaffordable in massive MIMO because their overhead increases with the number of BS antennas.The described time-domain and time-frequency orthogonal designs require total pilot overhead Ptotal = NgM.
  • Channel structure: Massive-MIMO channels are sparse in the virtual angular domain because the BS-side angle spread is small.The angular-domain support contains only a small part of the virtual coordinates carrying almost all multipath signals.
  • Channel structure: Subchannels on different OFDM subcarriers share spatially common sparsity because propagation characteristics remain nearly unchanged across the system bandwidth.This common support is illustrated by the virtual angular-domain channel vectors within the bandwidth.
  • Proposed approach: The proposed scheme combines CS-based adaptive CSI acquisition with closed-loop channel tracking using non-orthogonal pilots and temporal channel correlation.DSAMP jointly estimates multiple subcarrier channels, while the tracking stage adapts pilots according to previous channel estimates.

C. Temporal Correlation of Wireless Channels

Massive MIMO channels are modeled as quasi-static within each block but changing across blocks, while their virtual angular-domain support can remain common over multiple blocks. This temporal support stability motivates reusing the estimated support for lower-overhead channel tracking.

  • Block-fading model: Channels remain static across J consecutive time slots within a block and change from block to block.One time slot corresponds to one OFDM symbol.
  • Block-fading model: The coherence time can be shorter than the number of BS antennas, limiting conventional pilot-based estimation.For an example with M = 128, the coherence time is approximately J = 30 slots.
  • Temporal support correlation: Although channel gains vary between blocks, the channel angle spread changes more slowly, allowing the virtual angular-domain support to persist.The paper characterizes Q as the number of consecutive blocks over which common support holds.
  • CSI acquisition: During one block, pilots collected over G successive time slots can be jointly used to acquire downlink CSI.The channel is treated as quasi-static over those slots.
  • CSI acquisition: Orthogonal pilot designs require total overhead Ptotal = NgM, which grows with the number of BS antennas.The same overhead follows for both time-domain and time-frequency orthogonal pilots.
  • Motivation: The proposed approach combines channel estimation and feedback at the BS to exploit spatial sparsity and temporal correlation.This avoids relying on large user-side codebooks for fine-grain spatial channel structures.

III. SPATIALLY COMMON SPARSITY BASED ADAPTIVE CHANNEL ESTIMATION AND FEEDBACK SCHEME

The scheme adaptively acquires downlink CSI using non-orthogonal pilots and DSAMP, then tracks channels with lower overhead using the estimated support and sparsity level. Training continues only until a reliability criterion is met.

  • Adaptive CSI acquisition: The BS transmits non-orthogonal pilots, receives direct feedback, and uses DSAMP to jointly reconstruct sparse virtual angular-domain channels.The reconstruction uses low-dimensional feedback collected over multiple time slots.
  • Adaptive CSI acquisition: The BS stops pilot transmission when the estimated sparse channels satisfy a predefined reliability criterion; otherwise, it increases the training length.The adaptive loop enlarges the measurement vectors sequentially by adding another time slot.
  • Closed-loop tracking: After estimating the support and sparsity level, the BS uses LS channel estimation during the following Q − 1 blocks.The tracking overhead can be reduced to G = bSa, with pilots adjusted according to the estimated support.
  • Pilot design: The first-stage pilot is designed in advance, whereas tracking pilots are adaptively designed from the acquired CSI to reduce overhead and improve estimation MSE.The proposed pilots let different BS antennas share identical pilot subcarriers.
  • Assumption: The method assumes negligible feedback delay relative to the channel coherence time.This assumption supports direct use of the received feedback in the acquisition procedure.
  • Adaptive CSI acquisition: The initial overhead G0 is selected before acquisition, and DSAMP estimates the channel vectors for the current overhead Gi.The procedure can automatically determine an appropriate initial overhead for the next acquisition.

C. Proposed DSAMP Algorithm for Channel Estimation

DSAMP extends sparsity-adaptive pursuit to jointly recover multiple high-dimensional channel vectors that share a common support. Its staged support selection, least-squares updates, pruning, and residual tests guide adaptive recovery.

  • Algorithm principle: For each fixed-sparsity stage, DSAMP selects potential nonzero elements, performs LS estimation, and retains the most likely supports.These operations correspond to the algorithm’s support expansion, estimation, and pruning steps.
  • Algorithm principle: DSAMP jointly estimates multiple sparse channel vectors from different pilot subcarriers by exploiting their common support.It processes multiple low-dimensional received signals rather than recovering one signal from one measurement vector.
  • Performance claim: Compared with SAMP, SP, and OMP, DSAMP substantially reduces required time-slot overhead with similar computational complexity.The comparison is attributed to joint recovery using common support across multiple channels.
  • Support and residual processing: The algorithm computes residuals across the jointly processed channels and uses average energy to identify support elements.The support associated with the minimum average energy is selected according to the estimated channel coefficients.
  • Termination and staging: DSAMP terminates when the residual falls below a threshold or when continuing the current stage increases the residual relative to the previous stage.The latter condition advances the procedure to a new sparsity stage or stops unnecessary iterations.
  • Output: The final channel estimates are obtained after the staged iterations and are produced for all pilot subcarriers.The algorithm is applied to noisy feedback signals and outputs virtual angular-domain channel vectors.

D. Closed-Loop Channel Tracking with Adaptive Pilot Design

Closed-loop tracking reuses the support learned during an initial acquisition stage to estimate channels in subsequent blocks with LS. The pilot matrix is then designed from prior estimates to reduce overhead while targeting improved MSE.

  • Tracking mechanism: Because successive blocks share spatially common sparsity, the BS can use LS to estimate later channels from feedback pilots.This applies over the Q − 1 blocks following the initial acquisition block.
  • Tracking mechanism: The BS transmits a non-orthogonal pilot in each tracking block, and the user directly feeds the received pilot signal back.The procedure repeats the feedback operation used during initial acquisition.
  • Estimator properties: The LS estimator is unbiased and capable of approaching the CRLB when the pilot matrix and support-related quantities are known.The BS uses their estimates from the initial acquisition block in practice.
  • Adaptive pilot design: Tracking overhead can be reduced to Sa when estimating the channel in subsequent blocks.The tracking pilot uses G = Sa while targeting the best MSE performance.
  • Adaptive pilot design: The tracking pilot is designed differently from the initial pilot according to previous channel estimation.The design aims to minimize both the number of pilot time slots and CSI-estimation MSE.

IV. PERFORMANCE ANALYSIS

The scheme’s performance analysis covers pilot design, adaptive overhead, pilot-subcarrier placement, DSAMP complexity and convergence, performance bounds, and adaptive pilot tracking.

  • The analysis evaluates non-orthogonal pilot design for compressive-sensing-based adaptive CSI acquisition.
  • It derives the theoretical limit of required time-slot overhead for adaptive CSI acquisition.
  • It examines pilot-subcarrier placement and the computational complexity and convergence of DSAMP.
  • It analyzes the proposed scheme’s performance bound and the overhead and performance of adaptive non-orthogonal pilots.

A. Non-Orthogonal Pilot Design for CS based Adaptive CSI Acquisition

The proposed generalized multiple-measurement-vector formulation jointly recovers sparse signals with common support but different measurement matrices, providing diversity beyond conventional SMV and MMV recovery.

  • The problem jointly reconstructs multiple high-dimensional sparse signals with a common support set and different measurement matrices.
  • The formulation generalizes MMV and SMV: identical measurement matrices yield MMV, while one signal reduces it to SMV.
  • Different measurement matrices provide additional diversity beyond the diversity from multiple sparse signals in MMV.
  • For i.i.d. continuous measurement matrices, full-rank transformations relate each restricted matrix to a bridge matrix with high probability.
  • Recovery reliability improves with the diversity determined by the relevant rank; GMMV retains diversity even when multiple sparse signals are identical.

3) Pilot Design for CS Based Adaptive CSI Acquisition:

The pilot design uses randomized non-orthogonal signals so the sensing matrices satisfy a favorable Gaussian distribution, while the required training overhead adapts to channel sparsity and measurement-vector count.

  • Diversifying measurement matrices can improve sparse-signal recovery, motivating appropriate pilot-signal design.
  • Each pilot element uses an i.i.d. uniform phase, and the fixed design considers the worst case G = M.
  • The resulting sensing-matrix elements follow i.i.d. CN(0, 1), supporting reliable compression and recovery of sparse angular-domain channels.
  • The minimum required time-slot overhead is G = S_a + 1.
  • Increasing the number of measurement vectors P can reduce the overhead needed for reliable channel estimation.

C. Frequency-Domain Placement of Pilot Signals

This section addresses pilot-subcarrier placement, DSAMP implementation and stopping rules, computational complexity, and CRLB-based performance evaluation under random channel support.

  • Frequency-domain placement: The estimator directly estimates pilot-subcarrier channels; data-subcarrier channels are obtained using interpolation based on those estimates.
  • Complexity: DSAMP, OMP, SP, and SAMP have the same order of computational complexity for estimating one sparse signal per iteration.
  • Convergence: DSAMP stops when a low-energy angular coordinate falls below the noise floor or when the smallest residual identifies the likely exact sparsity level.
  • Performance bound: The CRLB is computed from the eigenvalues of the restricted pilot Gram matrix after the pilot design, array geometry, and channel support are specified.
  • Performance bound: Because channel support is random, the analysis considers the expected CRLB; simulations use oracle LS performance as a practical bound.

F. Adaptive Pilot Design and Required Time Slot Overhead for Closed-Loop Channel Tracking

The pilot design is guided by a lower-bound analysis: a unitary non-orthogonal pilot matrix attains the bound, while known sparsity can reduce acquisition overhead to the sparsity level.

  • Required time slot overhead: If the sparsity level Sa is known, the smallest CSI-acquisition overhead can be reduced to G = Sa.The analysis assumes that the true support set and sparsity level have already been acquired.
  • Analysis basis: The analysis uses the known support set Θ and sparsity level Sa to characterize the minimum overhead and pilot design condition.These assumptions connect the overhead result with the lower-bound pilot construction.
  • Adaptive pilot design: The lower bound is approached when the pilot Gram matrix is diagonal with identical diagonal elements.This condition follows from equality in the arithmetic-harmonic means bound.
  • Adaptive pilot design: A unitary matrix USa ∈ C^Sa×Sa in the non-orthogonal pilot matrix attains the lower bound.The passage identifies this pilot construction as achieving the lower bound of (32).

G. Selection of Thresholds for Algorithms 1 and 2

The threshold-selection procedures use estimated test-statistic distributions and simulated error or MSE behavior to choose operating thresholds for Algorithms 1 and 2.

  • Threshold pth in Algorithm 2: Algorithm 2 selects pth using estimated PDFs of the test statistic ρ2 under hypotheses indicating whether a candidate index belongs to the true support.The PDFs are obtained through Monte Carlo simulation because closed-form expressions are difficult to derive.
  • Threshold pth in Algorithm 2: pth = 0.02 achieves good MSE performance for Algorithm 2 at SNR = 15 dB and P = 64.The passage states that suitable pth values for other SNRs can be obtained similarly.
  • Threshold ε in Algorithm 1: Algorithm 1 selects ε using the estimated PDF of ρ1 and the Neyman-Pearson criterion to minimize false alarms for a specified miss probability.The criterion is applied to the simulated false-alarm and miss probabilities across ε.
  • Threshold ε in Algorithm 1: ε = 0.03 minimizes both false-alarm and miss probabilities for Algorithm 1 under typical Sa and G values.The simulation uses pth = 0.02 and evaluates the probabilities as functions of ε.

V. SIMULATION RESULTS

Simulations show that the proposed adaptive CSI acquisition and tracking schemes improve sparse recovery, MSE, BER, and overhead performance while approaching oracle or CRLB bounds.

  • CS algorithm comparison: The DSAMP algorithm achieves the best MSE among four CS algorithms and approaches the oracle LS estimator for Sa ≤ 14.It jointly estimates multiple sparse signals using their common sparsity, with only slightly higher complexity and the same complexity order as the alternatives.
  • Sparse signal detection: With P = 64 and unit detection probability, GMMV requires G = 11 compared with G = 17 for MMV, reducing overhead by approximately 35%.GMMV uses mutually independent measurement matrices while jointly recovering signals with a common support set.
  • Adaptive CSI acquisition: The adaptive CSI acquisition scheme approaches the oracle LS performance bound even when G ≤ 2Sa, unlike fixed-overhead DSAMP under insufficient overhead.The adaptive scheme adjusts G to obtain robust channel estimation, whereas unreliable recovery can degrade DSAMP MSE when overhead is too small.
  • Adaptive overhead: At Sa = 8 and SNR = 30 dB, the proposed scheme uses G ≈ 10, reducing required overhead by about 92% relative to conventional schemes with G = M = 128.The adaptive overhead follows the sparsity level while estimating channels associated with hundreds of BS antennas.
  • Closed-loop tracking: For Sa = 14, closed-loop tracking reduces average overhead from 20.43 to 14.02 while providing better MSE and approaching the CRLB.The tracking stage uses sparsity information from initial acquisition to adapt the pilot signal.
  • Overall system performance: Under Sa = 8 and varying SNRs, the proposed scheme approaches the CRLB with average overhead Ḡ < 2Sa and outperforms the compared acquisition schemes.The comparison includes J-OMP and fixed-overhead DSAMP; the resulting CSI also supports BER performance approaching the CRLB with ZF precoding.
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