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FDD Massive MIMO via UL/DL Channel Covariance Extrapolation and Active Channel Sparsification

Mahdi Barzegar Khalilsarai, Saeid Haghighatshoar, Xinping Yi, Giuseppe Caire

arXiv:1803.05754v2cs.ITmath.IT

TL;DR

In FDD massive MIMO, lack of UL-DL channel reciprocity makes downlink training and feedback costly, limiting the use of many BS antennas. The paper estimates DL covariance from UL pilots and designs a sparsifying precoder around that information. Simulations report better performance than J-OMP and JDLCM, especially at low DL pilot dimensions.

  • Problem

    FDD massive MIMO needs DL probing and CSI feedback because UL-DL channel reciprocity does not hold, creating substantial training and feedback overhead.

    Source: §I · pp. 2–3

  • Method

    The method estimates DL channel covariance from UL pilots using ASF reciprocity, then designs a covariance-based sparsifying precoder to limit effective-channel sparsity for a chosen DL pilot dimension.

    Source: B. Contribution · p. 5, §VI · p. 26

  • Results

    The proposed ACS method outperforms J-OMP and JDLCM in channel-estimation error and achievable sum-rate, especially at low DL pilot dimensions.

    Source: A. Comparisons · p. 23

  • Takeaways & Limitations

    The method uses DL covariance information to select beams and users for efficient communication with a limited DL pilot dimension.

    Source: §VI · p. 26

    Generalizing active channel sparsification to array geometries beyond UPAs requires efficient UL-DL covariance transformation and beam-space representation methods.

    Source: §VI · p. 26

Abstract

from arXiv · show

We propose a novel method for massive Multiple-Input Multiple-Output (massive MIMO) in Frequency Division Duplexing (FDD) systems. Due to the large frequency separation between Uplink (UL) and Downlink (DL), in FDD systems channel reciprocity does not hold. Hence, in order to provide DL channel state information to the Base Station (BS), closed-loop DL channel probing and Channel State Information (CSI) feedback is needed. In massive MIMO this incurs typically a large training overhead. For example, in a typical configuration with M = 200 BS antennas and fading coherence block of T = 200 symbols, the resulting rate penalty factor due to the DL training overhead, given by max{0, 1 - M/T}, is close to 0. To reduce this overhead, we build upon the well-known fact that the Angular Scattering Function (ASF) of the user channels is invariant over frequency intervals whose size is small with respect to the carrier frequency (as in current FDD cellular standards). This allows to estimate the users' DL channel covariance matrix from UL pilots without additional overhead. Based on this covariance information, we propose a novel sparsifying precoder in order to maximize the rank of the effective sparsified channel matrix subject to the condition that each effective user channel has sparsity not larger than some desired DL pilot dimension T_{dl}, resulting in the DL training overhead factor max{0, 1 - T_{dl} / T} and CSI feedback cost of T_{dl} pilot measurements. The optimization of the sparsifying precoder is formulated as a Mixed Integer Linear Program, that can be efficiently solved. Extensive simulation results demonstrate the superiority of the proposed approach with respect to concurrent state-of-the-art schemes based on compressed sensing or UL/DL dictionary learning.

I. INTRODUCTION

FDD massive MIMO requires downlink channel probing and feedback because UL-DL channel reciprocity does not hold, making training costly when many BS antennas are used. The paper estimates DL covariance from UL pilots and uses it to design a sparsifying precoder that supports many users with a chosen DL pilot dimension.

I. INTRODUCTION

  • When conventional training uses M antennas, a configuration with M ≥200 and T = 168 can consume the entire resource block, leaving no room for data.The M-dimensional feedback also creates significant uplink overhead.Source: §I · p. 3

A. Related works: compressed DL pilots

  • Prior compressed-pilot methods exploit beam-space sparsity, but their performance depends on channel sparsity, while some approaches require offline dictionary learning or discrete, separable paths.The dictionary-learning method must be rerun when the propagation environment changes; the cited sparse Bayesian method assumes a known number of discrete paths.Source: A. Related works: compressed DL pilots · pp. 4–5

B. Contribution

  • The proposed covariance extrapolation estimates each user’s ASF from UL pilots and uses angular reciprocity to infer DL covariance without assuming the ASF is sparse or discrete.It exploits the Toeplitz or block-Toeplitz covariance structure for ULA or UPA arrays.Source: B. Contribution · p. 6
  • The proposed sparsifying precoder uses DL covariance to maximize effective-channel rank while limiting each column’s sparsity to the assigned pilot dimension T_dl.Its optimization is formulated as a Mixed Integer Linear Program that can be solved with standard off-the-shelf solvers.Source: B. Contribution · pp. 6–7

II. SYSTEM SETUP

  • The system setup models FDD channels with an angular scattering function and frequency-dependent array responses, while assuming locally stationary scattering geometry over channel-codeword timescales.The channel is modeled as WSSUS with block fading over T time-frequency symbols.Source: §II · pp. 7–9

III. DL COVARIANCE ESTIMATION FROM UL PILOTS

The proposed DL covariance estimation relies on angular reciprocity: the channel ASF is treated as common to UL and DL. It avoids requiring sparse or discrete scattering and instead requires an ASF estimate that is positive and consistent with the UL covariance.

III. DL COVARIANCE ESTIMATION FROM UL PILOTS

  • The method assumes the channel ASF is the same for UL and DL and does not require it to be sparse or composed of discrete components.An ASF estimate that is real, positive, and consistent with the UL covariance is sufficient for DL covariance estimation.Source: §III · p. 9
  • The companion work states that any real, positive ASF estimate consistent with the UL covariance supports stable DL covariance estimation, regardless of sparsity.The passage points to [46] for this result.Source: §III · p. 9
  • For x in C^M, T(x) denotes a Toeplitz Hermitian matrix with first column x; it is positive semidefinite when x is a sampled autocorrelation function..Source: §III · p. 9

A. Uplink covariance estimation

The method estimates UL covariance from pilot observations, then estimates the ASF using a non-negative least-squares fit. Toeplitz positive-semidefinite structure and positivity and data consistency support the subsequent DL covariance estimate.

A. Uplink covariance estimation

  • The UL covariance estimation is decoupled across users under mutually independent user channels and AWGN, allowing the method to focus on a generic user.The projected sample covariance is based on the user’s UL pilot observations.Source: A. Uplink covariance estimation · p. 10
  • UL pilot observations are collected across N_ul coherence blocks, and their sample covariance is projected onto the Toeplitz, Hermitian, positive-semidefinite cone.The projection improves the estimate because the raw sample covariance is not necessarily Toeplitz; the resulting matrix is represented by its first column.Source: A. Uplink covariance estimation · p. 10

B. Estimation of the channel ASF

  • The ASF estimate is obtained by non-negative least squares, yielding a real, positive approximation whose generated UL covariance samples fit the estimated samples.The cited companion work identifies positivity and data consistency as the requirements for stable DL covariance estimation; NNLS can be computed with convex optimization techniques.Source: B. Estimation of the channel ASF · pp. 10–11

C. Covariance extrapolation via Fourier transform resampling

The method estimates the DL covariance by resampling the Fourier transform of an estimated angular measure at the DL frequency grid. It does not require the ASF to have sparse or discrete support.

C. Covariance extrapolation via Fourier transform resampling

  • DL covariance estimation resamples the Fourier transform of an estimated angular measure on the DL grid, using known samples on the UL integer grid.The estimated measure is obtained from the UL sample covariance; its Fourier transform is evaluated at grid points αm and used to form the DL antenna autocorrelation function.Source: C. Covariance extrapolation via… · p. 11
  • The resulting DL covariance matrix is formed by Toeplitz completion of the estimated DL autocorrelation function.Source: C. Covariance extrapolation via… · p. 11
  • The method does not require the ASF to have sparse or discrete support, unlike the cited ad-hoc methods.Source: C. Covariance extrapolation via… · p. 11

D. Circulant approximation of the DL covariance matrices

A circulant approximation gives users a shared DFT beam-space representation of their DL covariances. The paper then uses a fixed precoder to make effective channels sparse enough for limited DL probing while retaining a high-rank channel matrix.

D. Circulant approximation of the DL covariance matrices

  • For large M, the circulant approximation makes the DFT columns approximate common covariance eigenvectors across users.The covariance is represented through diagonal eigenvalue matrices in the shared DFT basis.Source: D. Circulant approximation of the DL… · p. 12
  • In this basis, each DFT column represents an angular direction and its user eigenvalue represents channel power along that direction.Limited local scattering and a large array can make only a few entries significant, yielding beam-space sparsity.Source: D. Circulant approximation of the DL… · p. 13

IV. ACTIVE CHANNEL SPARSIFICATION AND DL CHANNEL PROBING

  • The fixed precoder depends on channel statistics, not instantaneous channel realizations, which still require closed-loop DL probing and feedback.Source: §IV · p. 13
  • The precoder is designed so each effective user channel has low dimension while the collection retains high rank, supporting limited-pilot estimation and spatial multiplexing.The effective channel is the physical DL channel transformed by the precoder.Source: §IV · pp. 13–14

A. Necessity and implication of stable channel estimation

Stable channel estimation requires enough DL pilot dimensions for each channel’s effective sparsity. The optimization selects users and beam directions to meet that constraint while maximizing the rank of the effective channel matrix.

A. Necessity and implication of stable channel estimation

  • Tdl ≥ sk permits pilot matrices whose estimation error vanishes as noise power approaches zero; when Tdl < sk, the error remains positive.Source: A. Necessity and implication of stable… · p. 14
  • The sparsifying precoder constrains each effective channel’s sparsity to Tdl, allowing users to be served even when their intrinsic supports exceed that pilot dimension.The paper links nonvanishing high-SNR estimation error to SINR saturation and an interference-limited regime.Source: A. Necessity and implication of stable… · pp. 14–15

B. Sparsifying precoder optimization

  • The optimization selects a subgraph whose users have degree at most Tdl and whose incident edge weights exceed P0, while seeking a large-rank effective channel matrix.The graph represents users and beam directions; edge weights represent channel power along a beam.Source: B. Sparsifying precoder optimization · p. 16
  • The rank of the selected channel matrix is related to the largest matching, connecting the beam-user graph to the number of spatially multiplexed streams.The paper states that matrix rank equals the size of the largest intersection submatrix whose graph contains a perfect matching.Source: B. Sparsifying precoder optimization · p. 17
  • The resulting optimization is equivalent to a mixed integer linear program that can be solved with an off-the-shelf optimization toolbox.A regularization term can favor solutions with more active beams without changing the matching size when its factor is sufficiently small.Source: B. Sparsifying precoder optimization · pp. 18–19

C. Channel estimation and multiuser precoding

The section forms effective channels by selecting beam directions, then estimates them using a common DL pilot and feeds the measurements back to the BS. It uses the estimated channels for zero-forcing precoding and evaluates the resulting sum rate.

C. Channel estimation and multiuser precoding

  • The effective channel has nonzero entries only on selected beam directions that overlap the user's active scattering directions, with at most Tdl nonzero coefficients.Their positions and variances are known to the BS, enabling estimation from Tdl-dimensional DL pilots with MSE that vanishes as 1/SNR.Source: C. Channel estimation and multiuser… · pp. 19–20
  • A random unitary DL pilot matrix can satisfy the pilot-phase power constraint, and analog feedback returns the Tdl pilot observations to the BS.The BS estimates the UL channel and applies linear MMSE estimation to recover the feedback, with additional noise from the UL transmission.Source: C. Channel estimation and multiuser… · p. 20
  • The BS applies zero-forcing beamforming using the estimated effective channels and uses uniform power allocation across served user streams in the simulations.With imperfect channel estimates, the received signal can include inter-user terms; the sum-rate comparison assumes each receiver knows its channel coefficients.Source: C. Channel estimation and multiuser… · pp. 20–21

V. SIMULATION RESULTS

The simulations compare the proposed method with J-OMP and JDLCM for channel estimation and sum rate. The setup uses estimated UL covariances to design the proposed precoder and assumes noiseless feedback.

V. SIMULATION RESULTS

  • The proposed method is compared with J-OMP, which uses random Gaussian DL pilots and joint orthogonal matching pursuit, and JDLCM, which learns UL/DL dictionaries.JDLCM estimates channels with a norm-minimization program and assumes shared support across the UL and DL dictionaries.Source: §V · pp. 21–22
  • The comparison uses M = 128 antennas, K = 13 users, T = 128 symbols, and 1000 UL pilot observations for covariance estimation.For the proposed method, the BS approximates the covariance matrices as circulant and computes the sparsifying precoder with the MILP.Source: §V · p. 22
  • The simulations assume noiseless feedback, while J-OMP receives the channel sparsity order as a genie-aided input unavailable to the other methods.The paper reports that adding noisy feedback would increase the proposed method's relative gain over J-OMP because CS schemes are more noise-sensitive than covariance-based MMSE estimation.Source: §V · p. 22

A. Comparisons

The proposed ACS method outperforms J-OMP and JDLCM in channel estimation error and achievable sum rate, especially at low DL pilot dimensions. The pilot dimension that maximizes sum rate differs across the three methods.

A. Comparisons

  • ACS outperforms J-OMP and JDLCM in channel estimation error and achievable sum rate, particularly when the DL pilot dimension is low.Below the channel sparsity order, the CS-based methods perform poorly because they have fewer measurements than the inherent channel dimension.Source: A. Comparisons · p. 23
  • At SNR = 20 dB, Fig. 3 reports sum-rate-maximizing DL pilot dimensions Tdl ≈40 for ACS, ≈60 for JDLCM, and ≈70 for J-OMP.Fig. 3a plots normalized channel estimation error versus Tdl; Fig. 3b compares achievable sum-rate, with ACS outperforming JDLCM and J-OMP.Source: A. Comparisons · pp. 23–24

B. The effect of channel sparsity order

The proposed method accommodates different angular channel sparsity orders. In the reported setup, less-sparse channels generally yield higher sum rates at a fixed pilot dimension.

B. The effect of channel sparsity order

  • At a fixed Tdl, less-sparse channels generally achieve a higher sum rate with the proposed method.The simulations vary cluster interval sizes and corresponding approximate sparsity orders; the reported higher rates reflect greater beamforming gain when more scattering components contribute.Source: B. The effect of channel sparsity order · pp. 24–25
  • Unlike the proposed method, CS-based techniques relying on a small-sparsity assumption tend to perform poorly when that assumption is not met.The paper attributes the proposed method's flexibility across sparsity orders to active sparsification.Source: B. The effect of channel sparsity order · pp. 24–25

C. Relevance of the pre-log factor

The sum-rate slope in an intermediate SNR regime tracks the number of served DL streams, illustrating the relevance of maximizing the effective channel rank. The conclusion summarizes the covariance-based active channel sparsification approach and notes that broader array geometries require further development.

C. Relevance of the pre-log factor

  • In the intermediate SNR regime, the sum-rate slope agrees with the expected number of served DL streams, supporting rank maximization in the sparsifying-precoder design.Greedy ZF may serve slightly fewer than K = 13 users; the expected served-user count is therefore slightly below 13.Source: C. Relevance of the pre-log factor · p. 25

VI. CONCLUSION

  • The approach estimates DL covariance from UL pilots using angular reciprocity, then selects beams and users through active channel sparsification formulated as a mixed integer linear program.For ULA settings, the conclusion describes a common DFT eigenvector representation with vanishing error as M grows.Source: §VI · p. 26
  • For broader array geometries, efficient UL-DL covariance transformation and beam-space representation methods remain to be developed beyond the stated straightforward UPA generalization.The conclusion identifies these as directions for future work.Source: §VI · p. 26

A. Proof of Lemma 1

  • When T_dl ≥ s_k, a randomly generated Ψ achieves the rank equality with probability 1, and the estimation error vanishes as O(N_0) as N_0 decreases to zero.The proof states that the rank equality holds for any support set of size s_k.Source: A. Proof of Lemma 1 · pp. 27–28

B. Proof of Lemma 3

The proof links the rank of a weighted adjacency matrix to whether its bipartite graph contains a perfect matching.

B. Proof of Lemma 3

  • A perfect matching makes det(Q) nonzero with probability 1 and rank(Q) = r; without one, det(Q) = 0 and rank(Q) < r.The proof relies on non-identically-zero entries of W being drawn from a continuous distribution.Source: B. Proof of Lemma 3 · p. 28

C. Proof of Theorem 1

The proof encodes selective beam-user choice and maximum-cardinality matching as a mixed integer linear program, then establishes binary-valued solutions.

C. Proof of Theorem 1

  • The formulation selects beams and users while requiring each selected user to have no more than T_dl selected beams.The selected user’s beam weights must also sum to at least P_0.Source: C. Proof of Theorem 1 · pp. 28–29
  • The optimization maximizes matching cardinality over selected beam and user vertices, with constraints ensuring matched edges belong to the selected subgraph.The proof describes this as a modified maximum-cardinality bipartite matching problem with selective vertices.Source: C. Proof of Theorem 1 · pp. 29–30
  • Relaxing the matching variables to linear constraints still yields binary-valued optimal solutions when the beam and user selection variables are binary.The proof uses integrality of the bipartite matching polytope.Source: C. Proof of Theorem 1 · p. 30

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