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Millimeter Wave Channel Estimation via Exploiting Joint Sparse and Low-Rank Structures
Xingjian Li, Jun Fang, Hongbin Li, Pu Wang
TL;DR
MmWave channel estimation with one RF chain at each end must recover channels from limited beamformed measurements while accounting for both sparse scattering and angular spreads. The paper uses matrix completion followed by sparse recovery to exploit joint low-rank and sparse structure, achieving lower sample complexity than direct compressed sensing under the stated condition.
Problem
Channel estimation must recover mmWave channels from limited measurements in a one-RF-chain architecture, while existing approaches primarily exploit sparsity despite angular spreads.
Method
The method first applies low-rank matrix completion and then sparse recovery to exploit the channel’s joint low-rank and sparse structure.
Results
Theoretical analysis shows lower sample complexity than direct compressed sensing when L < p, and simulations corroborate the proposed method’s performance advantage.
Takeaways & Limitations
Angular spreads provide exploitable low-rank structure that, together with sparsity, can reduce the measurements required for mmWave channel recovery.
Abstract
from arXiv · showhide
We consider the problem of channel estimation for millimeter wave (mmWave) systems, where, to minimize the hardware complexity and power consumption, an analog transmit beamforming and receive combining structure with only one radio frequency (RF) chain at the base station (BS) and mobile station (MS) is employed. Most existing works for mmWave channel estimation exploit sparse scattering characteristics of the channel. In addition to sparsity, mmWave channels may exhibit angular spreads over the angle of arrival (AoA), angle of departure (AoD), and elevation domains. In this paper, we show that angular spreads give rise to a useful low-rank structure that, along with the sparsity, can be simultaneously utilized to reduce the sample complexity, i.e. the number of samples needed to successfully recover the mmWave channel. Specifically, to effectively leverage the joint sparse and low-rank structure, we develop a two-stage compressed sensing method for mmWave channel estimation, where the sparse and low-rank properties are respectively utilized in two consecutive stages, namely, a matrix completion stage and a sparse recovery stage. Our theoretical analysis reveals that the proposed two-stage scheme can achieve a lower sample complexity than a direct compressed sensing method that exploits only the sparse structure of the mmWave channel. Simulation results are provided to corroborate our theoretical results and to show the superiority of the proposed two-stage method.
I. INTRODUCTION
MmWave systems use large antenna arrays to offset severe path loss, but beam alignment and channel estimation create substantial training challenges. The paper exploits angular spreads alongside sparsity to reduce the measurements needed for channel recovery.
- Motivation: Large antenna arrays provide beamforming gain to compensate for mmWave path loss, but increase system-design complexity.Directional beams must be aligned between transmitter and receiver.
- Motivation: Exhaustive beam-pair search can require prohibitively long training when the BS and MS have many antennas.
- Prior approaches: Sparse-scattering models formulate mmWave channel estimation as sparse signal recovery and reduce training overhead.
- Angular spreads: Measurements show angular spreads across AoA, AoD, and elevation, producing block-sparse channel structure.Reported rms AoA spreads are 15.5° and 15.4°, while rms AoD spreads are 10.2° and 10.5° at two carrier frequencies.
- Contribution: The proposed two-stage method uses low-rank matrix completion followed by sparse recovery, requiring O(pL^2) measurements versus O(p^2L) for direct compressed sensing when L < p.L denotes scattering clusters and p measures maximum angular spread.
II. SYSTEM MODEL AND PRIOR WORK
The system estimates a mmWave channel from noisy scalar measurements collected with one RF chain at each end and analog beamforming or combining. A beam-space representation makes the channel amenable to sparse recovery.
- System model: The considered architecture uses one RF chain at both the BS and MS to reduce hardware complexity and power consumption.Analog transmit beamforming and receive combining use digitally controlled phase shifters.
- System model: Each training measurement is a noisy scalar formed by combining the channel with a transmit beamforming vector and receive combining vector.During training, the transmitted symbol is set to one and beamformer entries have constant modulus.
- Estimation problem: The receiver observes a noisy version of z^H H f rather than the full channel matrix, creating a channel subspace sampling limitation.
- Channel representation: A geometric channel model represents the channel through path gains and array responses parameterized by azimuth AoA and AoD.
- Channel representation: The beam-space channel H_v is sparse, with nonzero entries corresponding to channel path gains on discretized AoA and AoD grids.The representation uses overcomplete steering matrices A_BS and A_MS.
- Sparse recovery: The resulting estimation problem is formulated as sparse signal recovery and can be solved with compressed-sensing algorithms such as FISTA.
III. CHANNEL MODEL WITH ANGULAR SPREADS
Angular spreads arise from scattering clusters containing closely spaced paths. In beam space, these spreads create a channel that is simultaneously sparse and low-rank, motivating joint exploitation of both structures.
- Clustered channel model: The angular-spread model contains L scattering clusters, each contributing multiple paths around mean AoA and AoD values.Relative angular shifts describe the paths within each cluster.
- Joint structure: The clustered channel model remains compatible with compressed-sensing estimation while revealing additional low-rank structure.
- Joint structure: Each cluster’s virtual AoA and AoD representations are sparse because angular spread occupies only a small part of the full angular domain.
- Joint structure: The virtual channel H_v has at most p^2L nonzero entries, at most pL nonzero rows and columns, and rank L.Here p bounds the nonzero entries in each cluster’s virtual AoA or AoD vector.
- Recovery objective: The recovery objective is to estimate the joint sparse and low-rank virtual channel using as few measurements as possible.
IV. TWO-STAGE COMPRESSED SENSING SCHEME
The two-stage scheme first treats beamformed observations as samples of a low-rank matrix, then recovers the channel using its sparse structure. Matrix completion alone is theoretically valid but has suboptimal measurement scaling because it ignores sparsity.
- Matrix completion stage: Randomly selected, nonreused beam pairs turn the observations into sampled entries of the low-rank matrix Y = Z^H H F.The observed-entry set has cardinality T.
- Matrix completion stage: When the codebook matrices are full-rank square matrices, estimating H is equivalent to completing Y with nuclear-norm minimization.
- Matrix completion stage: After recovering Y, the channel H is estimated from the completed matrix.
- Two-stage motivation: Matrix completion alone requires O(L max{N_BS, N_MS}^5/4) measurements and ignores the channel’s sparse structure.Its measurement requirement grows approximately linearly with the larger antenna-array dimension.
A. Proposed Scheme
The proposed scheme separates low-rank and sparse recovery into two consecutive stages, using reduced-dimensional observations to recover the channel efficiently.
- Two-stage scheme: The first stage exploits low rank to recover the reduced matrix Y from sampled observations.The matrices Z and F use reduced dimensions, so Y is smaller than H.
- Two-stage scheme: The second stage uses compressed sensing on the reconstructed Y to recover the sparse channel structure.
B. Theoretical Results
The theoretical analysis establishes exact-recovery conditions for both stages and combines them into a high-probability guarantee for the full channel.
- Theorem 1: Theorem 1 assumes uniformly sampled observations and Gaussian random sensing matrices with dimensions reduced relative to the antenna arrays.
- Theorem 1: The complete channel is perfectly recovered with probability exceeding (1−c4n−3)(1−2e−c5NZ)(1−2e−c6NF ) when the stated conditions hold.
- Recovery stages: The algorithm first completes the low-rank matrix Y and then recovers the sparse virtual channel through compressed sensing.
- Matrix completion: Matrix completion recovers Y with probability exceeding 1−c4n−3 under its sampling condition.
- Sparse recovery: The sparse recovery guarantee applies to matrices with at most k nonzero columns and rows when the sensing matrices satisfy a 2k-restricted isometry property.
- Proof structure: The proof combines the matrix-completion guarantee with the sparse-recovery guarantee to obtain the full-channel result.
C. Discussions
The proposed method’s sample complexity scales favorably with angular spread and cluster rank, and can outperform direct sparse recovery when scattering clusters are fewer than angular support size.
- Sample complexity: The required measurements scale approximately linearly with p and quadratically with rank L.Here p measures maximum angular spread and L denotes the number of scattering clusters.
- Sample complexity: Because p and L are typically smaller than the antenna dimensions, the two-stage scheme can reduce overhead relative to low-rank matrix completion.
- Comparison with direct recovery: The two-stage method has lower sample complexity than direct compressed sensing when L < p.The paper associates L with scattering clusters and p with the maximum angular spread.
- Comparison with direct recovery: The comparison is especially favorable for a line-of-sight path, where L=1 while angular spread generally keeps p greater than one.
- Assumption: The theoretical analysis assumes Gaussian i.i.d. sensing matrices, although the actual beamforming and combining matrices are structured.The paper notes that structured matrices may nevertheless have suitable restricted isometry properties.
D. Extension To The Noisy Case
The two-stage method is extended to noisy observations by replacing exact recovery constraints with error-tolerant optimization problems.
- Noisy extension: The noisy-case algorithm adapts the two-stage compressed sensing scheme to observation noise.
- Noisy extension: The noisy formulation uses ε and ǫ as error-tolerance parameters in constrained optimization problems.
- Optimization: The constrained problems can be converted to unconstrained optimization using an appropriate regularization parameter λ.One resulting problem can be solved efficiently by fixed-point continuation.
V. SIMULATION RESULTS
The simulations compare two-stage compressed sensing with direct compressed sensing under varying measurements, angular spreads, and SNR. The proposed method generally performs better, especially as angular spread increases, while its implementation uses matrix completion followed by sparse recovery.
- Simulation setup: Two-stage CS uses SVT or FPC for matrix completion before applying a sparse-recovery algorithm.SVT is used for noiseless matrix completion, while FPC is used in the noisy case.
- Simulation setup: The simulations evaluate NMSE and success rate for two-stage CS and direct CS under specified channel, codebook, and SNR settings.The channel uses 64-antenna uniform linear arrays, two scattering clusters, and randomly selected beamforming and combining vectors.
- Results versus measurements: Two-stage CS has a clear performance advantage over direct CS as the number of measurements T varies, for both tested codebook designs.The RC codebook also performs better than the alternative beam-pattern design.
- Results versus angular spread: Two-stage CS improves over direct CS as angular spread increases, although direct CS performs better at a small spread of 6°.The tested equal AoA and AoD spreads range from 6° to 22°.
- Results versus SNR: Two-stage CS outperforms direct CS in moderate and high SNR regimes.The SNR comparison fixes NZ = NF = 24, T = 0.5NZNF, δθ = 15°, and δφ = 10°.
VI. CONCLUSIONS
The paper addresses channel estimation with one RF chain at both the BS and MS by exploiting jointly sparse and low-rank mmWave channels. Its two-stage method combines matrix completion and sparse recovery, and the analysis and simulations support lower measurement requirements than direct compressed sensing.
- The study considers mmWave channel estimation with only one RF chain at the BS and MS.
- Angular spreads are incorporated into the channel model and algorithm development, yielding a jointly sparse and low-rank channel structure.
- The proposed two-stage method first performs matrix completion and then applies sparse recovery to estimate the channel.
- Theoretical analysis shows that the two-stage method requires fewer measurements than direct compressed sensing that ignores the low-rank structure.
- Simulation results corroborate the theoretical analysis and demonstrate the superiority of the proposed two-stage compressed sensing method.
APPENDIX A PROOF OF LEMMA 1
The appendix proves exact recovery by combining restricted-isometry arguments with a structured error decomposition. The proof concludes that the reconstruction error is zero.
- The proof uses 2k-RIP properties for measurement matrices acting on matrices with at most 2k nonzero columns and rows.
- The reconstruction error E is decomposed into components supported on the true support and disjoint k × k blocks.
- Optimality of the recovered matrix and norm inequalities are used to bound the decomposed error.
- The proof establishes that the total error norm is zero, implying exact recovery X = ˆX.