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Spatially Sparse Precoding in Millimeter Wave MIMO Systems

Omar El Ayach, Sridhar Rajagopal, Shadi Abu-Surra, Zhouyue Pi, Robert W. Heath

arXiv:1305.2460v1cs.IT

TL;DR

High-cost, power-intensive mmWave mixed-signal hardware constrains practical precoding and combining. The paper exploits sparse channel structure with matching-pursuit-based designs, showing that mmWave systems approach theoretical spectral-efficiency limits with low-complexity, hardware-compatible processing.

  • Problem

    High cost and power consumption of mmWave mixed-signal hardware constrain practical precoding and combining.

  • Method

    The paper exploits sparse-scattering channel structure to formulate precoding design and applies matching pursuit to develop low-complexity solutions.

  • Results

    The proposed processing allows mmWave systems to approach theoretical spectral limits despite practical hardware constraints.

  • Takeaways & Limitations

    The framework supports low-complexity mmWave precoding and combining, including efficient quantization and limited-feedback operation.

Abstract

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Millimeter wave (mmWave) signals experience orders-of-magnitude more pathloss than the microwave signals currently used in most wireless applications. MmWave systems must therefore leverage large antenna arrays, made possible by the decrease in wavelength, to combat pathloss with beamforming gain. Beamforming with multiple data streams, known as precoding, can be used to further improve mmWave spectral efficiency. Both beamforming and precoding are done digitally at baseband in traditional multi-antenna systems. The high cost and power consumption of mixed-signal devices in mmWave systems, however, make analog processing in the RF domain more attractive. This hardware limitation restricts the feasible set of precoders and combiners that can be applied by practical mmWave transceivers. In this paper, we consider transmit precoding and receiver combining in mmWave systems with large antenna arrays. We exploit the spatial structure of mmWave channels to formulate the precoding/combining problem as a sparse reconstruction problem. Using the principle of basis pursuit, we develop algorithms that accurately approximate optimal unconstrained precoders and combiners such that they can be implemented in low-cost RF hardware. We present numerical results on the performance of the proposed algorithms and show that they allow mmWave systems to approach their unconstrained performance limits, even when transceiver hardware constraints are considered.

I. INTRODUCTION

MmWave’s severe pathloss motivates large antenna arrays and multi-stream precoding, but costly mixed-signal hardware necessitates hybrid RF/baseband processing. This paper exploits sparse mmWave scattering to design practical near-optimal precoders and combiners that approach unconstrained performance.

  • Motivation: MmWave signals experience orders-of-magnitude higher pathloss, while large arrays provide beamforming gain and enable multiple data streams.The decreased wavelength permits tightly packed arrays that can overcome pathloss and improve spectral efficiency.
  • Hardware constraints: High cost and power consumption of mmWave mixed-signal hardware force systems toward analog or RF processing with constant-modulus RF precoder constraints.Traditional digital processing requires dedicated baseband and RF hardware for every antenna element.
  • Limitations of prior work: Prior antenna-selection and beam-steering methods are limited because selection performs poorly in correlated mmWave channels and beam steering cannot capture dominant eigenmodes perfectly.Existing analog-processing algorithms also do not specialize in large-array mmWave channels with limited scattering and tight antenna packing.
  • Proposed approach: The paper formulates hybrid RF/baseband precoder design as a sparsity-constrained matrix reconstruction problem and solves it using orthogonal matching pursuit.The method approximates an optimal unconstrained precoder as a linear combination of RF-implementable beam-steering vectors.
  • Receiver processing and feedback: The framework extends to hybrid MMSE combiners through simultaneously sparse approximation and supports limited-feedback operation without requiring perfect transmitter channel knowledge.The paper also describes scalar and Grassmannian quantizers for compressing generated precoders, while deferring limited-feedback performance analysis.
  • Results: Simulation results show that the proposed strategy allows mmWave systems to approach unconstrained performance limits under practical transceiver constraints.The evaluation concerns the performance of the proposed strategy in the considered constrained architecture.

II. SYSTEM MODEL … RFHFRFFBBs + W∗

The paper models a single-user mmWave system with hybrid analog-RF and digital-baseband precoding and combining. Analog phase-shifter hardware imposes constant-modulus constraints, while the model assumes narrowband block fading and perfect instantaneous channel knowledge for precoding.

  • A. System Model: The single-user transmitter communicates N_s data streams from N_t antennas to a receiver with N_r antennas using N_RF,t RF chains.
  • N_RF: The transmitted signal is x = F_RF F_BB s, combining an analog RF precoder with a digital baseband precoder.The symbol vector s has dimension N_s × 1 and satisfies E[ss*] = 1/N_s I_N_s.
  • N_RF: RF precoder entries are implemented with phase shifters and therefore have equal norm, while total transmit power is enforced by normalizing F_BB.No hardware-related constraints are placed on the baseband precoder.
  • N_RF: The propagation model assumes a narrowband block-fading channel and produces a received signal with channel matrix H, average received power ρ, and i.i.d. complex Gaussian noise.
  • N_RF: The precoding model assumes perfect timing and frequency recovery and perfect, instantaneous channel knowledge at both transmitter and receiver.Receiver CSI may be obtained through training and shared with the transmitter through limited feedback.
  • RFHFRFFBBs + W∗: The receiver uses N_RF,r RF chains, analog phase shifters, and digital baseband combining to obtain the post-processing received signal.
  • RFHFRFFBBs + W∗: The resulting spectral efficiency accounts for the post-combining noise covariance matrix, while efficient channel estimation and frequency-selective channel treatment remain ongoing research topics.

B. Channel Model

The channel model reflects mmWave propagation through sparse scattering and tightly packed, highly correlated antenna arrays. The paper adopts a narrowband clustered representation based on the extended Saleh–Valenzuela model, with ULA and UPA response examples for beamforming analysis.

  • B. Channel Model: Sparse scattering and tightly packed arrays produce high antenna correlation, making traditional statistical MIMO fading distributions inaccurate for mmWave channels.The model therefore accounts for the spatial structure characteristic of mmWave propagation.
  • B. Channel Model: The paper adopts a narrowband clustered channel representation based on the extended Saleh–Valenzuela model to capture mmWave channel structure.The channel matrix H is modeled as contributions from Ncl scattering clusters, each containing Nray propagation paths.
  • B. Channel Model: Each path is characterized by a complex gain, azimuth and elevation arrival/departure angles, antenna element gains, and normalized transmit and receive array responses.Cluster powers and within-cluster angular quantities complete the statistical channel assumptions.
  • B. Channel Model: Uniform planar arrays are considered for mmWave beamforming because they reduce array dimensions, pack more elements, and enable elevation-domain beamforming.The paper also gives illustrative ULA and UPA array-response examples; the UPA has N = WH elements.

III. SPATIALLY SPARSE PRECODING FOR THE SINGLE USER MMWAVE CHANNEL

The section simplifies hybrid precoder design by approximating the optimal unconstrained precoder with feasible RF/baseband precoders. It reformulates this approximation as a sparsity-based problem and solves it using orthogonal matching pursuit.

  • Problem formulation: Hybrid precoding is designed by approximating the optimal unconstrained precoder Fopt = V1, which generally cannot be realized under constant-magnitude RF constraints.The feasible RF precoders consist of Nt × N_RF matrices with constant-magnitude entries, and no general solution is known for the resulting non-convex problem.
  • Approximate objective: Under Approximation 1, near-optimal hybrid precoders can be found by minimizing the Euclidean distance ∥Fopt − FRFFBB∥F.This approximation treats the hybrid and optimal precoders as sufficiently close, with effective SNRs in the dominant Ns subspaces assumed sufficiently high.
  • Approximate objective: The resulting task is the projection of Fopt onto hybrid precoders of the form FRFFBB with FRF ∈ FRF, but the feasible set is complex and non-convex.Consequently, the projection is difficult to obtain analytically or algorithmically in closed form.
  • Sparse reconstruction: The method exploits the structure of clustered mmWave MIMO channels to provide near-optimal solutions to the hybrid precoding problem.The section identifies the structure of the optimal precoder and uses channel structure to derive the approximation.
  • Sparse reconstruction: The joint design of FRF and FBB is reformulated as a sparsity-based problem and solved using orthogonal matching pursuit.This produces a sparsity solution based on the well-known concept of orthogonal matching pursuit.

N RF

Optimal precoders in practical mmWave channels generate spatially sparse beam patterns that can be accurately approximated with finite combinations of array response vectors. Algorithm 1 selects steering directions and linear combinations that closely reproduce optimal beam patterns, while more involved precoding can outperform simple RF-only beam steering in relevant regimes.

  • Optimal precoders generate spatially sparse beam patterns that finite combinations of array response vectors can accurately approximate.
  • Algorithm 1 selects the best N RF steering directions and forms linear combinations of the corresponding response vectors.
  • With 4 RF chains, the proposed sparse solution generates beam patterns closely resembling those of optimal unconstrained beamforming.This similarity is shown for a 256-element square array in an example channel with 6 scattering clusters.
  • Representative azimuth and elevation directions can replace geometric channel decomposition, making the approach naturally suited for limited feedback.
  • Although RF-only beam steering becomes optimal in very poor-scattering large-array limits, Section VI shows significant gains from more involved precoding strategies.

IV. PRACTICAL MILLIMETER WAVE RECEIVER DESIGN

The section develops practical hybrid mmWave receivers by replacing prohibitively complex optimal decoding with constrained linear MMSE combining. Exploiting clustered channel structure, it formulates receiver design as sparse recovery and solves it using orthogonal matching pursuit.

  • Motivation: Practical receivers use linear MMSE combining because optimal decoding can be prohibitively complex and cannot be realized before analog combination.Received signals must be linearly combined in the analog domain before detection or decoding.
  • MMSE Combiner Formulation: With fixed hybrid precoders FRFFBB, the receiver designs hybrid combiners WRFWBB to minimize mean-squared error under constant-gain phase-only RF constraints.The feasible RF combiner set consists of Nr × N constant-gain phase-only entries.
  • MMSE Combiner Formulation: The constrained MMSE problem is equivalent to projecting the unconstrained MMSE combiner onto hybrid combiners using an E [yy∗]-weighted Frobenius norm.The non-convex RF constraint prevents directly solving this projection problem.
  • Spatially Sparse Combining: Clustered mmWave channels enable near-optimal receivers by constraining RF combiner columns to array response vectors ar(φ, θ).The receiver-side formulation mirrors the spatially structured precoding approach while using representative arrival angles if exact angles are unavailable.
  • Spatially Sparse Combining: The resulting MMSE estimation problem is a multiple-measurement-vector sparse recovery problem solved with orthogonal matching pursuit in Algorithm 2.Auxiliary variables are used to recover the RF and baseband combiners.
  • Transceiver Design: The design decouples receiver combining from precoding by fixing FRFFBB, simplifying transceiver design while retaining practical low-complexity linear receivers.The section explicitly relaxes the perfect-receiver assumption of Section III.

N RF

Because a receiver can form a beam in only one direction, the design should account for the more-constrained terminal when choosing precoders or combiners. The proposed sequential approach starts with that side, updates the effective mmWave channel, and achieves near-optimal spectral efficiency, while joint optimization remains future work.

  • N RF: Limited receiver beam direction can reduce received power, motivating designs that account for the more-constrained terminal.The receiver can form a beam in only one direction, so beams pointing in different directions may lose actual received power.
  • N RF: The sequential design solves the hybrid precoder with Algorithm 1, then the hybrid combiner with Algorithm 2.The prescribed order is to solve FRFFBB first, then solve WRFWBB given FRFFBB.
  • N RF: Alternatively, the design starts with WRFWBB assuming FRFFBB = Fopt, then solves FRFFBB for the effective channel.This ordering starts from the more-constrained side and updates the effective mmWave channel before finding the remaining processing matrix.
  • N RF: The decoupled approach yields near-optimal spectral efficiency, while direct joint optimization of (FRF, FBB, WRF, WBB) remains future work.The paper identifies joint optimization as an interesting topic for future investigation.

V. LIMITED FEEDBACK SPATIALLY SPARSE PRECODING

The section addresses unavailable transmitter channel knowledge by using limited feedback from the receiver. The receiver computes a hybrid precoder approximation and separately quantizes its RF and baseband components while exploiting their mathematical structure.

  • V. LIMITED FEEDBACK SPATIALLY SPARSE PRECODING: Limited feedback fulfills the transmitter’s channel-knowledge requirement when perfect channel knowledge is unavailable in practical systems.The transmitter otherwise needs H to calculate Fopt and approximate it as a hybrid RF/baseband precoder FRFFBB.
  • V. LIMITED FEEDBACK SPATIALLY SPARSE PRECODING: The receiver acquires perfect knowledge of H, calculates Fopt, and computes a corresponding hybrid approximation FRFFBB before feedback.It then feeds information about FRFFBB to the transmitter.
  • V. LIMITED FEEDBACK SPATIALLY SPARSE PRECODING: The proposed feedback scheme separately quantizes FRF and FBB because hybrid precoders decompose naturally into RF and baseband components.The quantization exploits the mathematical structure present in each component.

A. Quantizing the RF Precoder

The RF precoder is efficiently represented through its azimuth and elevation angles, which are uniformly quantized and selected from predefined codebooks. Alternatively, Algorithm 1 can operate directly on quantized response vectors, jointly quantizing the angles and matching the baseband precoder.

  • A. Quantizing the RF Precoder: FRF is efficiently encoded by quantizing its 2N_RF azimuth and elevation angles.The angles provide a natural parametrization of FRF.
  • A. Quantizing the RF Precoder: The proposed method uniformly quantizes the N_RF azimuth and elevation angles using Nφ and Nθ bits, respectively.The resulting quantized angles are selected from the corresponding azimuth and elevation codebooks.
  • A. Quantizing the RF Precoder: The receiver quantizes FRF by selecting the entries of Cφ and Cθ closest in Euclidean distance to FRF’s angles.This provides a direct codebook-based quantization procedure.
  • A. Quantizing the RF Precoder: Alternatively, Algorithm 1 runs directly using the Nt × 2Nφ+Nθ matrix of quantized response vectors, with selected-angle indices fed back to the transmitter.This approach has higher search complexity but jointly quantizes all 2N_RF angles and automatically matches FBB to the quantized angles.

B. Quantizing the Baseband Precoder

The baseband precoder FBB is treated as an approximately unitary subspace quantity, enabling quantization on the Grassmann manifold. Codebooks can be designed with Lloyd’s algorithm and chordal distance using training precoders.

  • Baseband precoder structure: FBB is approximately unitary and can be made exactly unitary, providing structure for efficient quantization.The exact-unitarity construction is discussed in Remark 3.
  • Grassmannian quantization: Because spectral efficiency is invariant to Ns × Ns unitary transformations, FBB is a subspace quantity suitable for Grassmann-manifold quantization.This treats equivalent baseband precoders as the same subspace for quantization.
  • Codebook design: Suitable FBB codebooks can be designed with Lloyd’s algorithm on training precoders, using chordal distance as the metric.The construction follows established limited-feedback MIMO codebook methods.

VI. SIMULATION RESULTS

Simulations show that sparse precoding and combining closely approach unconstrained performance across practical and larger mmWave arrays, while outperforming simple beam steering. The framework also remains effective under rank adaptation, richer scattering, and quantized steering-angle feedback.

  • Larger arrays: For 256 × 64 arrays, the proposed precoding/combining solution achieves almost-perfect performance for both Ns = 1 and Ns = 2 and outperforms beam steering by approximately 5 dB.The result holds with six RF chains and despite beam steering’s expected asymptotic optimality for very large arrays.
  • Angle spread: The rate gap remains below 10% at an angle spread of 15° and is negligible around 5°, although richer scattering degrades performance.For Ns > 1 with smaller arrays, spectral efficiency degrades more rapidly as angle spread increases.
  • RF-chain flexibility: Increasing the number of RF chains mitigates the effect of increased scattering by enabling more flexible precoders and combiners.This mitigation is demonstrated for the same 64 × 16 system under increased angle spread.

VII. CONCLUSION

The paper develops a low-hardware-complexity spatially sparse precoding framework for mmWave systems, extending it to practical MMSE combining, quantization, and limited-feedback operation. Numerical results show that the approach can approach theoretical spectral-efficiency limits, while future work targets relaxed channel, array, and bandwidth assumptions.

  • MmWave precoder design is formulated as a sparsity-constrained signal recovery problem and solved using orthogonal matching pursuit.
  • The same framework designs practical MMSE combiners for mmWave systems.
  • Proposed precoders can be efficiently quantized and are well-suited for limited-feedback systems.
  • Spatially sparse mmWave processing allows systems to approach their theoretical limits on spectral efficiency.
  • Future work includes relaxing assumptions of perfect receiver channel state information, known antenna-array structure, and narrowband channels.
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